$B EMfl 315 J ■ ill jl i i 1 1 ! j, 1 ! ^ 1' 1 ^^^^^^ h if 1 ^^^^^ 1 1 M ■ 1 ^^^^w; i'.f 1 i ij! K ,'^^ZZ^,.,A^ Gninr of R, Tracy ^rawfor(a ELEMENTS OP PLANE TRIGONOMETRY A SERIES OF MATHEMATICAL TEXTS EDITED BY EARLE RAYMOND HEDRICK THE CALCULUS By Ellery Williams Davis and William Charles Brenke. ANALYTIC GEOMETRY AND ALGEBRA By Alexander Ziwet and Louis Allen Hopkins. ELEMENTS OF ANALYTIC GEOMETRY By Alexander Ziwet and Louis Allen Hopkins. PLANE AND SPHERICAL TRIGONOMETRY WITH COMPLETE TABLES By Alfred Monroe Kenyon and Louis Ingold. PLANE AND SPHERICAL TRIGONOMETRY WITH BRIEF TABLES By Alfred Monroe Kenyon and Louis Ingold. ELEMENTARY MATHEMATICAL ANALYSIS By John Wesley Young and Frank Millett Morgan. COLLEGE ALGEBRA By Ernest Brown Skinner. ELEMENTS OF PLANE TRIGONOMETRY WITH COM- PLETE TABLES By Alfred Monroe Kenyon and Louis Ingold. ELEMENTS OF PLANE TRIGONOMETRY WITH BRIEF TABLES By Alfred Monroe Kenyon and Louis Ingold. THE MACMILLAN TABLES Prepared under the direction of Earle Raymond Hedrick. PLANE GEOMETRY By Walter Burton Ford and Charles Ammerman. ^LANE AND SOLID GEOMETRY By Walter Burton Ford and Charles Ammerman. SOLID GEOMETRY By Walter Burton Ford and Charles Ammerman. CONSTRUCTIVE GEOMETRY Prepared under the direction of Earle Raymond Hedrick. JUNIOR HIGH SCHOOL MATHEMATICS By W. L. VosBURGH and F. W. Gentleman. This book is issued in a form identical with that of the books announced above ELEMENTS OF PLANE TRIGONOMETRY BY ALFRED MONROE KENYON PROFESSOR OF MATHEMATICS, PURDUE UNIVERSITY AND LOUIS INGOLD ASSISTANT PROFESSOR OF MATHEMATICS THE UNIVERSITY OF MISSOURI THE MACMILLAN COMPANY 1921 All rights reserved GOPYBIGHT, 1919, By the MACMILLAN COMPANY. Set up and electrotyped. Published April, 1919. ASTRONOvy 0!!Pt>. Norinooli i^resg J. 8. Cashing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE This book carries out the chief motives which guided the authors in their larger work on Plane and Spherical Trigonom- etry. On the other hand it has been entirely rewritten, and has been made still more elementary in character. The new text forms a treatment of Plane Trigonometry which is quite brief, but which nevertheless deals with the most essential topics in more than the usual detail. This has been accomplished by omitting or curtailing certain topics that are seldom used by the student except in some special line of work. Thus all of Spherical' Trigonometry and much of the detailed discussion of Trigonometric Identities and Equations is omitted. Such traditional topics as De Moivre's Theorem and infinite series were omitted from the author's larger work because they have few applications with- in the student's present grasp. These are of course omitted from the present book also. Thus this treatment contains a minimum of purely theoreti- cal matter. Its entire organization is intended to give a clear view of the immediate usefulness of trigonometry. The solution of Triangles remains the principal motive. As such, this problem is attacked immediately and it is pushed to a definite conclusion early in the course. More complete outlines than usual have been given for the solution of oblique triangles by means of right triangles. This method of solution was emphasized recently in the Syllabus of the War Department for instruction in the S. A. T. C. A very brief course could well close with this method of solving tri- angles. Other practical problems are introduced to furnish a motive for the treatment of the general angle, the addition theorems, radian measure, etc. Among other applications, the composi- vi PREFACE tion and resolution of forces, projections, and angular speed are introduced prominently. The tables are very complete and usable. Attention is called particularly to the table of squares, square roots, cubes, etc. ; by its use the Pythagorean theorem and the cosine law become practicable for actual computation. The use of the slide rule and of four-place tables is encouraged for problems that do not demand extreme accuracy. One edition of the book contains only the four-place tables. Many who use that edi- tion find it advisable to have students purchase also the five- place tables which are published separately bound under the title The Macmillan Tables. The authors have borne in mind constantly the needs of the beginner in trigonometry and have adapted the book to use in secondary schools as well as in colleges. Illustrative mate- rial abounds, and the explanations have been carefully worked out in great detail. The sample forms for the solu- tion of triangles is a striking instance of this tendency. A. M. Kenyon. Louis Ingold. CONTENTS PART I. ACUTE ANGLES AND RIGHT TRIANGLES Chapter I. Introduction § 1. Subject Matter § 2. Measurement § 3. Relations to Other Subjects. Applications § 4. Graphical Solution of Triangles § 5. Preliminary Estimate. Check § 6. Measurements in the Field § 7. Angles of Elevation and Depression § 8. Squared Paper § 9. Rectangular Coordinates Chapter §10. §11. §12. §13. §14. §15. §16. §17. §18. §19. Chapter §20. §21. §22. §23. §24. §25. §26. §27. Chapter §28. §29. 11. Definitions — Solution op Right Triangles Tables ...... Definitions of the Ratios Right Triangles .... Elementary Relations Construction of Small Tables . Functions of Complementary Angles Apphcations Directions for Solving Triangles The Question of Greater Accuracy The Use of the Large Tables . in. Trigonometric Relations Introduction 1 1 2 2 4 6 8 9 10 13 13 15 15 17 18 19 22 23 24 27 Pythagorean Relations 27 Functions of O"* and 90° 29 Functions of 30°, 45°, 60° 29 Trigonometric Equations 30 Inverse Functions 31 Projections 34 IV. Logarithmic Solutions of Right Triangles The Use of Logarithms Products with Negative Factors .... vii ; 37 38 viii CONTENTS PAOK Chapter V. Solution of Oblique Angles by Means of Right Triangles § 30. Decomposition of Oblique Triangles into Right Triangles 42 § 31. Case I: Given Two Angles and a Side .... 43 § 32. Case II: Given Two Sides and the Included Angle . 44 § 33. Case III: Given the Three Sides 45 § 34. Case IV: Given Two Sides and the Angle Opposite One of Them 45 PART n. OBTUSE ANGLES AND OBLIQUE TRIANGLES Chapter VI. Fundamental Definitions and Formulas § 35. Obtuse Angles 49 § 36. Reduction from Obtuse to Acute Angles ... 50 § 37. Geometric Relations 51 § 38. The Law of Cosines . 51 § 39. The Law of Sines 53 § 40. Diameter of Circumscribed Circle 54 § 41. The Law of Tangents 55 § 42. Tangents of the Half-angles . ^ .... 57 § 43. Radius of the Inscribed Circle 58 Chapter VII. Systematic Solution of Oblique Triangles § 44. Analysis of Data 60 60 62 63 65 66 68 § 45. Case I: Given Two Angles and a Side § 46. Case II: Given Two Sides and the Included Angle § 47. Logarithmic Solution of Case II . . . § 48. Case III : Given the Three Sides . § 49. Logarithmic Solution of Case III . § 50. Case IV: The Ambiguous Case Chapter VEIL Areas — Applications — Problems § 51. Areas of Triangles 72 § 52. Area from two Sides and the Included Angle . . 72 § 63. Area from Three Sides 72 § 54. Illustrative Examples 73 § 55. Composition and Resolution of Forces and Velocities . 75 § 56. Illustrative Examples 76 PART III. THE GENERAL ANGLE Chapter IX. Directed Angles — Radian Measure § 57. Directed Lines and Segments 82 § 58. Rotation. Directed Angles 82 CONTENTS ix PAGE § 69. Placing Angles on Rectangular Axes . ... 83 § 60. Measurement of Angles 85 § 61. Radian Measure of Angles ...... 85 § 62. Use of Radian Measure 85 § 63. Angular Speed 86 § 64. Notation 86 Chapter X. Functions of Any Angle § 66. Resolution of Forces. Projections .... 89 § 66, General Definitions. Trigonometric Functions of Any Angle 89 § 67. Algebraic Signs of the Trigonometric Functions . . 91 § 68. Reading of the Tables. Functions of - 6>, 90° + \ / / \ A / \ 1 / \ / /" \ y / c°, A /I ^\ 38 R i \ B 98 1 s^ ace 2J^ \jl s. Fig. 9. cross-section paper. The location of points is particularly- easy on such paper, so that a map, for example, is readily made by using it. By suitably placing the figure, required lengths can frequently be read off at once. Thus, if the triangle for the graphical solution of Ex. 1, § 4, be con- structed on cross-section paper, the required distance, PB., Fig. 9, can be seen at once to be about 56 yards. 10 PLANE TRIGONOMETRY [I, §9 9. Rectangular Coordinates. If any two perpendicular rulings OT and OX of the squared paper (see Fig. 10) are selected, the position of any point P in the plane is determined by means of the distances from these two lines to the point P, The paper can be so placed that these distances are vertical and horizontal, respectively; we shall usually suppose the paper in this position. ^ "~ "~ "~" ~* y ^Y ■" ^ ~" ^ )^ ~" ■"" -1- 2, 1-.^ )- .B (- 1, l.;2)^ SECiOND QUA DRANT f-;iRp Quadra:nt ^ 3 - ■X- ^ 7 1 UnitJ TH RD qu'adr; NT FOU RT H QUA.DF A[^T P 1 (1.4,- -.8)^ ,c \' -1 — 1) Fig. 10. Thus, in Fig. 10, the horizontal distance from OF to the point ^ is 1.2 units. To avoid confusion between points at the same distance above (or below) OX but on opposite sides of OY, it is customary to call distances measured to the right of OF positive, distances to the left of OF negative ; thus, B is said to be —1 unit from OY. Similarly, distances measured downwards from OX are called negative ; for example, D is — 0.8 from OX, and C is - 1 from OX and also - 1 from OY. The two distances to any point P from OF and OX are called the rectangular coordinates of P, and are frequently denoted I, § 9] INTRODUCTION 1 1 by the letters x and y, respectively. The horizontal distance x is called the abscissa of P; the vertical distance y is called the ordinate of P. In giving these distances it is generally under- stood that the first one mentioned is x, the last y. Thus A, Fig. 10, is briefly denoted by the numbers (1.2, 1.4); B is denoted by (-1, 1.2); C by (-1, -1); D by (1.4, -0.8). The lines OX, Fare called the axes of coordinates, or simply the axes, OX is called the aj-axis, Fthe ^/-axis. The point is called the origin. The four portions into which the plane is divided by the axes are called the first, second, third, and fourth quadrants, as in Fig. 10. To locate a point is to describe its position in the plane in terms of its distances from the coordinate axes ; e.g. (—5,2) is a point 5 units to the left of the ^/-axis and 2 units above the a>-axis. To plot a point is to mark it in proper position with respect to a pair of axes. EXERCISES III SQUARED PAPER 1. Locate and plot each of the following points with respect to some pair of axes : (a) (1,2), (&) (2, -3), (c) (4, -7), (d) (-5,2), (e) (-7, -7), (/) (7, 6), (9) (5, 12), (h) (8, -3), (i) (-6, -5), (j) (6, -2). 2. Show that the line joining (5, — 4) and (— 5, 4) is bisected by the origin. 3. On what lines do all points (1, 0), (2, 0), (-3, 0), (1.5, 0) lie? On what line do all the points (0, 0), (0, 1), (0, 2), (0, 5), (0, - 2) lie ? Make a general statement about such points. 4. Find the distance from the origin to each of the points in Ex. 1, by using the folded edge of another piece of squared paper. Compute the same distances by regarding each of them as the length of the hypotenuse of a right triangle, the lengths of whose sides can be read directly from the figure. Each of these methods can be used as a check on the other. Ans. (a) 2.2, (6) 3.6, (c) 8.1, (d) 5.4, (e) 9.9, (/) 8.6, (g) 13, (h) 8.5, (i) 7.1, (j) 6.3 5. Construct the triangle whose vertices are (6, 2), (8, 4), and (10, 12). Find its perimeter and its area. Ans. 21.8, 6. 12 PLANE TRIGONOMETRY [I, § 9 6. Find the lengtlis of the segments whose end points are : (a) (2, 4) and (5, 8) ; (6) (4, -3) and (-1, 3); (c) (1, -2) and (4, 2). Ans. 5, 7.8, 5. 7. Find the sides and diagonals of the parallelogram whose vertices are (2, 1), (5, 4), (4, 7), and (1, 4). Ans. SV% VlO, 2VT0, 4. 8. Plot the points A : (1, 0), JB : (- 3, 2), C : (1, 1), D : (7, 3) and determine the angle at which the line AB crosses the line CD. Ans. 46°. 9. Plot A: (2, 1), 5: (6, -1), C: (1, 3), D: (-2, -3) and find the angle at which AB crosses CD ; also find the area of the triangle whose sides are AB, CD, and BD. Ans. 90°, 16.8 10. Plot^: (5, -2), 5: (14, 8), (7: (2, 3) and find the distance from A to BC ; also find the area of the triangle ABC. Ans. 75/13, 37.5 11. A farm is described in the deed as N.E. J and E. \ of N. W. J, Section 5, Wayne Township, Tippecanoe County, Ind. Taking the center lines of this section as axes, make a map from the following data : A ditch crosses the farm through the points (—80, 40), (80, 80), (160, 136), distances being measured in rods. The house is at (152, 72). There are seven fields whose corners are : A, (- 80, 112), (- 80, 160), (— 16, 112), (-16, 160); 5, (-80, 40), (-16,56), (-16, 112), (-80, 112); G, (-80, 0), (0, 0), (0, 60), (-80, 40); D, (-16, 56), (80, 80), (80, 160), (-16, 160) ; E, (80, 80), (160, 136), (160, 160), (80, 160); F, (80, 0), (160, 0), (160, 136), (80, 80) ; G, (0, 0), (80, 0), (80, 80), (0, 60). Find the area of each field and the total length of fence. Ans. 19.2, 25.6, 25, 55.2, 26, 54, 35, (acres); 3 miles 68 rods. 12. Positions on a rectangular farm are given by their coordinates in rods, referred to two sides of the farm as axes, as follows : house (10, 4), barn (6, 4), gate of pasture (60, 20). A railroad passes between the house and barn, with a crossing at the point (3, 12). Draw a map showing these objects. Determine how much farther it is from the house to the barn by way of the crossing than along the straight line connecting them. How much farther is it from the barn to the pasture gate by way of the crossing than along a straight line ? Ans. 15.2, 9.78 13. A certain city park is bounded by a main street, two cross streets perpendicular to it, and a stream. The distances, in feet, to the stream measured perpendicularly from the main street at 100 ft. intervals are found to be 680, 650, 525, 450, 450, 460, 540. Draw a map of the park and determine approximately its area. Ans. 7 acres, 9580 sq. ft. 14. To determine the height of a tree OA standing in a level field the distance OB = 100 ft. from the base of the tree to a point B in the field, and the angle of elevation OB A = 37°, are measured. Find the height of the tree. Ans. 75 ft. CHAPTER II DEFINITIONS. SOLUTION OF RIGHT TRIANGLES 10. Tables. While the methods for solving triangles ex- plained in Chapter I are sufficient for all cases, they are really not convenient where great accuracy is desired, since for this purpose the figure would need to be drawn on a very large scale. The method usually employed when one desires greater accuracy than can be conveniently attained by the method of construction and measurement is the method of tables. Tables are constructed which give approximately the ratios of each pair of sides for all right triangles. To obtain the ratio of a certain pair of sides of a right triangle with a given acute angle it is then only necessary to consult the table. For example, it is known by geometry that if one angle of a right triangle is 30°, the side opposite this angle is one-half the hypotenuse. Hence if the hy- potenuse is given, that side, and hence also the other one, can be determined. If in Fig. 11, AB = 22.5, and ZA = 30°, then the side ^O = (1/2) (22.5)= 11.25 If, for an acute angle of every right triangle, the ratio of the opposite side to the hypotenuse were known to us, then we could solve every right triangle in the same manner. It will be shown later that all oblique triangles can be cut up into right triangles in such a way that the same tables can be used in all cases for solving oblique triangles. Since any triangle can be enlarged (or reduced) in size by drawing it on a larger (or smaller) scale, only the ratios of the sides are really important. 11. Definitions of the Ratios. As indicated in § 10, the ratio of two sides of a triangle does not depend upon the size 13 14 PLANE TRIGONOMETRY [II, § 11 of the triangle, but only upon the angles. Thus in the right triangles MPN, MP'N', MP^'N" of Fig. 12, in which PiV, P'N', P'^N'' are perpendicular to MN, the ratios NP/MP, N'P'/MP\ N''P "IMP " are all equal. Moreover, if piti^ni ig dra^n perpendicular to MP^ each of the ratios just mentioned is equal to N'"P"'/MP"\ (Why?) These ra- tios, then, depend only on the angle a at M. It is convenient to place the angle on a pair of axes so that the vertex falls at the origin 0, one side lies along the a^axis, to the right, and the other side falls in the first quadrant. On this side take any point P at random, except 0, and drop the perpendicular PM to the a?-axis (see Fig. 13). Let OP=r) then by geometry r = -Vx^ + 2/2,* where x and y are the coordinates of the point P. The various ratios of pairs of the three quantities x, y, r are the same for all points P taken in the side OP of the angle a. These are : (1) (2) (3) Fig. 13. y , called the sine of the angle a, written sin a. -, called the cosine of the angle a, written cos a. — , called the tangent of the angle a, written tan a. The reciprocals f of these ratios are also often used ; (4) r/y is called the cosecant of the angle a, written esc a. (5) r/x is called the secant of the angle a, written sec a. (6) x/y is called the cotangent of the angle a, written ctn a. * The radical sign is used to denote the positive square root. t The reciprocal of a number is unity divided by the number. The recipro- cal of a common fraction is the result of inverting it ; thus the reciprocal of y/r is r/y. Every number has a reciprocal except 0, which has not. II, § 13] DEFINITIONS 15 These six ratios are collectively called trigonometric ratios or also trigonometric functions of the angle. Other expressions derived from these are also frequently used ; for ex- ample, many engineers use the following combinations : (7) versed sine of a = 1 — cos a, written vers a ; (8) external secant of a = sec a — 1, written exsec a ; (9) haversine of a = half the versed sine of a = ^-^'^^'' ^ written hav a ; 2 and occasionally also the function coversed sine of a = 1 — sin a, written covers a. 12. Right Triangles. In the right triangle OFM, Fig. 13, y is the side opposite the angle a, x is the side adjacent to a, and r is the hypotenuse. From the definitions (l)-(3), we see that in any right triangle : (10) The sine of either acute angle = — — ; \ ^ ^ ^ ^ hypotenuse ' \ side adjacent \ (11) The cosine of either acute angle hypotenuse (12) The tangent of either acute angle = -r-z — ^? -; / and, after clearing of fractions, we find for either acute angle (13) The side opposite = hypotenuse x sine = side adjacent x tangent; (14) The side adjacent = hypotenuse x cosine = side opposite x cotangent; /-I i-\ TT ^ ^ side opposite side adjacent (15) Hypotenuse = f^ = r • ^ ^ sine cosine The student should so thoroughly learn these statements that he can apply them instantly and confidently to any right triangle that he sees, whatever its position in the plane. 13. Elementary Relations. The trigonometric functions are connected by many simple relations. Thus : ,^ />x ^ sin a . y y ^ (Id) tan a = , since = - -; — cos ot X r r 16 PLANE TRIGONOMETRY [11, § 13 Similarly, the student can easily show that sm a 1 (17) (18) sec a = ctn a = cos a cos a tana' (19) CSC a 1 sin a .^ -2 Other relations will be given later. The following examples illustrate a method of constructing an angle when one of its ratios is given. Example 1. Construct an acute angle whose sine is 2/7. To construct such an angle draw a right triangle whose hypotenuse is 7 and one whose side is 2. This can easily be done on cross-section paper. With a radius of 7 draw a circle and mark its intersection with the hori- zontal ruling 2 units above the center. The angle between the horizontal diameter and the radius to this intersection is the angle required. Example 2. Construct an acute angle whose tangent is 3/8. This is most easily done by drawing a triangle whose base is 8 and whose altitude is 3. The angle between the hypotenuse and base is the angle required. As in Example 1, it will be found conve- nient to draw the figure on cross- section paper. Fig. 15. Fig. 14. 3 8 \ EXERCISES IV. — TRIGONOMETRIC RATIOS 1. On cross-section paper construct angles whose sines are :• (a) 1/5; (6) 2/5; (c) 3/5; (d) 4/5; (e) 2/3; (/) 5/7; {g) 0.5 2. Is there an acute angle whose sine is any given positive number ? 3. Construct angles whose tangents are : (a) 3/10; (6) 1/2; (c) 2/3; (d) 1; (6) 10/3; (/) 2; {g) 7.5; (A) 3.4; (i) 1.7 4. Is there an acute angle whose tangent is any given number ? 6. How large, in degrees, is the acute angle whose tangent is 1 ? 6. How does the angle whose tangent is 2 compare with the angle whose tangent is 1 ? Check your answer by drawing an accurate figure. 11, § 13] DEFINITIONS 17 14. Construction of Small Tables. Approximate values of the trigonometric functions of a given acute angle may be 20 20 30 iO 60 HO 70 iO itO 100 Fig. 16. found by measurement as follows. On a sheet of squared paper, construct a quarter circle with its radius = 100, and 18 PLANE TRIGONOMETRY [11, § 14 with its center at the intersection of two heavy rulings. Draw a tangent to this circle perpendicular to the horizontal rulings. Given now any acute angle, a, lay it off above the horizontal axis with its vertex at the center of the circle. Call the points where its side crosses the circle and the tan- gent P and Q, respectively. Then the ordinate (y) of the point P can be read at least to units, and this divided by r = 100 gives the value of sin a to two decimal places. Similarly, the abscissa (x) of P can be read to units, and this divided by 100 gives cos a. Likewise the ordinate of Q can be read to units, and this divided by 100 gives tan a. Finally, ctn a, sec a, esc oc, can be computed as the reciprocals of tan a, cos a, sin a, respectively. The student will find it instructive to compute in this way, from Fig, 16, values to fill out the following table. a 5'' 10° 15" 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° 75° 80° 85° since cos a tan a etna 15. Functions of Complementary Angles. If all of this table is filled out correctly, it will be found that every number in it occurs twice ; once for an angle less than 45° and once for an angle greater than 45°. This result indicates that the sine of any angle is the cosine of its com- plement; and the tangent of any angle is the cotangent of its complement. These relations will now be proved for any acute angle a. Let p = 90° — a ; then a and fi are the acute angles of a right triangle. Denote the sides opposite a and fihj a and b. Fig. 17. II, § 16] SOLUTION OF RIGHT TRIANGLES 19 respectively j and the hypotenuse by c. Then by § 12, side opposite a sm a = -r f^ = - ; hypotenuse c ^ __ side adjacent __ a ^ "" hypotenuse c ' side opposite __a "~ side adjacent b ' ^ _ side adjacent a ctn fi = -r^ ^ — — =- ; side opposite b ' whence, remembering that /3 = 90° — a, (20) sin a = cos )8 = cos (90° - a), (21) tan a = ctn /? = ctn (90° - a). In the same way it can be shown that (22) sec a = esc (90° - a). 16. Applications. The values of the trigonometric ratios have been computed approximately for all acute angles, and recorded in convenient tables. These tables, together with the formulas just given, enable us to solve all cases of right triangles. On page 21 is printed a table giving the values of the ratios to three decimal places. If still greater accuracy is required, a four or a five-place table should be employed. In the following examples the three-place table is used. Example 1 . One angle of a right triangle is 38° and the hypotenuse is 12 ft. Find the lengths of each of the other sides. Draw a figure, mark the given parts, and indicate the parts to be found by suitable letters, say x and y. The sides x and y are then respectively the side ad- jacent and the side opposite. To find «, note that the hypotenuse is given ; hence by (14), § 12, X = 12 . cos 38°. The value of the cosine of 38° from the three place table is found to be .788 Using this value we find x=:12 (.788) .788 12 or X- 9.456 20 PLANE TRIGONOMETRY [11, §16 Example 2. Fig. 19. Similarly by equation (13) , § 12, 2/ = 12 . sill 38^ and from the three-place table the sine of 38° is found to be .616. Using this value we obtain y = 12 (.616) .616 12 y= 7.392 As a check, the Pythagorean theorem may be used, particularly if a table of squares is available. Thus, denoting the hypotenuse by h, we should have h = V(9.456)2 + (7.392)2 = 12.002 This agrees reasonably well with the given value h = 12. Another check that is more practical is given by measurement from a good figure. One side of a right triangle is 17 and the angle opposite this side is 27° ; what is the length of the hypote- nuse ? of the other side ? Denote the hypotenuse by u and the unknown side by v. Noting that the side opposite the given angle is given, find the side adjacent^ v, by (14), *§ 12. To find the hypotenuse, use (15), § 12 : v = 17. ctn 27° = 17(1.963) 1.963 17 13.741 19.63 V = 33.371 w = 17 - sin 27° = 17 ^.464 Performing the division we find ' u = 37.44 Check these answers by drawing an accurate figure. Example 3. The hypotenuse of a right triangle is 41 and one side is 13 ; find the opposite angle. ^ ^^ ;. Denote the opposite angle by a, then by equation (10), § 12, sin a = 13 -f- 41 = .317 From the table (p. 21) we see that sin 18° = .309 and that sin 19° = .326, so that sin a is very nearly halfway between sin 18° and sin 19°. We judge therefore that the angle a is about halfway between 18° and 19°; hence a = 18° .5 II, § 16] SOLUTION OF RIGHT TRIANGLES 21 TRIGONOMETRIC FUNCTIONS TO THREE PLACES OP DECIMALS a sin a sec a tan a ctn a. CSC a cos a 0° .000 1.000 .000 1.000 90° 1° .017 1.000 .017 57.290 57.299 1.000 89° 2° .035 1.001 .035 28.636 28.654 .999 88° 3° .052 1.001 .052 19.081 19.107 .999 87° 40 .070 1.002 .070 14.301 14.336 .998 86° 5^ .087 1.004 .087 11.430 11.474 .996 85° 6° .105 1.006 .105 9.514 9.567 .995 84° . 70 .122 1.008 .123 8.144 8.206 .993 83° 8^ .139 1.010 .141 7.115 7.185 .990 82° 9° .156 1.012 .158 6.314 6.392 .988 81° 10° .174 1.015 .176 5.671 5.759 .985 80° 11° .191 1.019 .194 5.145 5.241 .982 79° 12° .208 1.022 .213 4.705 4.810 .978 78° 13° .225 1.026 .231 4.331 4.445 .974 77° 14° .242 1.031 .249 4.011 4.134 .970 76° 15° .259 1.035 .268 3.732 3.864 .966 75° 16° .276 1.010 .287 3.487 3.628 .961 74° 17° .292 1.046 .306 3.271 3.420 .956 73° 18° .309 1.051 .325 3.078 3.236 .951 72° 19° .326 1.058 .344 2.904 3.072 .946 71° 20° .342 1.064 .364 2.747 2.924 .940 70° 21° .358 1.071 .384 2.605 2.790 .934 69° 22° .375 1.079 .404 2.475 2.669 .927 68° 23° .391 1.086 .424 2.356 2.559 .921 67° 24° .407 1.095 .445 2.246 2.459 .914 66° 25° .423 1.103 .466 2.145 2.366 .906 65° 26° .438 1.113 .488 2.050 2.281 .899 64° 27° .454 1.122 .510 1.963 2.203 .891 63° 28° .469 1.133 .532 1.881 2.130 .883 62° 29° .485 1.143 .554 1.804 2.063 .875 61° 30° .500 1.155 .577 1.732 2.000 .866 60° 31° .515 1.167 .601 1.664 1.942 .857 59° 32° .530 1.179 .625 1.600 • 1.887 .848 58° 33° .545 1.192 .649 1.540 1.836 .839 57° 34° .559 1.206 .675 1.483 1.788 .829 56° 35° .574 1.221 .700 1.428 1.743 .819 66° 36° .588 1.23() .727 1.376 1.701 .809 54° 37° .602 1.252 .754 1.327 1.662 .799 53° 38° .616 1.269 .781 1.280 1.624 .788 52° 39° .629 1.287 .810 1.235 1.589 .777 51° 40° .643 1.305 .839 1.192 1.556 .766 60° 41° .656 1.325 .869 1.150 1.524 .755 49° 42° .669 1.346 .900 1.111 1.494 .743 48° 43° .682 1.367 .933 1.072 1.466 .731 47° 44° .695 1.390 .966 1.036 1.440 .719 46° 46° .707 1.414 .1000 1.000 1.414 .707 46° COS a CSC a ctn a tana sec a sin a a 22 PLANE TRIGONOMETRY [II, §16 Example 4. The two perpendicular sides of a right triangle are 23 and 83 ; determine the acute angles and the hypotenuse. Denote the hypotenuse by h and the angle opposite the smaller side by a ; then by equation (12) § 12, tan a = 23 -r- 83. After performing the division it is found that tan a = .277 As in the example above it is noticed that tan a lies very nearly halfway between tan 15° and tan 16° ; we have, therefore, very approximately, a = 15°.5 17. Directions for Solving Triangles. In the solution of triangles, use the following procedure : (a) Draw a diagram approximately to scale, indicating the given parts. Mark the unknown parts by suitable letters, and estimate their values. (6) If one of the given parts is an acute angle y consider the re- lation of the known parts to the one which it is desired to find, and apply the proper one of formulas (10) ••• (15), § 12. (c) If two sides are given, and one of the acute angles is desired, think of the definition of that function of the angle which employs the two given sides. (d) Check each result. EXERCISES v. — SOLUTION OF RIGHT TRIANGLES 1. One side of a right triangle is 21 ; the adjacent angle is 42° ; de- termine the remaining side and the hypotenuse. Check. 2. One side of a right triangle is 21 and the opposite angle is 42° ; de- termine the remaining side and hypotenuse. Check. 3. The hypotenuse of a right triangle is 28 ; one angle is 32°. Deter- mine the two perpendicular sides. Check. 4. What is the angle of inclination of a roof which has half pitch ? 1/3 pitch? [Note. The pitch of a roof is equal to the height of the comb above the eaves divided by the total distance between the eaves. ] 5. In the following triangles h denotes the hypotenuse ; the angle A is opposite the side a and the angle B is opposite the side b. Use the table to compute the unknown parts from the given parts. Check. II, §19] SOLUTION OF RIGHT TRIANGLES 23 (a) A = 61^ b = 41. (d) A = 32°, a = 330. (6) a = 421, 6 = 401. (e) a = 313, h = 720. (c) a = 62, /I = 125. (/) B = 49°, h = 24. 6. Determine the height of a tower MN, if the horizontal distance EM to it is 450 ft. and the angle of elevation MEN is 27°. Check. 7. A vertical pole 35 ft. high casts a horizontal ^ •^^^^ shadow 45 ft. long. Determine the angle of elevation of the sun above the horizon. Check. 8. An object known to be 100 ft. in height stands on the bank of a river; from the opposite bank of the river the angle of elevation of the top of the object is found to be 24°; find the width of the river. Check. 9. The radius of a circle is 7 ft. What angle will a chord of the circle 11 ft. long subtend at the center ? Check. 10. From the top of a cliff 92 ft. in height the angle of depression of a boat at sea is observed to be 20°. How far out is the boat ? Check. 11. To find the distance between two objects A and B, where 5 is in a swamp, the distance AG = 350 ft. is measured at right angles to the fine joining them. At G an observer holds an ordinary rake with the end of the handle at his eye and with the center of the rake directed toward A. There appear then to be 6 teeth of the rake between A and B. If the teeth are one inch apart and the handle of the rake is five feet long, de- termine the distance between A and B. 18. The Question of Greater Accuracy. The degree of accuracy of the results obtained by using the values of the trigonometric functions to three places of decimals, while sufficient for many ordinary applications, is not satisfactory for some purposes ; for example, in extended surveys, in astronomy, and in any work for which the data must be deter- mined by using instruments of precision. More accurate values have been calculated. The values for angles at intervals of V are given to five decimal places in five- place tables.* * Throughout this book, page references to Tables are to The Macmillan .Tables. These tables may be had separately bound. They are bound with this book in the edition with complete tables. The edition of this book with brief tables contains only four-place tables, for the convenience of those who prefer the full tables separately bound. 24 PLANE TRIGONOMETRY [II, §19 19. Use of the Large Tables. Five-place tables are used in precisely the same manner as the small table of p. 21. Example 1 . One angle of a right triangle is 42° 20' and the hypotenuse is 28 ft. 6 in. long. Find the remaining sides and the other angle. Draw a diagram to illustrate the problem, indicating the given parts. Denote the unknown parts by the letters a and 6, as in Fig. 22. To find 6, note that it is the side adjacent to the given angle, and that the hypotenuse is given. Hence, by (14), § 12, b = 28.5 cos 42^20' = 28.5 x .73924 = 21.07 Note that a is opposite the given angle; hence by (13), § 12. a = 28.5 sin 42° 20' = 28.5 x .67344 = 19.19 the sine and the cosine of 42° 20' being found in the Tables, p. 43. The angle /3, being the complement of 42° 20', is 47° 40'. Example 2. .The perpendicular sides of a right triangle are 22 ft. 6 in. and 54 ft.^ respectively. Find the hypotenuse and the angles. Draw a diagram, indicating the given parts and lettering the parts to be found, as in Fig. 23. To find a, note that the given parts are the sides opposite and adjacent to it ; hence by the definition of tangent, we write tan a = 22.5 -- 54 = .41667 From the Tables, p. 33, tan 22° 37'. = .41660 and tan 22° 38' --= .41694 whence a = 22° 37'+ and /3, its complement, is 67° 23'-. By the Pythagorean theorem of plane geometry, using a table of squares and square roots, Tables p. 94, /i2 = 54^ + 22^5^ = 3422.25 whence, /t = 58.5 Tables, p. 103. Another method of finding h is the following: Having found a = 22" SI', h = 54/cos 22° 37' = 54/.92310 = 58.498 by (15) § 12. However, this method is open to the objection that any error made in computing a vitiates the resulting value found for h. In general, com- pute each unknown part from the given parts ; i.e. do not use computed parts as data if it can he avoided. In solving right triangles, observe carefully the directions of § 17, p. 22, and use five-place values of the functions (Tables, pp. 22-44 and pp. 94-111) as illustrated in the preceding examples. II, §19] SOLUTION OF RIGHT TRIANGLES 25 EXERCISES VI. — RIGHT TRIANGLES 1. Solve the following right triangles. The hypotenuse is denoted by /i, other sides by other small letters, and any angle by the capital letter corresponding to the small letter that denotes the side opposite it. (a) A = 61° 17', b = 1.4 (d) M= 49° 49', /t=24.6 (g) p = lS.2, g = 50. (6) A = 32° 31', a = 33. (e) b = 4.848, h = 10. (h) u = 11.65, h=2^, (c) A = 62.12, h = 254. (/) C7= 63° 2', u = 40. (i) m=34.2, h =100. Ans. (a) 2.56,2.91; (6) 51.77, 61.39; (c) 14° 9'.4, 75° 50'. 7, 246.29 ; (d) 18.80, 15.87 ; (e) 61°, 29°, 8.746 ; (/) 20.35, 44.88 ; (g) 20°+, 70°-, 53.21 ; (h) 27° 46'.5, 62° 13'.5, 22.12 ; (Q 70°, 20°, 93.97 2. In the following right triangles find the side not given : (a) (h) (c) (d) (e) (f) (9) (h) (0 (J) (k) G) side 2.19 45.6 5.82 53.4 73.6 25.6 46 17.5 46.5 6.83 13.5 106 ^JV. 7.75 9.43 54.4 45.5 9.92 35.1 535.3 side 82.5 19.2 138 110.4 42.7 ans. 7.43 94.26 7.42 56.75 156.4 48 119.6 42 63.13 7.19 32.4 524.7 3. In each of the following right triangles find the three parts not given and the area. (a) a = 30.2, h = 33.3 Ans. 24° 55' .1, 65° 4'.9, 14.03, 211.85 (b) A = 35°, b = 100. Ans. 70.021, 122.07, 3501. (c) h = 4S, B = 27°. Ans. 19.52, 38.31, 373.98 (d) h:=z 176, A = 32°. ^ns. 93.26, 149.25, 6959.68 (e) /i = 425, 6 = 304. Ans. 45° 40', 297, 45144. 4. The base of an isosceles triangle is 324 ft., the angle at the vertex is 64° 40'. Find the equal sides and the altitude. Ans. 302.89, 255.93 5. The shadow of a tower 200 ft. high is 252.5 ft. long. What is the angle of elevation of the sun ? Ans. 38° 23'. 6. A chord of a circle is 21.5 ft., the angle which it subtends at the center is 41°. Find the radius of the circle. Ans. 30.7 7. To determine the width AB of a river, a line BC 100 rods long is laid off at right angles to a line from B to some object A on the opposite bank visible from B. The angle BCA is found to be 43° 35^ Find AB. Ans. 95.17 8. What is the angle of elevation of a mountain slope which rises 238 ft. in one-eighth of a mile (up the slope)? Ans. 21° 8'+. 26 PLANE TRIGONOMETRY [II, §19 9. Two ships in a vertical plane with a lighthouse are observed from its top, which is 200 ft. above sea level. The angles of depression of the two ships are 15^^ 17' and 11° 22^ Find the distance between the ships. Arts. 262.96 10. A flagstaff stands on the top of a house. At a point 100 ft. from the house the angles of elevation of the bottom and top of the staff are respectively 21° 60' and 33° 3'. Find the height of the staff. Ans, 26. 11. A 24-foot ladder can be so placed in a street as to reach a window 16 ft. high on one side and by turning it over on its foot it will reach a window 14 ft. high on the other side. Find the width of the street. Arts. 37.38 12. The length of one side of a regular pentagon is 24 ft. Find the lengths of the radii of the inscribed and circumscribed circles and the area. Ans. 16.62, 20.42, 991.2 13. The side of a regular decagon is 10 in. long. Find the radii of the inscribed and circumscribed circles and the area. Ans. 15.39, 16.18, 769.6 14. A round silo 21.6 feet in diameter subtends a horizontal angle of 6°. Find the distance from the observer to the silo. Ans. 236.7 15. In an isosceles right triangle show that lines from either base angle to the points of trisection of the opposite side cut off respectively, one-fifth and one-half the altitude from the hypotenuse to the vertex of the right angle. CHAPTER III TRIGONOMETRIC RELATIONS 20. Introduction. A few simple trigonometric relations have been given in § § 12, 13, and 15. In this chapter we shall obtain others. The student should first review those already given. 21. Pythagorean Relations. The following equation be- tween the abscissa x, the ordinate y, and the radius r is true for every point in the plane : ^ (1) x^+y'^ = r\ Dividing by r^, we obtain but by § 11, at least when a is acute, xjr = cos a, yjr = sin a ; hence (2) sin2 a + cos2 a = 1 ; ^^^e. tlie, sum of the squares of the sine and cosine of any acute angle is equal to unity. \ Dividing (1) by x^, and then by y'^, we obtain respectively : (3) 1 + tan2 a = sec^ «, (4) 1 + ctn2 a = csc2 a. Formulas (2), (3), and (4) are examples of trigonometric identities. An identity in any quantity, a, is an equation con- ^x Fig. 24. * Formulas (2), (3), and (4) are called the Pythagorean relations because they are obtained from this equation, which is the Pythagorean theorem of plane geometry. t This statement, as well as (3) and (4) below, will later be found to hold for all angles, for the general definitions of sine and cosine. 27 28 PLANE TRIGONOMETRY [HI, §21 taining a which is satisfied by every value of a for which both members are defined. Many other examples of identities will be found in the pages that follow. These formulas and those of § 13 are often useful in simplify- ing expressions or in verifying equations. Other interesting relations are given in exercises that follow. Example 1. To show that sin^ a — cos* a = sin2 a — cos2 a. The expression on the left is the difference of two squares and can therefore be factored ; hence we have sin^ a — cos* a = (sin2 a + cos2 a) (sin2 a — cos2 a) which is equal to sin2 a — cos2 a, since sin2 a + cos2 a = 1. The formulas may also be used to compute the value of one of the trigonometric functions from that of another. Example 2. Given tan 6 = 5/12, to find cos 0. Analytic Method, By (3), 1 + tan2^ = sec2^ ; hence, sec2^ = 1 + 25/144 = 169/144, or sec d = 13/12. Hence, cos = 12/13, since cos = l/sec 0. Geometric Method, The following method is much more practical, and is easily applied to any example of this sort. Draw a right triangle whose base is 12 and whose altitude is 5. The hypotenuse is easily found to be 13. It follows that ^ ^ ^ ^ > ^ 5 ^ *«^ ^ L^ e 12 Fia. 25. ^^^^^sMe^djacent^-^2/13. hypotenuse EXERCISES VII. — PYTHAGOREAN RELATIONS. IDENTITIES 1. In exercises (a) — {%) determine the values of the remaining func- tions of the acute angle by each of the methods of Example 2, above. (a) sin ^ = 3/5. (&) sin <9 = 1/3. (c) cos ^ = 1/3. (d) sin = 5/13. (e) tan = VS. (/) tan = 3/4. (gr) tan = 1/m. (h) sin =b/c. (i) sec = 2. Prove the following relations for any acute angle 0: 2. (sin ^ + cos 0)2 zz: 1 + 2 sin cos 0. 3. cos tan = sin 0. 4. tan + ctn ^ = sec ^ esc 0. 5. sin ^ sec ^ = tan 0. 6. (sec — tan 0) (sec + tan ^) = 1. 7. (sin3 + cos3 0) z= (sin ^ + cos ^) (1 - sin cos 0). 8. cos2 - sin2 ^ = 1 - 2 sin2 = 2 cos2 ^ - 1. 9. sec2 csc2 = tan2 -f ctn2 -\- 2. Ill, §23] TRIGONOMETRIC RELATIONS - 29 22. Functions of 0° and 90°. If an angle of 0° be placed on coordinate axes and the construction of page 14 be made, the point P will lie on the ic-axis, and we shall have x=ry y = 0. The functions sine, cosine, tangent, and secant of 0° are defined by the same ratios as are the corresponding functions of acute angles : hence as in (1), (2), (3), and (5), page 14, sin 0° =^= 0, cos 0° =-= 1, tan 0° =^= 0, sec 0° =-= 1. r r X X The definitions of cotangent and cosecant given for acute angles cannot be applied to 0° because y = 0, and therefore the divisions x/y and r/y, which occur in those definitions, are impossible. Similarly if the angle of 90° be placed on the coordinate axes and the construction of page 14 be made, the point P will lie on the y-axis, and we shall have x = 0, y = 'i^' The sine, cosine, cotangent, and cosecant of 90° are defined by the same ratios as are the corresponding functions of acute angles ; hence by the definitions sin 90°= ^ =1, cos 90° = - = 0, ctn 90° = - = 0, esc 90°= - = 1. r r y y The definitions of tangent and secant given for acute angles cannot be applied to 90°, because x = 0, and the divisions y/x and 7'/x are impossible. We say that 0° has no cotangent or cosecant, and 90° has no tangent or secant.* 23. Functions of 30°, 45°, 60°. In plane geometry it is shown how to construct a right triangle in which one acute angle is 30°, or 45°, or 60°. From these triangles the sine, cosine, tangent, etc., of these angles can be computed. * It is often said that the tangent of 90°, for example, is infinite; this ex- pression does not give any value to the tangent at 90°, but merely describes the fact that the tangent becomes and remains larger than any number we may name as the angle approaches 90°. Similar statements hold for the others. 30 PLANE TRIGONOMETRY [HI, §23 To find the functions of 45°, construct an isosceles right triangle with the equal sides some convenient length m. By the Pythagorean Theorem compute the hypotenuse = m^s/% Then by the definitions (10, 11) § 12, m _ 1 _ J ,_ / \ sin 45°= / /^ N SsTTl and /45° 4X cos 45°= mV2 Fig. 26. m ==-A_^i-/o m V2 V2 =iV2, whence by means of the relations (16, 17, 18, 19), § 12, tan 45° = ctn 45°= 1, and sec 45° = esc 45°= V2. To find the functions of 30° and 60°, construct an equilateral triangle of side m, and divide it into two right triangles by a per- pendicular from one vertex to the opposite side. Apply the definitions (10), (11), § 12, to obtain the values of the functions of 30° and 60° given in the following table. 0^ 30° 45° 60° 90° V2 = 1.414 V3 = 1.732 I/V2 = V2/2 l/\/3 = V3/3 sin 1/2 V2/2 V3/2 1 cos 1 V3/2 V2/2 1/2 tan V3/3 1 V3 These values should be memorized, since the angles 0°, 30°, 45°, 60°, and 90° occur frequently. It is easy to show that all of the relations proved in §§13, 15, 21, hold for the values given in this table. 24. Trigonometric Equations. An equation that is not an identity (§ 21) is sometimes called a conditional equation. Thus the equation sin a -f cos a = 1 is not an identity since there are many values of a for which it is not true ; there are values of a, however, which do satisfy the equation: for Ill, §25] TRIGONOMETRIC EQUATIONS 31 example, if 0° is substituted for a it will be found that the left- hand members reduce to 1 since sin 0° = and cos 0° = 1. This equation is therefore a conditional equation but not an identity. The simplest trigonometric equations are of the form sin a = 1/2, tan a = 1/3, etc., i.e, equations in which the angle a is to be determined from the value of one of the trigonometric ratios. We have already found solutions of such equations in Examples 3 and 4, § 16, and Example 1, § 19. The method there employed of looking up the value of the angle in a table can always be used for this form of equation. A trigo- nometric equation is therefore considered to be practically solved when it is reduced to one of these simple forms. Eor the present we shall consider only positive solutions not greater than 90°. Later it will be found that such equations have other solutions. (See §§ 36 and 68.) If a trigonometric equation contains more than one of the trigonometric functions, all but one can usually be eliminated ; the resulting equation may then be solved algebraically for the function which remains ; the solutions may then be found by the methods explained above. Example 1. Solve the equation sin2 t — cos^ i = 3 sin i — 2. In this equation cos2 t may be replaced by its equal 1 — sin2 t ; the equation then becomes a quadratic in sin t, viz. : 2 sin2 i ~ 3 sin i + 1 =0. This equation is equivalent to the given one; i.e. every solution of either is a solution of the other. The solutions may now be found by factoring: (2 sin t - 1) (sin « - 1) = 0. Hence we have either sin i — 1 = 0, whence sin < = 1, and t = 90° ; or else, 2 sin « - 1 = 0, whence sin ^ = 1/2, and t = 30°. There are no other solutions which do not exceed 90°. 25. Inverse Functions. A notation is sometimes needed for the angle whose sine (or any other ratio) is a given number. A notation quite frequently employed is sin"^ x where x is the given number. In this notation the equation sin a = 2/7 could 32 PLANE TRIGONOMETRY [III, § 25 also be written in the form a = sin~^ (2/T). This equation is to be read, a = the angle whose sine is 2/7. It should be carefully noted that the (—1) of this notation is not an exponent although it is written in the position usually occupied by an exponent. Any other character written in the same position would be regarded as an ordinary exponent; thus the expression sin^/? would be understood to mean, the square of the sine of the angle /S. Many prefer the notation arcsinx to the one given above, and this notation, though not so frequently employed as the other, is nevertheless used to a considerable extent. We shall therefore throughout this book use either notation in order to familiarize the student with both. EXERCISES VIII. — SIMPLE TRIGONOMETRIC EQUATIONS 1. Solve the following equations by constructing a figure for each. (a) sin« = 2/5. (g) cosx = .63 (6) sin X = 1/2. (h) cos x = V3/2. (c) sin ic = .8 (i) sin x = 0. (d) sin aj = .866 (j) cos x = 0. (e) sin x = AS (Jc) sin x = 1. (/) cosx = 1/2. (I) cosx = 1. 2. Prove that there is always an acute angle solution of the equation sin x=c, if c is any number between and 1. 3. Prove that there is always an acute angle solution of the equation tan X = c, if c is any positive number whatever. 4. Find sin-i (2/5) graphically. [Hint. Compare Ex. 1(a).] 5. Express the answer to each of the exercises 1(a) to \(l) by means of the notation sin-i or cos-i (or arcsin, arccos, etc.). 6. Find sin-i(2/3), and also tan-i (1/2) graphically. 7. Find arcsin (.66667), and also tan-i (.60000) by the Tables. Solve each of the following equations for x. 8. 2 sin2 X + sin X = 1. [Hint. Solve this quadratic for sin x. There are, of course, no solu- tions corresponding to values of sin x greater than 1.] 9. (a) 2 sin2 x — 5 sin x + 2 = 0. (6) 4 cos2 ^ -f 8 cos ^ = 5. Ill, §25] TRIGONOMETRIC EQUATIONS 33 10. (a)tanaj=l. (d) tan x =— 2.6 (6) tanx =— 1/2. (e) tan x = 5.3 (c) tan X = 2. (/) tan x = 0. 11. (a)taii2x = 3. (6)tan2^ = 6J. (c) tan2 (? = 6 - 4 V27 12. 2 sin2 X — cos x = 1. 13. cos2 x = sin2 x. 14. 5 sin X + 2 cos2 x = 5. 15. sec2 x + tan x = 3. 16. If a and b are the sides of a right triangle, c the hypotenuse, and A the angle opposite a, show that the area of the triangle is equal to either of the expressions ac cos A he sin A Fig. 27. 17. Two straight pieces of railroad track MA and NB are to be con- nected by a circular track AKB with a radius of 500 ft. and center, 0, tangent to MA and NB. The straight portions of the track pro- duced intersect at a point V at an angle of 100°. (a) How far back from F should the track begin to turn ? (6) How far from V along the , bisector OF of the angle AVB is the center ? (c) Find the shortest distance from V to the curved portion. 18. If, in a figure similar to that of Ex. 17, Z. AVO is any angle, and Z VOA is denoted by a, and OA = r, show that (a) J.F= r tana; (6) -fiTF = r exsec a ; (c) ^-B = 2 r sin a. 19. The side b of the triangle in Ex. 16 is extended beyond ^ to a point D, making AD = c, so that ^-BD is isosceles. Show that (a) ZADB]=A/2; (6) ^i) = 2^"ccos (^/2). (c) From the right triangles DOB and ACB, show that c sin J. = a = 2 c cos (A/2) sin (^/2) ; hence sin -4 = 2 sin (A/2) cos (^/2) ; (d) Likewise, show that c cos A=b = 2 c cos2 (A/2) — c ; hence cos A = 2 cos2 (A/2) -1 = cos2 (A/2) ~ sin2 (A/2). 34 PLANE TRIGONOMETRY nil, §27 Fig. 29. 26. Projections. The projection of a line segment AB upon a line I is defined to be the portion MN of the line I between perpendiculars drawn to it from A and B, respectively. The length of this projection is easily found if the length of AB and the angle a which the line AB makes with I are known. Eor, draw a parallel to I through A, meeting BN at C. Then AOB is a right triangle and the angle at ^ is a ; hence by (14), § 12, MN = AB cos a or, the projection of a segment upon a given line is equal to the product of the length of the segment and the cosine of the angle the segment makes with the given line. The projections of a segment upon the coordinate axes are frequently used. If the segment makes an angle a with the horizontal, the pro- jections on the X and y axes are, respectively, (5) Tto]^AB = AB cos a, Projj, AB = AB sin a, where Tto^^AB and 'Pro] ^AB denote the projections of AB on the a>axis and the y-axis, respectively. 27. Applications of Projections. In mechanics and related subjects, forces and velocities are represented graphically by line segments. A force, say of 10 lb., is represented by a seg- ment 10 units in length in the direction of the force. A veloc- ity of 20 ft. per sec. is represented by a segment 20 units in length in the direction of motion. The projection upon a given line ?, of a segment represent- ing a force, represents the effective force in the direction I ; this is called the component of the given force in the direction L Fia. 30. Ill, §27] PROJECTIONS 35 Example 1. A weight of 50 lb. is placed upon a smooth plane in- clined at an angle of 27° with the horizontal. What force acting directly up the incline will be required to keep the weight at rest ? Draw to some convenient scale a segment 60 units in length directly downward to represent the force exerted by the weight. Projectjthis seg- ment upon^a line inclined at an angle of 27° with the horizontal. The length of this projection WQ, Fig. 31, is 50 cos 63° = 22.7 nearly. This repre- sents the component of the force down the plane. Fig. 31. Therefore, a force of 22.7 lb. acting up the plane will be required. Example 2. A ladder 30 ft. long, when lying horizontal supported at its ends, will carry a safe load of 150[lb. on its middle round. Is it safe for a man weighing 190 lb. to mount it when it is so placed as to reach a window 18 ft. above the ground ? We have to find the component, perpendicu- lar to the ladder, of the man's weight when he stands on the middle round. Let TTP, drawn vertically downward from the middle point of AB, Fig. 32, represent 190 (which need not be on the same scale as ^J5 which repre- sents 30) . Then the component perpendicular to AB is Fig Now by (11) § 12, whence WQ = 190 cos PWQ = 190 cos GAB, cos CAB = AC/AB = 4/6, WQ = 190 X 4 . 6 :162, which is greater than the safe load. Example 3. A traveling crane moves with uniform speed down a shop 297 ft. long and 60 ft. wide in 1 min. 41 sec. It carries a load from one corner along the diagonal to the opposite corner. Find the speed of the crane and of the car which runs on it. Let AP = the speed of the load along the diagonal which by the data of the problem = 3 ft. per sec. (AP need not of course be on the same scale as ^5 and AD). 3 cos PAQ = 2.94+ and QP Fig. 33. Then AQ = the speed of the crane = the speed of the car = 3 sin PAQ = .69+ 36 PLANE TRIGONOMETRY pil, § 27 EXERCISES IX.— PROJECTIONS 1. Find the horizontal and vertical projections of the segments : (a) length 42, making an angle of 37° with the horizontal. (6) length 5.5, making an angle of 50° with the vertical. Ans. (a)33.54, 26.28 ; (6) 3.54, 4.21 2. A straight railroad crosses two north and south roadways a mile apart. The length of track between the roadways is 1 J mi. A train travels this distance in 2 min. Find the components of the velocity of the train parallel to the roadways and perpendicular to them. Find the angle between the track and either roadway. Ans. |, J, 53° 7.8' 3. The eastward velocity of a certain train is 24 mi. per hour. The northward velocity is 32 mi. per hour. Find its actual velocity along the track and the angle the track makes with the east and west direction. Ans. 40, 53° 7.8' 4. A car is drawn by means of a cable. If a force of 5000 lb. exerted along the track is required to pull the car, what force will be required when the cable makes an angle of 15° with the track ? Ans. 5176.4 5. Find the horizontal and vertical components of a force of 30 lb. making an angle of 40° with the horizontal. Ans. 22.98, 19.28 6. Find the horizontal and vertical projections of the segment which joins the points (8, — 3) and (—2, 7). Ans. 10, 10. 7. The stringers for a stairway are 20 ft. 7.8 in. long. The steps are to have 7 in. risers and 12 in. treads (which includes 1 in. overhang) . Determine the number of steps, using the horizontal and vertical projec- tions of the stringer to check the result. Ans. 19. 8. Five forces act on the point A: (—4, 0) viz.: AB, AC, AD, AE, AF, and the points A, B, (7, D, E, F are the vertices of a regular hexa- gon, center at the origin. Show that the vertical com- ponents balance, and find the sum of the horizontal components. Ans. 24. 9. Determine the width and height of a crate for the chair shovm in Fig. 34. Ans. 35+, 48f+. 10. In surveying, the projection of a line on a north and south line is called the latitude of the line and the projection on an east and west line is called the departure of the line. Find the latitude and de- Fig 34 parture of the following lines: (a) length 41 rods, bearing N 26° 15' E. Ans. 36.772, 18.134 (6) length 487 feet, bearing E 32° 30' S. Ans. 259.66, 410.73 (c) length 17.32 rods, bearing N 40° 45' W. Ans. 13.053, 11.247 CHAPTER IV LOGARITHMIC SOLUTIONS OF RIGHT TRIANGLES 28. The Use of Logarithms. Logarithms may be used to shorten computations involving multiplications , divisions, rais- ing to powers or extracting roots, but not involving additions or subtractions. In much of the numerical work which follows, the use of logarithms is very advantageous in saving time and labor, but the student should bear in mind that logarithms are not necessary. They are merely convenient, and they belong no more to trigonometry than to arithmetic. One of the ques- tions which a computer has to decide is whether or not it will be advantageous to use logarithms in a given problem. At the end of this book will be found a table of the logarithms of numbers (Tables, p. 1), and a table of the logarithms of the trigonometric functions (Tables, p. 45), with explanations of their use (pp. v-xvii)."* In case a review of the principles of logarithms is desired, this explanation should be studied before proceeding with the rest of this chapter. The notation log tan 62° 51' means the logarithm of the tangent of 62° 51' ; the tangent of 62° 51' is a number, 1.9500, and the logarithm of this number is 0.29003, as may be seen by looking up log 1.9500 in Table I. This last result is found in Table III, p. 73, which enables us to avoid the labor of looking in Tables II and I, in succession. A formula which has been arranged so as to involve only products and quotients of powers and roots of quantities either known or easily computed from the known quantities, * In the edition of this book with brief tables, only four-place tables are given. Those using that edition should refer to The Macmillan Tables, to which all page references made here apply. 37 38 PLANE TRIGONOMETRY [IV, §29 is said to be adapted to loganthmic computation. Thus the formula h = va2 + 62, which gives the hypotenuse /i of a right triangle in terms of the sides a and 6, is not adapted to logarithmic computation. On the other hand, the formula 6=V/i2-a2=>/(/i + a) Qh — a) which gives one side in terms of the hy- potenuse and the other side, is adapted to logarithmic computation because (/i -f a) and (/i— a) are easily obtained from h and a. Thus, if the hypotenuse is 17.34 and one side is 12.27, the other side is x = V(5.07) (29.61) log 5.07 =0.70501 log 29.61 = 1.47144 log X2 : Tables, p. 10 Tables, p. 5 : 2.17645 logx = 1.08822 X = 12.252 Tables, p. 2 The formulas (10 to 19), §§ 12, 13, are all adapted to loga- rithmic computation. Example 1. Find a = 29.45 sin 46° 23 log 29.45 = 1.46909 Tables, p. 5 log sin 46° 23' = 9.85972 - 10 Tables, p. 89 log a = 1.32881 a = 21.321 Tables, p. 4 675.4 Example 2. Find a from tan a -. log 675.4 = 2.82956 log 423.7 = 2.62706 log tan a = 0.26250 a = 57° 53'.9 42.98 423.7 Example 3. Find h = cos 15° 20' log 42.98 = 11.63327 -10 log cos 15° 20' = 9.98426 - 10 logh= 1.64901 h = 44.567 Tables, p. 13 Tables, p. 8 Tables, p. 78 Tables, p. 8 Tables, p. 61 Tables, p. 8 29. Products with Negative Factors. To find by use of logarithms the product of several factors some of which are negative, the product of the same factors, all taken positively, is first obtained, and the sign is then determined in the usual IV, § 29] RIGHT TRIANGLES BY LOGARITHMS 39 manner by counting the number of factors with negative sign. Example 1. Find x = {- 115) (23.41) (- .6422) (- .1123) Noticing first that there are an odd number of negative factors, we may ^^^*^ -x = (115) (23.41) (.6422) (.1123); and we may compute — x as follows. log 115 = 2.06070 log 23.41 = 1.36940 log .6422 = 9.80767 - 10 log .1123 = 9.05038 - 10 log (-X) =2.28815 — X = 194.15 whence x = — 194.15 The use of logarithms in numerical calculation is further illustrated in the following examples. Example 2. Find x -f 740050 2 log 87 = 3.87904 i log 3241 = 1.75534 5.63438 5.86926 log 740050 log X3 whence 29.76512-30 logx= 9.92171-10 x= 0.83504 o -I.- ^ / 5.62(4.8)i-' Example 3. Findx^x/ / aWs whence \- (.e log 5.62= 0.74974 1.5 log 4.8= 1.02186 11.77160-10 2.3 log 0.684 = 9.62064-10 ^ logx2= 2.15096 logx= 1.07548 x = 11.898 Tables, p. 17 Tables, p. 6 Tables, p. 14 Tables, p. 16 Tables, p. 10 Tables, p. 9 Tables, p. 13 Tables, p. 2 EXERCISES X. — LOGARITHMS. RIGHT TRIANGLES 1. Make the following computations by logarithms (a) .001467 X 96.8 x 47.37 Ans. 6.7268 (6) .0631 X 7.208 x .51272 Am. 0.23317 (c) 2v^5/3^ Ans. 0.1364 (d) \/- 0.00951 Ans. -0.5142 (e) 15.008 X (- 0.0843)7(0.06376 x 4.248) Ans. - 4.671 (/) y/EM^x \/6l72/v/298:54 Ans. 3.076 40 PLANE TRIGONOMETRY [IV, § 29 (9) (18.9503)11 (- O.l)i^ Ans. 1.134 (h) (- 0.1412)2/^-0.00476 Ans. - 0.11858 (i) 1/(72.32)J Ans. 0.05761 (j) V(0-00812)* (471.2)vV(522.3)8 (0.01242)* Ans. 0.8929 2. The following formula d = 0.479-v/— r^ is used to determine the diameter d, of water pipe in terms of the coefficient of friction c, the length I, the flow/, and the head h. Compute d when c = 0.02, I = 500, /zi:5, /t=:10. ^ns. 0.91136 3. A wire 0.1066 cm. in diameter and 27.1 cm. long is stretched 0.133 cm. by a weight of 454 grams. Find the modulus of elasticity by the formula e = — , in which I = length, a — area of cross section, and s = the elongation produced by a weight w. Ans. 1.0365 x 10^. 4. The flow of water over a weir is given by the formula /= ^V2^6 Find/ when k = 4.736, g = 32.2, h = 1.2 Ans. 399.32 5. A steel bar 98.75 cm. long between supports 0.96 cm. wide and 0.74 cm. deep is deflected 1.48 cm. by a weight of 5000 grams at the middle. Find the modulus of elasticity by the formula e = . , ,„, , in which I = 4 od^fi length, b = breadth, d = depth, and h = the deflection due to the weight w. Ans. 2.0908 x 109. 6. The pressure p and the volume v of a gas at constant tempera- ture are connected by the relation pv^ = k. Find p when v = 36.36, a = 1.41, k = 12600. Ans. 79.414 7. The period of a conical pendulum is given by the formula T = 2t \^^^^^^ . Find T when m = 0.347, I = 96.8, a = 9° 20^ w = 340. Ans. 1.9618 8. The volume (gal.) of a conical tank of height h (in.) and vertical angle 2 ais v = irh^ tan^ 06/693. Find the capacity of such a tank whose angle at the vertex is 42° 30' and whose height is 12 ft. 5 in. Ans. 2267.8 9. If a ball of radius r is rolled inside a spherical surface of radius i?, the time of oscillation is given by the formula T = 2ir'^ — ^ -. Find the radius of a concave mirror in which a | in. steel ball makes an oscilla- tion in 1.4 sec. Take g = 384. Ans. 13.805 rv, § 29] RIGHT TRIANGLES BY LOGARITHMS 41 10. Solve by means of logarithms the following right triangles, where h denotes the hypotenuse, other small letters the sides, and the corre- sponding capital letters the angles opposite those sides. (a) J. = 63° ; h = 28.54 Ans, 25.429, 12.957 (6) P = 65° 25'.2 ; p = 69.25 Arts. 31.676, 76.152 (c) A = 28° 25' ; h = 29.36 Ans. 25.822, 13.972 Id) Cr= 28° 40'.4 ; v = 20.71 Ans, 11.326, 23.605 (e) a = 735.1 ; h = 846.2 Ans, 60° 18^6, 419.14 (/) r = 9.328 ; s = 6.302 Ans. 55° 57^4, 11.257 (g) a = 59.68 ; h = 69.27 Ans, 59° 29^4, 35.17 (/i) G = 36° 21' ; /i, = 41.376 Ans, 33.325, 24.524 11. Solve the following right triangles having given (a) hypotenuse = 431.8, side = 127.3 Ans, 17° 8'. 7, 412.61 lb) angle = 43° 48^ side adj. = 67.92 Ans, 94.104, 65.133 (c) angle = 55° 11', side opp. = 68.34 Ans. 83.242, 47.527 (d) hyp. = 61.14, side = 48.56 Ans, 37° 25', 37.149 (e) angle = 49° 13', side adj. = 72.3 Ans. 110.68, 83.810 (/) sides = 126 and 198. Ans. 234.72, 32° 28J'. (g) angle = 57° 46', side opp. ^ 0.688 Ans. 0.4338, 0.8134 Ih) angle = 32° 15'.4, side opp. = 547.25 Ans, 867.12, 1025.4 12. A tree stands on the opposite side of a small lake from an observer. At the edge of the lake the angle of elevation of the top of the tree is found to be 30° 58'. The observer then measures 100 ft. directly away from the tree and finds the angle of elevation to be 18° 26'. Find the height of the tree and the width of the lake. Ans, 74.973, 124.94 13. From a point 250 ft. from the base of a tower on a level with the base the angle of elevation of the top is 62° 32'. Find the height. Ans, 480.93 14. To determine the height of a tower, its shadow is measured and found to be 97.4 ft. long. A ten-foot pole is then held in vertical position and its shadow is found to be 5.5 ft. Find the height of the tower and the angle of elevation of the sun. Ans, 177.09, 61° 11'.4 15. Find the length of a ladder required to reach the top of a building 50 ft. high from a point 20 ft. in front of the building. What angle would the ladder in this position make with the ground ? Ans. 53.85, 68° 12'. 16. The width of the gable of a house is 34 ft. ; the height of the house above the eaves is 15 ft. Find the length of the rafters and the angle of inchnation of the roof. Ans. 22.67, 41° 25'.4 17. Assuming the radius of the earth to be 3956 mi. find the distance to the remotest point on the surface visible from the top of a mountain 2J mi. high. Ans. 140.67 mi. CHAPTER V SOLUTION OF OBLIQUE TIOANGLES BY MEANS OF RIGHT TRLA.NGLES 30. Decomposition of Oblique Triangles into Right Tri- angles. A general method for solving oblique triangles in all cases consists in dividing the triangle into two right triangles by a perpendicular from a vertex to the opposite side ; these right triangles are then solved by the methods of the previous chapter. In all except the three side case the perpendicular can be drawn so that one of the resulting right triangles con- tains two of the given parts. It may sometimes happen that the perpendicular will fall outside the given triangle. 31. Case I : Given Two Angles and a Side. It is im- material which side is given, since the third angle can be found from the fact that the sum of the three angles is 180°. Drop the perpendicular from either extremity of the given side. Example 1. An oblique triangle has one angle equal to 43°, another equal to 67°, and the side opposite the unl«iown angle equal to 61. De- ^ termine the remaining parts. It is immediately seen that the third angle is 180°-(43° + 67°) = 70^. To solve this triangle draw the figure approximately to scale and drop the perpen- ^^ dicular CD=p from one extremity C of the known side to AB, the side opposite C. Denote the unknown side CB by a. In the right triangle J. CD, the hypot- ^A enuse and one angle are known ; hence by (13), § 12, p = 51 sin 67° = 46.95 An angle and the side opposite, in the right triangle BCD, are now known; hence by (15), § 12, a = p/sin 70° = 46.95/.9397 = 49.96 The side AB may be found in the same manner. Check as in § 5, p. 4. 42 V,§32] SOLUTION OF OBLIQUE TRIANGLES 43 If in the equation a = p/sin 70° we substitute the value p = 51 sin 67° previously found, we obtain for a the equation 51 sin 67° "*- sin 70° This formula is adapted to logarithmic computation. Apply- ing the principles of logarithms we obtain log a = log 51 + log sin 67° — log sin 70°. Eemembering that subtracting a logarithm is equivalent to adding the co-logarithm of the same number, we may arrange the numerical work as follows : log 51 = 1.70757 log sin 67'^ = 9.96403 -10 colog sin 70° = 0.02701 log a = 1.69861 a = 49.959 In this solution, p was eliminated. Even if the equations are used without eliminating p^ the actual value of p need not be found, since only log p is needed to complete the solution. 32. Case II : Given Two Sides and the Included Angle. The triangle can be divided into two right triangles, one of which contains two known parts, by a perpendicular from eithei' extremity of the unknown side to the side opposite. Example 1. Two sides of a triangle are 26.5 and 32.8 ; the included angle is 52° 18'. Find the remaining parts. In the figure let J.5 = 32.8, ^.0 = 26.5, and the angle at J. = 52° 18'. Drop a per- pendicular p from B to the opposite side. Denote the unknown side by a and the seg- ments of ACbj X and y as in Fig. 37 ; then p, x, ?/, and tan G can be computed in the following order : p = 32.8 sin 52° 18' = 32.8 x .79122 = 25.952 X = 32.8 cos 52° 18' = 32.8 x .61153 = 20.058 y = 26.5 -x = 26.5 -- 20.058 = 6.442 tan C =p H- 2/ = 25.952 x 6.442 = 4.0286 44 PLANE TRIGONOMETRY [V,§33 Hence from the tables, (7=76° 3'.6 a = y-^cosG = 6.442 -- .24101 = 26.73 These formulas are not well adapted to logarithmic compu- tation. The values of p and x may be computed separately by logarithms, after which y and tan C may be found. We use the formulas p = c sin A, x = c cos A, y = b ^ Xy tan C = p -T- y. The work can be conveniently arranged in two columns, as follows : log 32.8 = 1.61687 log 32.8 = 1.61687 log sin A = 9.89830 log cos A = 9.78642 logp = 1.41417 log X = 1.30229 log y = 0.80902 x = 20.068 log tan C = 0.60515 y = h^x= 6.442 C=76°3'.6 \ogy= 0.80902 a = y-^ cos G log cos C = 9.38190 a = 26.738 log a = 1.42712 33. Case III : Given the Three Sides. In this case it is not possible to divide the triangle into two right triangles in such a way that one of them contains two of the given parts ; how- ever, if a perpendicular is dropped to the longest side from the vertex of the angle opposite, the segments into which this side is divided by the perpendicular are easily computed. g Example 1. The sides of a triangle are > ^/TV • a = 36.4, 6 = 50.8, and c =72.6 Determine <^^ p, ^^y ^^^ angles. ^y^ I ^^y Draw a figure and drop a perpendicular A-^^- 72^5 ^^ from B upon AG, Denote the segments of Fig. 38. ' the base by x and y as in Fig. 38 ; then p2 = 50.8^ - x2 = 36^^ - y2 ; hence ajz _ 2/2 = 608^ - 36^^^ = 1266.68 ; that is, {x -y) (x + y) = 1266.68 Since x + y = b = 72.6, we have x-y = 1266.68 -^ 72.6 = 17.32 ; whence, adding, x = 44.91, and, subtracting, 2/ = 27.69 V,§34] SOLUTION OF OBLIQUE TRIANGLES 45 Since we now know x and ?/, the angles A and C are easily found. The student may complete the solution by using the formulas cos ^ = aj -f- 50.8 cos G = y -^ 36.4 Logarithms may be used as in the previous case to compute the separate products and quotients. The following is a con- venient arrangement : x2 ^y2 = 50.8' - 36.r = c2 - a2. Factoring both sides gives (x 4- y) (X — y) = b(x-y) = (c-{- a) (c - X — ?/ = (c 4- a) (c — a) -f- 6 a) c = 50.8 a = 36.4 c -f a = 87.2 c — a = 14.4 x + y=zb = 72.5 X- 2/ = 17.32 X = 44.91 2/ = 27.59 cos A = x -^ c log X = 1.65234 log c = 1.70586 log cos ^ = 9.94648 A = 27° 51^9 5 = log (c -I- a) = 1.94062 log (c-a) =1.15836 colog b = 8.13966 log (x-y) = 1.23854 cos C = y -^ a log y = 1.44075 log a = 1.56110 log cos C = 9.87965 C = 40°42'.9 111° 25'.2 34. Case IV : Given Two Sides and the Angle Opposite One of Them. The triangle is solved by dropping the perpen- dicular from the vertex of the angle included hy the given sides. Example 1. One angle of a triangle is 37"" 20' ; one side adjacent is 25.8 and the side opposite is 20.8. Solve the triangle. First construct the given angle A and on one side of A lay off ^5 = 25.8 With B as center and radius = 20.8 describe an arc of a circle meeting the opposite side in two points C and C^ Either of the triangles ABC, ABC satisfies the given conditions; the case is on this account called the ambiguous case. Fio. 39. 46 PLANE TRIGONOMETRY [V,§34 The student should note that the triangle 5 CC is isosceles and that the interior angle of ABC at C is equal to the exterior angle of ABC at C ; hence the interior angles C and C are supplements of each other. To solve ABC draw the perpendicular BD =p from B; then determine p from the right triangle ABB. p = 25.8 sin 37° 20' = 15.6464 Next determine C from the right triangle BD G; . ^ p 15.6464 ^^„„^ «^^^ = a = -20:8-=-^^''^' hence C is the acute angle whose sine is .75223 ; i.e. (7=48° 47'. The student can complete the solution as follows: AC = AD + DC; B = 180° -(A+ C). Also for triangle ABC ^ a = 180° - C ; •5' = 180°- (^+ C); AC = AD -CD. For the logarithmic solution we use the formula . n P <^ sin A sm C = -= a a Then the work may be arranged as follows : logc = 1.41162 log sin A = 9.78280 colog a = 8.68194 log sin (7=9.87636 C = 48° 47M C = 131° 12'.9 5 = 93°52'.9 jB' = 11°27M 6 = a sin B/sin A b' = a sin B' / sin A log a = 1.31806 log a = 1.31806 log sin B = 9.99900 log sin B' = 9.29785 colog sin A = 0.21720 colog sin A = 0.21720 log 6= 1.53426 log 6' = 0.83311 h = 34.218 b' = 6.8094 If, in a given problem, the side opposite the given angle is less than the perpendicular let fall upon' the unknown side, there is no solution, and if it is greater than the other given side there is one solution only. The construction indicated in Ex. 1 will in all cases show the number of solutions. V,§34] SOLUTION OF OBLIQUE TRIANGLES 47 EXERCISES XL — SOLUTION OF TRIANGLES Find the remaining parts of the following triangles by suitably divid- ing each into two right triangles. Capital letters represent angles ; small letters the sides opposite them. 1. (a) A = 17° 17', B = Sr 37^ c = 174 ; Ans. 63.186, 129.81 (6) A = 24° 14^ a = 43" 13', c = 240 ; Ans, 143.86, 323.69 (c) L = 28°, M = 51°, I = 6.3 Ans. 10.429, 13.173 2. (a) a = 41, 6 = 51, C = 62° ; Am. 48° 44'.7, 69° 15' .3, 48.152 (6) 6 = 3.5, c = 2.6, ^ = 33°; ^ns. 99° 58'.9, 47M'.l, 1.9356 (c) u=z22, v = 12, ir=42°. ^ns. 106° 27'.6, 31° 32'.4, 15.35 3. (a) a = 7, b = 12, c = 15 ; ^ns. 27° 16', 51° 45'.2, 100° 58'.8 (6) i = 10, m = 14, n = 20 ; ^ns. 27° 39'. 6, 40° 32'.2,111° 48'.2 (c) u = 3, V = 4, ?/; == 5. ^?is. 36° 52'. 2, 53° 7'.8, 90° O'.O 4. (a) a = 50.8, 6 = 35.9, u4 = 64° ; ^ns. 39° 26'.0, 76°34'.0, 54.973 (6) , = 6.22, A. = 7.48, (? = 26° ; ^r^. j^'^'^'f ' ^''^ ^^ '^^ ^^'^^ ^^ ' ' [148°ll'.l, 5°48'.9, 1.438 (c) b = 23.4, g = 19.8, B = 109° ; Ans. 53° 8'.1, 17° 51'.9, 7.5922 Id) a = 213, b = 278, 5 = 100°. Ans. 48° 59'.2, 31° 0'.8, 145.45 5. To determine the distance from a point A to an inaccessible object J5, a base line J.C = 300 ft. and the angles BAG = 40°, BGA = 50° are measured. Find the distance AB. Ans. 229.8 6. To determine the distance between two trees. A, B, on opposite sides of a hill, a point C is chosen from which both trees are visible; the dis- tances ^C = 400 ft., BC = 361 ft., and the angle ACB = 55° are then measured. What is the distance between the trees ? Ans. 353.08 7. The sides of a triangular field are 43 rods, 48 rods, and 57 rods, respectively ; determine the angles between the sides. Ans. 47° 24', 55^ 15', 77° 21'. 8. A 50-ft. chord of a circle subtends an angle of 100° at the center. A triangle is to be inscribed in the larger segment, having one of its sides 40 ft. long. How long is the other side ? Is there only one solution ? Ans. 65.22 9. A triangle having one of its sides 60 ft. long is to be inscribed in the segment of Ex. 8. Determine the remaining side. How many solu- tions are there in this case ? Ans. 18.88, 58.25 10. Find the length of a side of an e(iuilateral triangle circumscribed about a circle of radius 15 inches. Ans. 51.96 in. 48 PLANE TRIGONOMETRY [V,§34 11. The angle of elevation of the top of a mountain is observed at a point in the valley to be 60^ ; on going directly away from the mountain one half mile up a slope inclined 30° to the horizon, the angle of elevation of the top is found to be 20°. Find the height of the mountain. Ans. 4529.5 ft. 12. The base of an isosceles triangle is 245.5 and each of the base angles is 68° 22^ Find the equal sides and the altitude. Ans. 332.96, 309.51 13. The altitude of an isosceles triangle is 32.2 and each of the base angles is 32° 42'. Find the sides of the triangle. Ans. 100.31, 59.60 14. A chord of a circle is 100 ft. long and subtends an angle of 40° 42' at the center. Find the radius of the circle. Ans. 143.78 15. From a point directly in front of a building and 150 feet away from it, the length of the building subtends an angle of 36° 44'. How long is it ? Ans. 66.40 16. Find the perimeter and the area of a regular pentagon in- scribed in a circle of radius 12. Ans. 70.534, 342.38 17. Find the perimeter and the area of the regular octagon formed by cutting off the corners of a square 15 inches on a side. Ans. 49.705, 186.39 18. Find the perimeter and the area of a regular pentagon whose diagonals are 16.2 inches long. Ans. 50.06, 172.466 19. Find the perimeter and the area of a regular dodecagon inscribed in a circle of radius 24. Ans. 149.08, 1728. 20. Two chords subtend angles of 72° and 144° respectively at the center of a circle. Show that when they are parallel and on the same side of the center, the distance between the chords is one-half the radius. 21. Devise a formula for solving an isosceles triangle when the base and the base angles are given ; when the base and one of the equal sides are given ; when one of the equal sides and one of the base angles are given. PART n. OBTUSE ANGLES AND OBLIQUE TRIANGLES CHAPTER VI FUNDAMENTAL DEFINITIONS AND FORMULAS 35. Obtuse Angles. The solution of oblique triangles in- volves obtuse * as well as acute angles. For this reason we need to be able to determine the values of the trigonometric ratios for such angles ; it is not necessary, however, to enlarge our tables for this purpose, for, as will now be shown, every ratio for an obtuse angle can he expressed in terms of some ratio of an acute angle. Let an obtuse angle a be placed on the coordinate axes with the vertex at the origin and one side along the ic-axis to the right ; then the other side will fall in the second quadrant. The ratios sin a, cos a, etc., are defined in terms of x, y, and r = Vx^ -\- y^ precisely as they were for acute angles in § 11. It should be noticed, however, that since x is negative while y and r are positive, every ratio which involves x is negative for an obtuse angle ; thus x/r = cos a, y/x = tan a, and their reciprocals, sec a and ctn a, are all negative for obtuse angles. We now proceed to obtain equations similar to the equations sin (90° — a) = cos a, etc. (proved in § 15), which enabled us to find the values of the ratios of acute angles greater than 45° in terms of the ratios of angles less than 45°. * An obtuse angle is an angle which is greater than 90° and less than 180°. B 49 >x 50 PLANE TRIGONOMETRY [VI, § 36 36. Reduction from Obtuse to Acute Angles. Let a be placed on coordinate axes as described above, and let the supplement of a be denoted by /3 (which is an acute angle). Lay off j8 in the first quadrant with one side along the aj-axis. From a point P in the side of a (in second quadrant) and a point P in the side of /3 (in first quadrant) at the same dis- tance r from the origin, draw the perpendiculars FM, F'M', as in Eig. 41. The value of x for the point F will be negative since F is in the second quadrant. Let its co- ordinates be (—a, b) ; then, since the triangles OFM, OFM' are symmetric, the coordinates of F are (a, h) . As in § 11, we have ky (-a, b) \^ M' (a. b) b Fig. 41. sin a = - = sin p, r :180° cos a = - a r -cos ^, or, since /3 - (1) sin a = sin (180° - a) ; (2) cos a = - cos (180° - a) . In a similar manner it can be shown that (3) tan a = -- tan (180° - a). It follows that if a is an obtuse angle we find its sine by looking for the sine of its supplement, which is an acute angle, and similarly for the other functions, always having regard for the proper sign. EXERCISES XII. -FUNCTIONS OF OBTUSE ANGLES AY 1. From the accompanying fig the following relations: (a) sin (90*^ + a) = cos a. (6) cos (90° + a) = - sin a. (c) tan (90° -\-a) =- ctn a. {(i) ctn (90° + a) = - tan a. ure prove p .?o' p' Fig. 42. VI, § 38] LAW OF COSINES 51 2. Construct obtuse angles whose functions have the following values : (a) sin e = 1/3. (5) tan (9 = - 3/4. (c) cos ^ = - 3/.5. (c!) sin e = 1/2. (e) sin = V2/2. (/) sin = V3/2. 3. Find the values of the remaining functions of the angles of Ex. 2. 4. Express the following as functions of an angle less than 45°, and look up their values in a table. (a) sin 121°. (6) cos 101°. (c) tan 168°. (d) sin 99°. (e) ctn 178°. (/) cos 154°. (g) cos 133° 11'. (h) tan 144° 38'. (i) sin 92° 3'. 5. Solve the equation 6 cos2 x + 7 cos ic + 2 = 0. [To solve an equation of this type one should first regard it as an algebraic (quadratic) equation in which the unknown is cos x : replacing cos X by the letter t we have the equation 6^24.7^4.2 = 0. The solu- tions of this equation are i (or cos x) = — 1 or ^ = — |. Then find from the tables the angles x satisfying the equations cos x = — J and cos X = — I ; they are x = 120° or x = 131° 48'. 6] 6. Show that the equation tan x = c has an obtuse angle solution if c is any negative number. 7. Show that the equation sin jc = c has both an acute and an obtuse angle solution if c is any positive number less than 1. 8. Show that the equation cos x = c has a solution between 0° and 180° if c Hes between + 1 and — 1, and that this solution is an acute angle if c is positive and an obtuse angle if c is negative. 9. Find all of the solutions between 0° and 180° for the following equations : (a) 3 sin2 x - 2 sin x — 1 = 0. (6) 4 sin2 x — 3 sin x - 1 = 0. (c) 6 sin2 X + sin X — 1 = 0. (d) 6 sin2 x — sin x - 1 = 0. 37. Geometric Relations. In the following sections certain fundamental geometric and trigonometric relations connecting the sides and angles of any triangle are given. Upon these is based a systematic method of solution of oblique triangles, which is given in the following chapter. 38. The Law of Cosines. In any triangle, the square of any side is equal to the sum of the squares of the other two sides minus twice their product into the cosine of their included angle. Denote the sides of a triangle by a, 6, c, and the angles opposite by A, B, C ] and express the square of side a in terms 52 PLANE TRIGONOMETRY [VI, §38 of by c, and C as follows. Drop a perpendicular, p, from B to the opposite side and denote the segments of this side by x and y. By (13, 14) § 12, we have in Fig. 43, p = c sin A, X— c cos A, y = b — x=b — c cos A a^ = y^^p^= (b —c cos Ay + c2 sin2 ^ = 62 _ 2 &c cos ^ 4- c2 (cos2 ^ + sin2 A) whence, since sin^ A + cos^ A = l (4) a2 = 62 _^ c2 — 2 &c cos -4. If as in Fig. 44 the side a to be found is opposite an obtuse angle A, y=b + x', but by (2) § 36, x=c cos (180°-^) = — c cos A ; hence y =b — c cos A and p= c sin (180° — ^) = c sin J[, exactly as in the case considered above. The law of cosines can be used to compute one side of a triangle when the other two sides and one angle are known, and also to find the angles when the three sides are known. Example 1. One angle of a triangle is 66° 25' and the including sides are 3 and 6. Find the third side. ic2 = 32 + 52 - 30 (.4) =22, ,'.x = a/22 = 4.69 Example 2. Two sides of a triangle are 7 and 8 and the angle opposite the former is 60°. Find the third side. 72 = x2 + 82-16x (J) whence « = 3 or « = 6 and there are two solutions. Example 3. The sides of a triangle are 3, 5, and 7. Find the greatest angle. 72 = 32 + 62-30 cos X whence cos x = — \ and x = 120°. VI, §39] LAW OF SINES 53 EXERCISES XIII. — THE COSINE LAW 1. Two sides of a triangle are 1.5 and 2.4, and their included angle is 36°. Find the third side. Arts. 1.48 2. Two sides of a triangle are 5 and 8 and the included angle is 135°. Find the third side. Arts. 5.69 3. Two sides of a triangle are 3 and 4 and the angle opposite the former is 30°. Find the tliird side. Arts. 2 V3 + V5 or 2 V3 - V5. 4. The sides of a triangle are 3, 5, and 6. Find the smallest angle. Ans, 29° 55'.6 5. The sides of a triangle are 10, 14, and 17. Find the angles. Ans, 36° 1', 55° 25', 88° 34'. * 6. Two sides of a triangle are 11 and 17, and the angle opposite the former is 30°. Find the third side by the law of cosines. 7. Devise a method for finding the angle between two lines without an instrument for measuring angles. Could the law of cosines be used for this purpose ? 39. The Law of Sines. Any two sides of a triangle are to each other as the sines of the angles opposite. Denote the sides and angles of a triangle by a, &, c, A, B, C, as above. Prove that a __ sin A 6 sin 5 as follows : Drop a perpendicular from C (the angle included by the sides a and b) to the opposite side. In Fig. 45, where the Fig. 45. Fig. 46. angles A and B are both acute, by (13), § 12 p = a sin B and also p = b sin A, whence a sin B = b sin A 54 PLANE TRIGONOMETRY [VI, § 40 and (5) and dividing through by h sin B, a sin^ h sin JB In Eig. 46, where one of the given angles is obtuse, p = a sin 5' = a sin (180° — B)= a sin B and also p = b sin A, exactly as above. If the perpendicular is drawn from one of the other vertices, say from A, the above procedure leads to ^ ^ c^sinC' Erom equation (5), dividing each side by sin A and multi- plying each side by 6, we see that g __ b sin A sin B Erom (6) we see, similarly, that each of these ratios is equal to c/sin (7. It follows that we have a b c (7) sin A sin B sin C 40. Diameter of Circumscribed Circle, it can be shown that each of the ratios in (7) (where a, 6, c, stand for the numerical measures of the sides) is equal to the numerical measure of the diameter of the cir- cumscribed circle ; and this furnishes another proof of the law of sines. Circumscribe a circle about the triangle ABC, draw the diameter BA'= d, and con- nect A' C. Then angle A' CB is a right angle and A'=A since each is measured by one-half the arc BC, Therefore by (16), § 12, a a B sin A' sin A b c and similarly d = , d = ^ sinB' sinC Fig. 47. If the angle A were obtuse we should have A' = 180° - A, but since sin (180° — A') = sin A, the same result holds in this case also. Therefore in general, the diameter of the circle circumscribed about a triangle is equal to any side divided by the sine of the opposite angle. VI, § 41] LAW OF SINES 55 The law of sines can be used whenever three parts of a tri- angle are known, of which two are a side and the angle opposite. Example 1. Two angles of a triangle are 10° 12' and 46° 36' and the shortest side is 10. Find the longest side. The angle opposite the longest side is 123° 12' and g ^ 10 sin 123° 12' sin 10° 12' whence ^ ^ 1 0(.83676) ^ .17708 Example 2. The three sides of a triangle are 3, 5, and 7. We have seen in Ex. 3, p. 62, that the largest angle is 120"'. Find the smallest angle. sin X sin 120° whence sinx =?^= .37115 14 and, since x must be acute, x = 21°47'.2 EXERCISES XIV. — THE SINE LAW 1. Two angles of a triangle are 19° and 104° and the side opposite the former is 20. Find the other two sides. Ans, 61.5, 59.6 2. The sides of a triangle are 8, 13, and 15. Find the angle opposite the second side by the law of cosines and the other two by the law of sines. Ans. 60°, 32° 12', 87° 48 . 3. The sides of a triangle are 21, 26, 31. Find the angles as in Ex. 2. Ans. 56° 7', 42° 6', 81° 47'. 4. Compute the length of the radius of the circumscribed circle for each of the triangles in Exs. 1-3. 5. Two angles of a triangle are 38° 12' and 61° 10', and the included side is 350.6 Find the other two sides. Ans. 219.7,311.3 41. The Law of Tangents. In any triangle the difference of any two sides is to their sum as the tangent of one-half the differ- ence of the angles opposite those sides is to the tangent of one-half their sum. Let ABC be any triangle having two sides a and b unequal, say a>b; the included angle C may be acute, right, or obtuse. 56 PLANE TRIGONOMETRY IVI, § 41 With a radius bj the shorter of the given sides, and center (7, the vertex of the included angle, describe a circle through A k a-b H Fig. 48. which cuts the side CB in a point Z) between B and C and also at a second point E beyond C. Draw EA, and at B erect a perpendicular which meets i^^l produced at F, On Di^ as a diameter construct a circle ; this circle will pass through A and By for FAD is a right angle since it is the supplement of DAE which is inscribed in a semicircle, and FBD is sl right angle by construction. This construction is possible for any triangle in which a>b. Angle BFE = ^(A + B) since it is the complement of angle CEA = i (7; and -i ^ + i J5 + i C= 90° since the sum of the angles of a triangle is 180°. Angle DFA = B since each is measured by one-half the arc AD ; therefore BFD = BFE — DFA = ^(A + B)-B=i{A-B). In the right triangles DBF smd EBFhj (13), § 12, a - 6 = J5i^. tan ^{A - B), a + b = BF' tan ^(A + B), whence (8) a_& ^ tan|(>t-g) a + b tani(A + B)' VI, § 42] LAW OF TANGENTS 57 This formula is still true but trivial, if a = b, since in that case each side reduces to zero ; if a < 6, the result would ob- viously be * /Q\ b — a __ tan ^(B — A) ^ ^ ^ 6+~a""tani(B-+-^) Since ^(A -{- B) is the complement of i (7, (8) can be re- duced to the form (10) t^nUA^B)- a-i-b ctn J C. 42. Tangents of the Half-angles. The tangent of one-half any angle of a triangle can be expressed in terms of the sides as follows. Bisect the angles of the triangle ABC and draw the in- scribed circle tangent to the sides at P, Q, and B. Let r be the radius of this circle and let 8 stand for one-half the perime- ter of the given triangle, i.e. 2s = a + b-{'C. Then AP=AB, BE = BQ, CQ=CF, and BE-^BQ+CQ+CF = 2BQ-{-2QC=2a, whence 2AP=2s-2a and AP=AE = s-a, Similarly, BE = BQ = s--b and (7Q=CP = s-c. In the right triangle AFO, by (12), §12, tan i A= s— a A similar result holds for the other two angles. Hence we have the three formulas : Fig. 49. (11) tanl>l = tani5 = . s — a -b' tanlC=- ' 5 — C 68 PLANE TRIGONOMETRY [VI, § 43 Fia. 50. 43. Radius of the Inscribed Circle. It remains to express r in terms of the sides of the triangle. In the triangle ABG produce the sides AB and AO, Bisect the angle A and the exterior angles at B and (7. These bisectors meet in a point / which is the center of a circle which touches the side a, and the sides h and c produced. This circle is called an escribed circle of the triangle. Denote its radius by r' and mark the points of tangency P, Q, R. Then we have Aq = AP, BQ=BB, CP=CB, therefore AB + BE = AC+CB = s, where s denotes half the perimeter of the given triangle. It follows that AQ = s and BQ^AQ-^AB^S'-c. In the right triangle BQI, angle IBQ = 1(180° - B)= 90° - i 5 and therefore angle BIQ - § 42, in triangle BQI, r B ; then by (13, 14), § 12 and (11), :(s-c)ctniB = ^'-^^^'~'\ and in triangle AQI, r^ = s tan ^ A = - Equating these two values of r' and solving for r, we have, (12) • r=-\/5E^(fEMEf). The symmetry of this result in a, &, c shows that we shall get the same result if we produce sides c and a^ or a and &, VI, § 43] HALF ANGLE FORMULAS 59 Example 1. The sides of a triangle are 145/13, 119/13, and 156/13. Find the radius of the inscribed circle and the angles of the triangle. We first compute the values of s, s — a, s — 6, and s — c s = 1 (a + 6 + c) = 210/13, s-a = 65/13 = 5, s - 6 = 7, s - c = 54/13. Substituting in the formula for r we obtain ^ /7x5x(54/13)^^^^ ^ 210/13 tani^=— ?^ = 3/5, tan* 5 =-^^ = 3/7, tan 1 C = —^ = 13/18 ; s — a s — b s — c hence from the tables we find ^/2 = 30°57^8, i?/2 = 23° 11^9, C/2 = 35°50'.3 Example 2. Two sides of a triangle are 12 and 8 and the included angle is 60^. Find the remaining angles. Denoting the unknown angles by A and B we have A+ B = 180° - 60° = 120°, then by the law of tangents we have 12 - 8 ^ tan K^ - -B) _ tan ^(A - JB) 12 + 8 tan 60° ~ V3 ' hence tan l(A - B) = V3/5 and } (A- B)= 19° 6'.4 Adding this result to i(A + 5) = 60° we obtain A = 79°6'.4, and sub- tracting we get B = 40° 53'.6 EXERCISES 1. The three sides of a triangle are 7, 12, and 15. Find the radius of the inscribed circle and thQ angles. 2. Determine the angles of the following triangles : (a) a = 5, 6 = 9, c = 11. (c) a = 10, 6 = 12, c = 15. (6) a = 4, 6 = 8, c = 10. (d) a = 6, 6 = 8, c = 10. 3. Determine the angles and third side of the following triangles : (a) a = 4, 6 = 8, C = 20°. (c) a = 10, 6 = 12, G = 35°. (6) a = 4, 6 = 8, C = 40°. (d) a = 13, 6 = 17, C = 44°. 4. To determine the distance between two objects A and B separated by a barrier, the distances J.0 = 40 rd., BC = 4S rd. are measured to a third point C The angle ACB = 68° is then measured. Find the dis- tance AB and the other angles of the triangle ABC. CHAPTER VII SYSTEMATIC SOLUTION OF OBLIQUE TRIANGLES 44. Analysis of Data. In the solution of oblique triangles the following cases arise : Case I. Given two angles and a side. Case n. Given two sides and the included angle. Case III. Given the three sides. Case IV. Given two sides and an angle opposite one of them. The direction '•'■Solve a triangle^'' tacitly assumes that a suf&cient number of parts of an actual triangle are given. A proposed problem may violate this assumption and there v^^ill be no solution. Thus there is no triangle whose sides are 14, 24, and 40. An attempt to solve such an impossible problem gives rise to a contradiction such as, for example, the sine or cosine of some angle greater than 1. Any triangle which can be constructed can be solved. 45. Case I. Given Two Angles and a Side. In this case it is immaterial which side is given, since the third angle can be found from the fact that the sum of the three angles is 180°. There is one and only one solution, provided the sum of the given angles is less than 180°. TJie other two sides can be found, one at a time, by the law of sines (§ 39). Example 1. Given one side of a triangle a = 2.903 and two of the Q angles B = 79° 40^ C = 33^ 15' ; find the remaining parts. A = 180° - (79° 40' + 33° 16') = 67° 6'. By the law of sines b _ sin 79° 40^ Fig. 51. 2.903 sin 67° 6'' 60 VII, § 45] SOLUTION OF OBLIQUE TRIANGLES 61 Many of the computations in the solution of triangles are of the follow- ing type. To find one term of a proportion, - = - , when the other three b d are known, no matter in which of the four positions the unknown stands. The student should master this problem. The following rule applies. Imagine the means, and also the extremes, to be connected by straight lines crossing at the = sign. Multiply together the pair of knowns thus con- nected and divide by the known opposite the unknown. Applying this rule to the computation of b, the work may be written down as follows : sm 79° 40' = .98378 sin 67° 5' = .92107)2.85691334 |3.1007 2.903 2 76321 295134 92703 885402 92107 196756 59634 2.85591334 .-. 6 = 3.1007 This work can be shortened by the use of logarithms. In all cases where the product of two or more numbers is to be divided by other numbers we can use the following principle (Tables, p. x). Subtracting the logarithm of a number is equivor lent to adding its cologarithm. The computation of 5 by logarithms may be written as follotvs : log 2.903 = 0.46285 log sin 79° 40' = 9.99290 - 10 colog sin 67° 6' = 0.03571 log b = 0.49146 6 = 3.1007 The side c is found similarly from the proportion c ^ sin 33° 15' 2.903 sin 67° 5'* To check, apply the law of sines (§ 39), or the Law of tangents (§ 41) to the computed sides b and c. EXERCISES XV. — CASE I Solve the following triangles. SmaU letters represent sides and cor- responding capital letters the angles opposite. 1. B = 50° 30', C = 122° 9', a = 72. Ans. 334.28, 476.51 2. F = 82°20', G« = 43°20', /=48. Ans. 33.097,39.165 3. M = 79° 59', iV^ = 44°41', p = 477. Ans, 340.73, 398.39 4. P = 37° 58', 5. A = 70° 55', 6. A= 51° 47', 7. A = 48° 10', 8. B = 38° 12', 9. Z7=46°36', 10. B = 21° 16', 11. B = 62° 42', 12. B = 58°20', 13. G = 43° 50'.4, 14. G^ = 75°2'.7, 15. Two observers 62 PLANE TRIGONOMETRY [VII, § 46 Q = 65°2', r = 133.2 Ans. 84.103,110.679 ir=:52°9', a = 48.09 Ans. 42.645, 40.031 5 = 66° 20', c = 337.6 Ans. 300.73,350.58 5 = 54° 10', c = 38.7 ^ns. 29.516, 32.116 C = 61°10', a = 70.12 Ans. 43.949,62.257 F=124°18', w; = 1001. Ans. 4598.6,5228.4 C=113°34', d = 20.93 Ans. 10.705,27.053 ikf=52°22', a = 39.75 Ans. 38.995,34.753 Gf = 61°2'.3, ^ = 8.75 Ans. 8.512, 8.715 Q = 69°30'.2, c = 73.05 Ans. 96.685,97.123 ir=43°44'.3. A: = 81.5 Ans. 103.32,113.89 Two observers, facing each other 3 kilometers apart and at the same altitude, find the angles of elevation of a Zeppelin to be 57° 20' and 64° 30', respectively. Find the height. Ans. 2.683 16. A diagonal of a parallelogram is 18.56 and it makes angles 26° 30' and 38° 40' with the sides. Find the sides and the area of the parallelo- gram. Ans. 9.125, 12.777, 105.81 17. A lighthouse was observed from a ship to be N. 16° W. ; after sail- ing due east 4.5 miles, the lighthouse was N. 48° W. Find the distance from the lighthouse to the ship in both positions. Ans. 5.682, 8.163 18. The side of a hill is inclined at an angle of 22° 37' to the horizon. A flagstaff at the top of the hill subtends an angle of 13° 17' from a point at the foot of the hill, and an angle of 18° 2' from a point 100 ft. directly up the hill. Find the height of the flagstaff. Ans. 95.053 19. To find the distance from a station A to an inaccessible point 5, a base fine AC = 600 ft., and the angles ACS = 68° 18', CAB = 58° 28' are measured. Find the distance AB, 20. To find the height of an inaccessible object AB, a base line CD = 250 ft. is measured directly toward the object : also the angles of eleva- tion ADB = 48° 20' and ACB = 38° 40'. Find the height AB. 46. Case II. Given Two Sides and the Included Angle. There is always one and only one solution. The obvious method of solution is to find the third side by the law of cosines (§ 38), and then the other two angles by the law of sines (§ 39). Example 1. Two sides of a triangle are 10 and 11, and the included angle is 36° 24' . Find the other parts. VII, § 47] SOLUTION OF OBLIQUE TRIANGLES 63 Draw a figure, denote the unknown side by a, and the unknown angles by jB, G. Then we may write a2 = 10^ + TT^ _ 2(10) (11) cos 35° 24', a2=221-(220)(.81513). Then a2 _ 41.6714, whence a = 6.4553 (Tables, p. 104) . ^ To find B and C by the law of sines, we have sin B 11 and sin G 10 sin 35^ 24' 6.4553' sin 35° 24' 6.4553' whence on computing (see Example 1, § 45) 5=80^47'.0, C=63°48'.8 Check : A + B -{- G = 179"" 59'. 8 Example 2. Two sides of a triangle are 138.65 and 226.19, and the included angle is 69° 12'. 9. Find the third side. Construct the triangle as in Fig. 53. a2 = 138-652 + 226.19^ - 2(138.65) (226.19) cos 59° 12'.9 While this is not adapted to logarithms, neverthe- less logarithms can be used to compute separately the three terms on the right ; for the moment call the third term, x. log 138.65 =2.14192 2 4.28384 (138.65)2 = 19224 51161 70385 x= 32102 a2 = 38283 a = 195.66 (Tables, p. 95) log226.19 = 2.35447 2 (226.19)2 4.70894 : 51161 log 2 = 0.30103 log cos 59° 12'.9 = 9.70911 2.14192 2.35447 lOgX : ; 4.50653 X = 32102 47. Logarithmic Solution of Case 11. When two sides and •the included angle are given, a triangle can be completely solved by logarithms by finding first the other two angles by the law of tangents (§ 41). 64 PLANE TRIGONOMETRY [VII, § 47 Example 1. In a triangle MPT, side m = 138.65, side t = 226.19, and the included angle P = 69° 12^9. rind the other parts. Applying the law of tangents to the given sides, noting that i>m, t-m _ t2ini(T-M) t + m tanj(r+^* In this proportion three terms are known since T-\- M= 180° — P. The work may be set down as follows. t = 226.19 m = 138.65 «~m= 87.54 t + m = 364.84 i(T+M)= J(180° - P) = 60° 23'.55 i(T-M) = 22° 53^5 .-. r= 83° 17' ^1^= 37° 30' log (i-m)= 1.94221 colog lt + m)= 7.43790 - 10 log tan i(T+ M) = 0.24646 log tan i( T - if) = 9.62567 - 10 The side p can now be found by solving the proportion p ^ sin69°12'.9 138.66 sin 37° 30' log 138.66 = 2.14192 log sin 69° 12'.9 = 9.93404 ~ 10 colog sin 37° 30' = 0.21666 logp =r 2.29161 from which p = 196.66 Compare Example 2, § 46. 1. (a) W (O (d) (O (/) (9) (7i) 2. 53° 8'. EXERCISES XVI. — CASE II Solve the following triangles by using the law of cosines : a = 22, 6 = 12, O = 42°. Arts. 106° 27'.7, 31° 32'.4, 16.35 5 = 62°. Ans. 66° 13', 71° 47', 13.27 iV^=126°. Ans. 23° 46' .6, 31° 13'.4, 66.89 a = 14, I =28, a = 21, a = 2.2, I =13, M = 41, 5 = 3.5, c = 16, m = 36, 6=24, h = 4.2, m = 16, tj = 51. c = 28. Ans. 46°61'.6, 66°30'.3, 76°38'.l c = 5.5 Ans. 21°16'.9, 43°61'.4, 114°61'.7 w = 20. Ans. 40°27'.l, 62°69'.6, 86°33'.3 W=61°, Ans. 69°67\3, 49° 2'.7, 47.48 c = 2.6, ^ = 33°. Ans. 47°1'.3, 99°58'.7, 1.935 Two sides of a triangle are 2.1 and 3.5 and the included angle is Determine the remaining parts. Ans. 36° 52', 90°, 2.4 VII, §48] SOLUTION OF OBLIQUE TRIANGLES 65 3. . How long is a rod which subtends an angle of 60° at a point which is 6 ft. from one end of the rod and 8 ft. from the other ? Arts. 7 ft. 4. How long is a rod which subtends an angle of 120° at a point 3 ft. from one end and 5 ft. from the other ? Ans, 7 ft. 5. Solve each of the following triangles, using logarithms : (a) a = 52.8, 6 = 25.2, 0=124° 34'. ^ns. 17° IIM, 70.233 (6)6 = 55.1, c = 45.2, ^ = 16° 16'. ^ns. 47° 14'.1, 17.246 (c)i=131, m = n, JV=39°46'. ^ns. 31° 19'. 9, 88.568 (d ) a = 35, 6 = 21, = 48° 48'. Ans. 36° 44'.4, 26.415 (e) u = 604, V = 291, W= 106° 19'. Ans. 22° 9'.5, 740.45 (/) a = 23.45, 6 = 18.44, D = 81° 50'. Ans. 56°56'.4, 41°13'.6, 27.696 ( gr) u = .6238, V = .2347, C = 108° 30'. Ans. 53°49'.2, 17°40'.8, 0.7329 6. Two sides of a triangle are 22.531 and 34.645 ;• the included angle is 43° 31'. Determine the remaining parts. Ans. 40°16'.7, 96° 12'.3, 23.716 7. To determine the distance between two objects A and B separated by a hill, the distances J. C = 300 ft., BC = 277 ft., and the angle ACB = 65° 47', are measured. From these measurements find the distance AB. Ans. 313.94 8. Two objects, A^ B, are separated by an impassable swamp. A station C is selected from which distances in a straight line can be meas- ured to each of the objects. These distances are found to be CA = 341 ft. 7 in., CB = 237 ft. 5 in., and the angle ACB is found to be 53° 11'. Find the distance AB. Ans. 275.4 9. Two objects, J., B^ are separated by a building. To determine the direction of the line joining them, a point C is taken from which both A and B are visible and the distances J. C = 200 ft., BC = 137 ft. 9 in., and the angle A CB = 52° 25' are measured. Determine the angle which AB makes with AC. Also the distance AB. Ans. 43° 15'.9, 159.27 10. To determine the distance between two ob- jects A and B, a base line CD = 350 ft. in the same plane as A and B is measured, and the angles B CD = 40° 42', ACB = S0^ 30', ADB = 6r 12', ADC = 32° 41', are observed. Find the distance AB. Ans. 273.4 ' 48. Case III. Given the Three Sides. There is one and only one solution, provided the sum of any two of the given sides is greater than the third side. 66 PLANE TRIGONOMETRY [VII, § 49 The law of cosines, applied to the side opposite the required angle, will always give a solution ; and if the sides are small, or if only one angle is required, it is often the best method. ^ Example 1. Find the angles of the triangle whose sides are 5, 7, 8. V5 By the law of cosines : 52 = 72 + 82 -2. 7. 8 cos ^, A^ ^B 72 = 52 + 82 - 2 . 5 . 8 cos J5, 82 = 52 + 72 - 2 . 5 . 7 cos 0, cos ^ = Ji = .84615+ cos ^ = i, cos C = | = .14286- ^ = 32°12'.3, 5 = 60°, C = 81°47'.2 Check : A-\- B+ C= 179° 59'.5 Example 2. The sides of a triangle are 2431, 3124, and 2314. Find the largest angle. 3124^ = 23l4^ + 243l^ - 2(2314) (2431) cos a, 2(2314) (2431) Call the numerator x and the denominator y. Then the solution may be carried out by logarithms as f olfows : ^2314 = 3.36436 2 6.72872 5354500 5909600 11264100 9759250 1504850 logx: logy: log COS a log 2431 = = 6.17750 = 7.05117 = 9.12633- = 3.38578 2 6.77156 -10 log 3124 = 3.49471 2 6.98942 2314^ = 2431^ = log 2 = 0.30103 log2314 = 3.36436 3124^ = x = log2431 = 3.38578 log?/ = 7.05117 •. a = 82° IS'.B 49. Logarithmic Solution of Case III. To compute by the aid of logarithms the three angles of a triangle whose sides are known, we first find the radius of the inscribed circle by the formula of § 43 : .^J (s-ct)(s-b)(s-c) ^ VII, § 49] SOLUTION OF OBLIQUE TRIANGLES 67 and then compute the angles by the formulas of § 42 : r tan ^A = , tan ^B = , tan ^ C = - s — a s — b s — c Example. Find the angles of the triangle whose sides are 2314, 2431, and 3124. The work may be arranged as follows a = 2314 s = 3934.5 h = 2431 s -a = 1620.5 c = 3124 s- 6 = 1503.5 2s = 7869 s- c= 810.5 Computation of log r colog s = 6.40512 - 10 log (s- a) =3.20965 log (s- 6)= 3.17710 log(s -c)= 2.90875 log 7-2 = 5.70062 2s = 7869.0 (^Check) log r = 2.85031 log r = 2.85031 log (s-a)= 3.20965 log (s - &) = 3.17710 log tan 1 ^ = 9.64066 - 10 log tan i ^ = 9.67321 - 10 J ^ = 23° 36' .8 i J5 = 25° 13'.8 log r = 2.85031 log (s- c) = 2.90875 log tan A C = 9.94156 - 10 iC = 41°9'.4 Then ^ = 47° 13'.6, 5 = 50°27'.6, 0=82° 18'^ Check : i(A + B + G) =90'' OO'.O EXERCISES XVIL — CASE III 1. In each of the following triangles, the three sides are given. Find the smallest angle. (a) 1, 2, 3. - Ans. 0° (6) 3, 5, 7. Ans. 68°12'.8 (c) 3,4,5. Ans. 36°52'.2 (d) 13, 14, 15. Ans. 53°7'.8 (e) 35,41,47. Arts. 46° 15M (/) 4.7, 5.1, 5.8 Ans. 50° 35 .3 (g) 48.3, 53.2, 62.7 -^^«- 48°24'.4 (h) 1.9,3.4,4.9 ^^«- 16°25'.6 (1)32.1,36.1,40.2 ^ris. 49°24'.0 U) 5.29, 6.41, 7.02 ^^^s- 46°7'.0 2. Solve each of the following triangles, using logarithms : (a ) a = 22.2, h = 31.82, c = 40.64 Ans. 32°54'.6, 51°8'.8, 95° 56'.6 (6) a =27.53, 6 = 18.93, c = 30.14 Ans. 63°31', 37°59M, 78°29'.9 68 PLANE TRIGONOMETRY [VII, § 50 (c) a =523.8, 6 = 566.2, c = 938.4 Ans. 29°17'.3, 31° 55' .5, 118°47'.3 (d) I =3.171, m = 5.331, n = 5.101 Ans. 35°18'.3, 76° 18' .6, 68°23'.l (e) u =40.04, v = 50.56, to = 70.12 Ans. 34°7'.2, 45°5'.9, 100°46'.8 (/) p = 38.2, b = 45.36, d = 26.54 Ans. 57° 14' .7, 87°, 35°45'.2 (g) m = .126, n = .3226, c = .253 Ans. 21° 11, 112°17'.8, 46°31'.2 (A) a =.0506, 6 = .1234, c = .0936 Ans. 21° 56', 114°2V.4, 43°42'.6 (i) w = 167, v = 321, to = 231. ^ns. 29°56'.4, 106°24'.3, 43°39'.3 ( j ) u = 196,1, ?J = 264.1, w = 135.4 Ans. 46° 3'.6, 29° 48'.8 3. Find the angle subtended by a rod 16.2 ft. long at the observer's eye, which is 11.9 ft. from one end and 17.6 ft. from the other. Ans. 73° 44'. 4. To determine without an instrument for measuring angles the angle between two lines meeting at C, the distances CA = 500 ft. and CB = 700 ft. are measured ; AB is then found to be 633 ft. Find Z A CB. Ans. About 61°. 5. A piece of land is bounded by three intersecting streets, on which the property has a frontage of 312 ft., 472 ft., and 511 ft. respectively. Find the angles at which these streets cross. Ans. 64°28'.4, •77°40'.4, 37°51'.4 6. In Fig. 57 AB = 316.8 ft., BC = ^^^'^'^' 226.4 ft., ^(7 = 431.6 ft., and AD = 280.4 ft. Find BD. Ans. 576.1 50. Case IV. The Ambiguous Case. Here we have given two sides and the angle opposite one of them ; i.e. an angle, a side adjacent, and the side opposite. The number of solutions (two, one, or none) is best deter- mined by the geometrical construction of the triangle from the data. Construct an angle AGQ, equal to the given angle which we shall at first suppose to be acute ; on one of its sides lay VII, § 50] SOLUTION OF OBLIQUE TRIANGLES 69 off GA equal to the given adjacent side and drop a perpen- dicular AP, to the other side OQ. Then with A as center and with a radius equal to the given opposite side draw an arc. If, as in Fig. 6^ (a), this arc does not reach GQ, there is no solution; if it is tangent to GQ, as in Fig. 58 (b), there is one solution; if it cuts GQ twice, as in Fig. 58 (c), there are two solutions; if it cuts GQ once, as in Fig. 58 (d), there is one solution; and finally if the given angle is obtuse, there is no solution when the radius of the arc is less than GA and one solution when it is greater. Fig. 58. The results may be collected for reference as follows : Let G = the given angle, (adj.)= the given adjacent side, (opp.) = the given opposite side ; then I. WJien G is acute, compute p=:(adj.) sin G; then if (ppp.)

(adj.) one solution. II. JVhen G is right or obtuse, if (opp.) ^ (adj.), there is no solution but if (opp.) > (adj.), one solution. The practical method, however, in the case of any given problem is to construct the triangle approximately to scale. Having determined the number of solutions, the unknown parts can be computed by the law of sines. 70 PLANE TRIGONOMETRY [VII, § 50 Fig. 59. Example 1. Two sides are 12.56 and 10.54 and the angle opposite the latter is 64° 20'. Solve the triangle. Construct the angle G = 64° 20' and lay- off GA = 12.56 and draw AP. A glance at the tables (p. 34) shows that -sin G > .9, whence p > .9 x 12.56 > 11. Therefore, no solution. Example 2. In the triangle ABC^ a = 301.35, c = 352.11, and A = 33° 17'. Determine the remaining parts. Construct angle A = 33° 17', lay off AB = 352.11, and draw BP. Without any tables whatever, we know that sin 33° 17' < .7 and there- fore p < .7 X 360 < 260, and therefore there are two solutions. sin G 352.11 sin 33° 17' 301.35 log sin 33° 17' = 9.73940 - 10 log 352. 11 =2.54668 colog 301.35 = 7.52093 - 10 log sin = 9.80701-10 (7 = 39°53'.0 Fig 60. There are two angles less than 180° having a given sine ; therefore (7i = 39° 53' and C^ = 140° 7'. From this i)oint on we have to solve two distinct triangles, viz. : ABCi and ABC2. Call AGi, 61, and AC2, 62 ? angle ABC\^ Bi and angle ABG2 , ^2. Then B^ = 106° 50' and B^ = 6° 36'. 61 ^ sin 106° 50' 301.36 sin 33° 17' * log301.35 = 2.47907 log cos 16° 50' = 9.98098 - 10 colog sin 33° 17' = 0.26060 log 61 = 2.72065 61 = 525.59 sin 6° 36' 301.35 sin 33° 17' log 301.35= 2.47907 log sin 6° 36' = 9.06046 - 10 colog sin 33° 17' ±= 0.26060 log 62 = 1.80013 &2 = 63.114 Example 3. Two sides of a triangle are 5 and 7 and the angle oppo- site the latter is 120°. Solve the triangle. Construct the angle G = 120°, lay off GA = 5. It is at once obvious that a circle center at A, of radius 7, will cut G Q once and only once, at B. Let the student complete the solution, finding by the law of sines, angle B and the side GB. Ans. 38° 12'.8, 3. 1. a = 17.16, b = 14.15, 2. a = 54, 6 = 48.6, 3. u = 971, V = 1191, 4. I = 281, m = 152, 6. b = 13.12, c = 7.22, 6. P= 48, q = 36.1, 7. m = 10.08, n = 5.82, 8. i = 93.99, 8 = 91.97, 9. a = 309, b = 360, 10. k = 91.06, m = 77.04, 11. One diagor lal of a para VII, § 50] SOLUTION OF OBLIQUE TRIANGLES 71 EXERCISES XVIII. — CASE IV Solve each of the following triangles, using logarithms ; if two solu- tions exist, obtain both of them. 5 = 42^. Ans. 83°45'.7, 21.022 A = Sr 14'. Ans. 120° 56'.9, 89.314 Z7=51°15'. Ans. 55° 41'. 8, 1028.5 L = 103°. A71S. 45° 11'.6, 204.61 5 = 39° 54'. Ans. 20° 40'. 2, 17.814 Q = 45°50'. Ans. 61° 39'. 5, 44.293 if =21° 31'. Ans. 146° 15^4, 15.264 r=120°35'. Ans. 2° 1'.3, 3.85 ^ = 21°14'.4 Ans. 133°47'.7, 615.67 Ji = 51°9'.l Ans. 87°37'.9, 116.82 ilelogram is 68 ft. long and makes an angle of 30° 20' with the other diagonal ; one side is 22 ft. long. Find the length of the other side. Ans. 48.107 or 74.450 12. In a certain town the streets intersect at an angle of 82° 14'. It is desired to know the distance between two objects, A and 5, which lie on a line parallel to one set of streets and which are separated by a large building. A line AG = 200 ft. is measured along a side line parallel to the other set of streets, and CB = 222 ft. is then measured. De- termine AB. Ans. 127.09 13. The pilot of a ship S sees a lighthouse H on the shore ; by measm-ing the angle of elevation of the top of the lighthouse, and knowing its height, he de- termines that it is 8950 ft. from his ship. At the ship ^^^' "^* an angle of 2° 40' is subtended by a line connecting the lighthouse with a light L on the shore known to be 575 ft. from the lighthouse. Find the angle SLH and thus determine exactly the position of the sliip with reference to the shore. Practically, how may he tell which of the two possible solutions is actually correct ? Ans. 46° 24', 133° 36'. 14. Suppose a, 6, and A are given ; let x represent the third side. Apply the law of cosines to side a and determine under what conditions the resulting equation in x will have (1) no real root, (2) one positive real root, (3) two positive real roots. Consider separately the two cases when A is acute and when A is obtuse and compare results with the statements of § 50, p. 68. CHAPTER VIII AREAS —APPLICATIONS — PROBLEMS 51. Areas of Triangles. It is shown in plane geometry that the area A,"^ of a triangle is equal to one-half the product of any side and the altitude from the opposite vertex. (1) The area of a triangle is equal to one-half the product of the base and altitude. 52. Area from Two Sides and the Included Angle. If we have two sides and the included angle, a, b, and C, and drop a perpendicular upon one of the given sides, as p upon b, then p= asin C and by (1) A = ^ 6 (a sin C) ; whence (2) The area of a triangle is equal to one-half the product of any two sides into the sine of their included angle, 53. Area from Three Sides. If the three sides are given, draw lines from the vertices to the center of the inscribed circle dividing the tri- angle into three triangles having a common altitude, r. By (12), § 43, . ^ l (s-a)(8-&)(g-c) _ Fig. 65. ♦The area is denoted by the boldface type A in distinction from the angle A. 72 VIII, § 54] AREAS 73 The sum of the bases of the three triangles is a-|-6 + c = 2s. Therefore their combined area is, by (1), (3) A = rs = V = 260 ft., in a ^horizontal plane with the base A of the object, is measured, also the angle of eleva- tion ACB = 13° 22', and the angles DCA = 35° 37' and CDA = 64° 28'. Determine the height AB. Ans. 64.44 80 PLANE TRIGONOMETRY [VIII, § 56 15. A tall building stands at the foot of a hill. From a point on the side of the hill the angle of depression of the base of the building is ob- served to be 14° 36^ and the angle of elevation of the top is 21° 43'. A level line from the instrument meets the building 19 ft. 7 in. above the base. Find the height of the building. Ans. 49.62 16. A balloon is observed, at the moment it passes over a level road, from two points in the road an eighth of a mile apart. The angles of ele- vation from the two points are 33° 11' and 42° 6'. Find the distances of the balloon from the two observers. Arts. 374.31, 427.26 17. In surveying, it is sometimes desired to extend such a line as AB in the figure beyond an obstacle. If at J5 a right turn of 58°, BE = 126 ft., and at J5; a left turn of 110° are laid off, compute EC^ and the angle (right turn) at C. Ans. 135.6, 52°. 18. To find the distance PQ in Fig. 76, a base line AB is measured = 518 ft. At A the angles PAQ = 43° 18' and QAB = 48° 32' are measured and checked by measuring P^B = 91°50', and at 5, ^J5P = 38° 43', P5Q = 41°28', ABQ = SO"" 11'. Find PQ by two methods. Ans. 451.39 19. Find the distance A (7, Fig. 77, through a thicket, having measured AB = 20.71 rods, BG= 18.87 rods, angle Fig. 76. ABC = 6^° 12'. Ans. 18.40 Fig. 77. 20. From two points A and B, 300 ft. apart on the deck of a ship, a second ship, S, is observed. The angles ABS = 85° 18', and BAS = 83° 47' are measured. What is the distance between the ships ? Ans. 2496, 2502, av. 2499. 21. How far to the side of a target 1300 ft. away should a gunner aim from a ship going 15 mi. per hour, if the speed of the bullet is 2000 ft. per second and he fires when he is directly opposite ? Ans. 14.3 22. From a railway train going 50 mi. per hour a bullet is fired 1000 ft. per second at an angle of 75° 28'. 3 with the track ahead. Find its speed and direction. Ans. 71° 29'.2, 1020.9 ft. per second. 23. A man in a railway car going 45 mi. per hour observes the rain- drops falhng at an angle of 30° with the vertical. Assuming that the raindrops are actually falling vertically, find their speed. Ans. 77.9 VIII, § 56] MISCELLANEOUS PROBLEMS 81 24. The resultant of two forces is 10 lb. ; one of the forces is 8 lb. and makes an angle of 36° with the resultant. Find the magnitude of the other force. Ans. 6.88 25. A horse pulls a canal boat by a rope which makes an angle of 25° 36' with the tow path. What size of engine would propel the boat at the same speed ? (Assume that the horse is doing one '* horse power.") Ans. 0.9+ 26. A man climbs a hill inclined (on the average) 32° with the hori- zontal. His pocket barometer shows that at the end of 2 J hr. he has increased his elevation 2760 ft. Find his average speed up the slope. Ans. 2076.8 27. The sides of a triangular field are 82.7 rods, 91.4 rods, and 104.3 rods. Determine the area of the field and the angles between the sides. Ans. 226.39 A. , 49° 27^4, 67° 7^6, 73° 26^ 28. Find the area of a triangular piece of ground, having two angles, respectively, 73° 10' and 90° 60', and the side opposite the latter 160.6 rods. Ans. 18.7 A. 29. Find the areas of triangles which have the following given parts j (a) a = 116.082, 6 = 100, C=118°16'.7 (6)6 = 100, A = 76° 38'.2, C = 40° 6'. (c) w = 31.326, « = 13°67', i7=63°ll',3 (d) a = 408, 6 = 41, c = 401. (e) a =.9, 6 = 1.2, c = 1.6 Ans. (a) 6112.1 (6) 3606.8 (c) 136.13 (d) 8160 (e) .54 30. Three circles whose radii are 2, 3, 10, respectively, are tangent externally. Find the area of the triangle formed by joining their centers. Ans. 30. 31. Prove that the area of the triangle formed by joining the centers of any three circles which are tangent externally is a mean proportional between the sum and the product of their radii. See § 63. 32. Prove that one-half the product of the three sides of any triangle is equal to the product of its area into the diameter of its circimiscribed circle. See §§40 and 62. 33. Prove that the area of any triangle is equal to the product of the radii of its inscribed and circumscribed circles into the sum of the sines of its angles. See §§40 and 63. PART m. THE GENERAL ANGLE CHAPTER IX DIRECTED ANGLES — RADIAN MEASURE 57. Directed Lines and Segments. As explained in ele- mentary algebra, it is often convenient to select one direction on a straight line as the positive direction; the other is then called the negative direction. Thus, if two forces act along the same line, but in opposite directions, it is convenient to call one positive and the other negative. Two segments are said to have the same sense if they lie on the same line or on parallel lines, and if both are positive or both are negative. Two segments are said to be of opposite sense if they lie on the same line or on parallel lines, and if 4- Q Q Q one is positive and the other is negative. ^"■^"^I; " Thus, in Fig. 78, AB = EF, while ^ +--+— ^ ^ ^(7 = -G^^. and CB= - FQ, The numerical measure of a directed seg- ment is the number of units in its length with the sign + or — , according as the segment is positive or negative. 58. Rotation. Directed Angles. In describing rotation, it is convenient to regard angles as positive or negative in a manner analogous to that explained in § 57 for line-segments. An angle is thought of as generated by the rotation of one of its sides about the vertex as center; its first position is called the initial side, the final position is called the terminal side. An angle generated by a rotation opposite to the motion of the hands of a clock (counterclockwise) j is said to he positive; 82 IX, § 59] DIRECTED LINES AND ANGLES 83 an angle generated by a clockwise rotation, is said to be negative.* Angles may be of any magnitude, positive or negative. Thus, in Fig. 79, a, /3, 8 are positive angles ; y is negative ; fi is greater than a straight angle ; and 8 is greater than 360°, or a complete revolution. In rotating parts of ma- chinery, such angles have a very vivid meaning. Thus, a wheel which rotates 370° per second has a very different speed from that of a wheel which rotates 10** per second. yig. 79. 59. Placing Angles on Rectangular Axes. To place any given angle on a pair of rectangular axes in the plane of the angle, put the vertex at the origin and the initial side on the a>-axis extending to the right ; the terminal side will then fall in one of the four quadrants (or, if the angle is a multiple of a right angle, on one of the axes). If the terminal side falls in the first quad- rant, the angle is said to be an angle in the first quadrant, etc. In Fig. 80, a is a positive angle in the first quadrant, ^ is a negative angle in the fourth quadrant, 8 is a positive angle in the fourth quadrant. Quad. II f /Quad. (\ Act \ \ Quad.V. A Q^ad. A" Fig. 80. EXERCISES XXII. — DIRECTED LINES AND ANGLES 1. What angle will the minute hand of a clock generate in 2 hr. 24 min. 10 sec. ? 2. A flywheel is running steadily at the rate of 450 revolutions per minute. What angle does one of its spokes generate in 2 sec? In 1.2 sec? * Either of these directions may of course be chosen as the positive direc- tion of rotation, the other is then the negative direction. The choice here made is the customary one for angles ; but in many kinds of machinery, the other sense of rotation is considered positive, as in the case of a clock. 84 PLANE TRIGONOMETRY [IX, § 59 3. Find the sum, or resultant, of two forces that act in the same line whose intensities (measured in pounds) are — 6 and + 10, respectively. Draw a figure to represent the solution. 4. If three forces of intensities +7, — 16, -f 2 (lb.), respectively, act on a body in the same line, find the resultant force. Draw a figure. 5. If a man walks with a speed of 4 mi. per hour toward the rear of a train going 35 mi. per hour, find his actual speed. Draw a figui-e. 6. A man's gains and losses (indicated by — ) in business in succes- sive months are |260, — $118, |35, |712, — |15. Find the total gain and the average gain per month. Draw a figure. 7. By means of a ruler and a protractor, construct the following angles and their sums ; check by adding their numerical measures. (a) _ 76° and 125°. (6) 66"" and - 30°. (c) 45° and 30°, and 70° (d) - 60° and - 36°. (e) 485° and 55°. (/) - 750° and 30°. 8. With some two of the angles just given verify a -f j3 = ^ + a. 9. (a) Construct 27° + 85° + ( - 45°) + 135°. (6) Construct - 150° + 96° + 24° + (- 80°). 10. If a wheel is rotating 120° per second, how many revolutions does it make per minute ? how many per hour ? How many degrees does it turn through per minute ? 11. Express an angular speed of 2.5 revolutions per second in degrees per second ; in revolutions per minute ; in degrees per minute. 12. A flywheel rotates at the rate of 40 revolutions per minute. Through what angle does one of its spokes turn in a second ? 13. Eeduce an angular speed of 3.4 revolutions per second to degrees per second ; to degrees per minute ; to revolutions per minute. 14. Find the angular speed of the rotation of the earth on its axis (a) in revolutions per minute ; (6) in degrees per second. 15. Construct a right triangle whose sides are 3 and 4 ; construct an angle which is 3 times the smaller angle of this triangle. 16. Construct the following angles and place them on the axes, (a) - 150° ; (6) 285° ; (c) 480° ; (d) 670° ; (e) - 225° ; (/) - 450°. 17. In what quadrant is each of the following angles : 459°, 682°, 725°, - 100°, - 1090°, ± 85°, ± 95°, ± 175°, ± 185°, di 265°, ± 276°, ± 355° ? 18. Taking a = 60°, i3=-300°, 7=-50^ 5 =r 310° draw a figure showing that a differs from /3, and also that y differs from 5 by 360°. 19. Find the angle between 0° and 360° which differs from each of the following angles by a multiple of 360° : (a) -42° 13'; (b) - 842° j (c) 364° 23'; (d) 2700°. EX, §621 RADIAN MEASURE 85 60. Measurement of Angles. An angle may be named and used belore it is expressed in any system of measurement. Thus, we may refer to an angle ^ of a right triangle whose perpendicular sides are 16 in. and 24 in., respectively ; and we can compute tan A = 24/16 = 1.5, etc., without measuring A in terms of any unit angle. General theorems like the law of sines remain true in any system of measurement. The unit angle (see § 2) chiefly used in Geometry and Trigonometry is the degree with its subdivisions minute, tenth of minute, second, with which the student is familiar. It is often convenient to use another unit angle called the radian, 61. Radian Measure of Angles. =^ A radian is a positive angle such that when its vertex is placed at the center of a circle, the intercepted arc is equal in length to the radius. This unit is thus a little less than one of the angles of an equilateral triangle ; in fact it follows from the geometry of the circle, since the length of a semicircum- ference is rrr, that (1) TT rac/zans = 180°, where 17 = 3.14159, F^^i. whence 1 radian = 57° 17' 44".806, or 57°.3 approximately. It is easy to change from degrees to radians and vice versa by means of relation (1), which should be remembered. Con- version tables for this purpose are printed in Tables, pp. 91-93. 62. Use of Radian Measure. It is shown in geometry that two angles at the, center of a circle are to each as their inter- cepted arcs ; therefore if an angle at the center is measured in radians and if the radius and the intercepted arc are measured in terms of the same linear unit, their numerical measures satisfy the simple relation : (2) arc = angle x radius. * Sometimes also called circular measure. 86 PLANE TRIGONOMETRY [IX, §64 In other words, the number of linear units in the arc is equal to the product of the number of radians in the angle by the number of linear units in the radius. Example 1 . Find the difference in latitude of two places on the same meridian 200 mi. apart, taking the radius of the earth as 4000 mi. Angle = arc/radius = 1/20 in radians = 2° 51' 63", approximately. 63. Angular Speed. In a rotating body a point P, which is at a distance r from the axis of rotation, moves through a dis- tance 2 7rr during each revolution or through a distance r while the body turns through an angle of one radian. Therefore if v is the linear (actual) speed of P (in linear units per time unit, e,g, feet per second), and if w is the angular speed of the ro- tating body (in radians per time unit, e,g, radians per second), then their numerical measures satisfy the relation (3) i; = r . CD ; hence the angular speed of a rotating body is numerically equal to the actual speed of a point one unit from the axis of rotation. Engineers usually express the angular speed of the rotating parts of machinery in revolutions per minute (R. P. M.) or revolutions per second (R. P. S.). These are easily reduced to radians per minute (or per second) by remembering that one revolution equals 2 tt radians. Example 1. A flywheel of radius 2 ft. rotates at an angular speed of 2.5 R. P. S. Find the hnear speed of a point on tlie rim. In radians per second, w = 2.6x27r = 5 7r, and for a point 2 ft. from the axis of rotation v ~2 x ^ir = 31.416 ft. per second. Example 2. Find the angular speed of a 34-inch wheel on an auto- mobile going 20 mi. per hour. Every time the wheel turns through a radian the car goes forward 17 in. (the length of the radius), and 20 mi. per hour = 362 in. per second ; therefore the wheel turns through 352/17 = 20.7, radians per second. 64. Notation. In measuring angles in radian measure we shall adopt the practice universal in advanced work and write only the numerical measure of the angle in terms of the unit IX, § 64] RADIAN MEASURE 87 one radian. Thus in the expression tan x, the letter x will de- note a number (the numerical measure of an angle) rather than the angle itself. See § 2. When necessary, to call attention to the fact that radian measure is intended, the symbol (^''^) is appended to the nu- merical measure, thus : 1(^> = 1 radian = 57° 17' 44".8, 2<'"> = 2 radians = 114° 35' 29".6, TT^'^^ = TT radians = 180° = 2 rt. A, (7r/2)('> = 7r/2 radians = 90° = 1 rt. Z, and so forth. As it happens that the acute angles whose trigonometric functions are most easily recalled without consulting tables are simple fractional parts of 180°, the number tt often appears as a factor of the numerical measure of angles. In this system, for example, sin (7r/2) = 1, cos (tt/S) = 1/2, tan (7r/4) = 1, etc. The use of pure numbers, such as 2 or tt in place of an angle is pre- cisely similar to the use of 10 for 10 feet or 10 inches in expressing lengths. The student should supply the unit of measurement (radians or feet or inches), and should not confuse the number tt (= 3.14159 •••) with the angle whose measure is tt radians^ as he should not confuse the number 10 with the distance 10 feet. EXERCISES XXIII. — ANGULAR SPEED — RADIAN MEASURE 1. Express the following angles in degrees, minutes, and seconds : (a) 7r(^)/4; (6) 7rW/6 ; (c) 2 7r(»-)/3; (d) 3K 2. Express the following angles in radians : (a) 25°; (&) 30° ; (c) 35° ; ((2) 28° 39'; (e) 114° 36'. 3. How far short of one revolution is 6(»") ? 4. To gain ability to judge the size of angles in circular measure, express approximately (to within 1°) angles whose sizes are l^*"), 4('"), 5(»">, S^*"). Draw an angle which is about your impression of an angle of 2(*">, and measure it with a protractor. Do not revise your figures. 5. If a vehicle moves at the rate of 15 ft. per second, through what angle does one of its wheels, 3 ft. in diameter, revolve in 1 sec. ? Ans. IOC'). 88 PLANE TRIGONOMETRY [IX, § 64 6. If the linear speed of a vehicle is 30 mi. per hour, what is the angular speed of one of its wheels which is 4 ft. in diameter ? Ans. 22 radians per second. 7. A wheel 6 ft. in diameter is connected by a belt 40 ft. in length with a wheel 4 ft. in diameter. If the large wheel makes 30 revolutions per minute, how often does the seam of the belt pass this wheel ? What is the angular speed of the smaller wheel? Ans. 6y^ sec, 3}| radians per second. 8. Find the angular distance on the earth between two points whose distance from each other, on the arc of a great circle, is 800 miles. [Take the radius of the earth to be 4000 miles. ] Ans. 11° 21' 33". 9. Find the distance in miles between two points on the earth's sur- face whose angular distance is 1° ; between two points whose angular distance is 0.25 radians. Ans. 69.81,1000. 10. Find the length of the subtended arc of an angle of 3.46 radians at the center of a circle of radius 5. Ans. 17.3 11. Find the length of the subtended arc of an angle of 55° at the center of a circle of radius 3. Ans. 2.8798 12. Find the angle at the center which subtends an arc of 3 ft. on a circle of radius 4 ft. Express the angle in radians and in degrees, and compare the work done in the two cases. Ans. | radian = 42^.97+ 13. Reduce to radian measure by means of Tables IV, p. 91 : (a) 23° 40' ; (6) 68° 45' 20" ; (c) 138° 35' 15". Ans. 0.4130612, 1.2000109, 2.4188082 14. Reduce to degree measure by means of the Tables pp. 92-93 : (a) 3.46W ; (b) .256(»-) ; (c) .0127(^) ; (d) 8.240-). Ans. 198° 14' 36".2, 14° 40' 3".8, 43' 39".5, 472° 7' 2" 15. Reduce the following angular speeds to degrees per second ; to revolutions per second ; to revolutions per minute : (a) 4.5(*"> per sec. ; (6) 2.48^^) per sec. ; (c) 10.54(*"> per sec. Ans. (a) 257.83, 0.7162, 42.972 ; (6) 142.09, 0.3947, 23.682; (c) 603.90, 1.6775, 100.66 CHAPTER X FUNCTIONS OF ANY ANGLE 65. Resolution of Forces. Projections. In § 26, p. 34, we saw how to find the components of a force, or a velocity, on any line, as the projection of the force on that line ; and we saw that the components of y f:(x.y) a force F on each of two per- pendicular axes, even when the angle a is obtuse, are (1) F^ = 'PY0J^F=FG08a, Fy = Pro j ^ F = i^ sin a. Fig. 82. If several forces occur in the same problem, some of them may make an angle a greater than 180° with the positive direction OX. It is convenient to define cos a and sin a for angles greater than 180° so that the equations (1) remain true. If we do so, the projection on the two axes of any directed segment of length r joining the origin to a point P are (2) X = Proj^ r = r cos a, y = Proj^ r = r sin a, where a is the angle between the positive direction OX and the positive direction OP, and may be an angle of any size, posi- tive or negative. Hence the desired definitions are : (3) cosa = -, sma = -- These definitions are consistent with those already given, §§ 11, 35, for the sine and the cosine ; i.e. in case 0°^ a ^ 180°, they determine the same values as the earlier definitions. 66. General Definitions. Trigonometric Functions of Any Angle. The definitions of sin a and cos a given in § 65 have, 90 PLANE TRIGONOMETRY P^,§66 of course, no necessary dependence upon forces. Each is a number which depends only on the magnitude and sign of the angle. A purely geometric definition of these and of the other trigonometric functions of any angle a, consistent with the definitions of §§ 11, 35, and with the fundamental relations between them, such as tan a = sin a/cos a, sin^ a -\- cos2 oc = 1, the reciprocal re- lations, etc., may be made as follows : Place the given angle on a pair of rectangular axes, and select any ^^' ' point P whose coordinates are {x, y) terminal side at a distance r > from the origin. P (x.y) P (x.y) on the Then (4) (5) (6) (J) (8) (9) sina = cos a = - = r y __ ordinate r "" radius ' abscissa radius ' y ordinate tan a = - = -^ — ; — X abscissa X abscissa ctn a = - = — -T-. — -r~ > y ordinate r radius sec a = - = -^j — -. — , X abscissa CSC a = - = radius y ordinate ' provided x4^()\* provided ?/ =5^ ; provided a; ^^fc ; provided y ^0. Three additional functions sometimes used are : (10) The versed sine of a : vers oc = 1 (11) (12) and also the coversed sine of a = 1 — sin a. cos a. The haver sine of a : hav a = ^(l — cos a). The external secant of a : exsec a = sec a - * The exceptions noted are based on the general principle that a fractional expression does not represent a number if its denominator is zero. X, § 67] FUNCTIONS OF ANY ANGLE 91 By these definitions every angle has a sine and a cosine, be- cause in the ratios yjr and xjr the denominator r is never zero. There is no secant or tangent ^ for 90°, or for 270°, or for any angle whose terminal side coincides with either the positive or negative end of the ^/-axis, because the denominator x in the ratios r/o;, ylx^ is zero. Similarly, there is no cosecant or co- tangent for 0° or for 180°, or for any angle whose terminal side coincides with the positive or negative end of the ic-axis. There exists a tangent, cotangent, secant, and cosecant for every angle except those just mentioned. If two angles differ by any multiple of 360° it is evident that any one of the trigonometric functions will have the same value for both of them because the initial sides of the two angles (when placed on the axes) will coincide, and also their terminal sides. It follows that for a point P on the common terminal side the values of a?, ?/, and r are the same for both angles ; hence the ratio which defines any given function will be the same for both angles. For example : sin (- 295°) = sin ^h"", cos (- 315°) = cos 45°, tan 1476°= tan 36°, sin ((9 - 180°) = sin (180°+ ^), cos (x - 90°) = cos (270° + x), tan (360° - y)= tan (- y). 67. Algebraic Signs of Trigonometric Functions. The sine of any angle in the first or second quadrant is positive, because the ordinate of any point above the aj-axis is positive ; the sine of any angle in the third or fourth quadrant is negative, be- cause the ordinate of any point below the a;-axis is negative. The cosine of any angle in the first or fourth quadrant is positive, because the abscissa of any point to the right of the * To say that 90° has no tangent does not mean that the tangent of 90° is zero. When we say that an article has no value we mean that it has a value and that value is zero. Not so here. Since the general definition of tangent 4oes not apply to 90°, we could, if we found it convenient, define tan 90°, but we do not ; we leave it undefined. Often it is said tan 90° = oo , but this does not mean that 90° has a tangent ; it means that as an angle a increases from 0° to 90°, tan a increases without limit, and that before a reaches 9(P. 92 PLANE TRIGONOMETRY [X,§67 2/-axis is positive ; similarly, the cosine of any angle in the second or third quadrant is negative. Similarly, the signs of tan a, ctn a, sec a, esc a, etc., may be determined directly from a figure; they are as follows : Quadrant sin a cos a tana etna I sec a CSC a 1st + + + + + + 2d + - - - - + 3(1 - - + + - - 4th - + - - + - Note. (1) tana is positive (negative) v^hen sin a and cos a have hke (unhke) signs ; (2) reciprocals have the same sign. EXERCISES XXIV. — FUNCTIONS OF THE GENERAL ANGLE 1. By placing the angles on the axes, show from the definitions that ( a) sin 225° = - V2/2, cos 225° = - \/2/2. ( 6 ) sin 150° = 1/2, cos 150° = - \/3/2. (c ) sin 330° = - l/2,_cos 330° = V3/2. (d) sin (- 315°) = V2/2, cos (- 315°) = V2/2. ( e) sin (- 1020°) = V3/2, cos (- 1020°) = 1/2. (/) sin 180° = 0, sin (n • 180°) = ; for n = ± 1, ± 2, ± 3, .... . {g) cos 90° = 0, cos[(2n- 1)90°]= 0; forn=± 1, ±2, ±3, .... 2. Which of the following are positive and which negative ? sin 72°, sin 352°, sin 850°, tan 128°, sec 260°, sin (-20°), cos (-380°), sin (-260°),. cos 160°, ctn 280°, cos 33°, csc91°, cos (- 40°), tan (- 140°), cos(-400°). 3. Prove for any angle a that sin2 a -f cos2 a = 1. [Use x"^ + y^ = r2.] Prove each of the other P3rthagorean relations for any angle a : 1 + tan2 a = sec2 a, if cos cc :^ ; 1 -|- ctn2 a = csc2 a, if sin a =^ 0. 4. Prove that ctn a, sec a, esc a are the reciprocals of tan a, cos a, sin a, respectively, for all values of a for which both are defined. 6. (a) Prove that the sine of any angle in the first or second quad- . rant is between and 1. (6) Prove that the cosine of any angle in the 1st or 4th quadrant is between and 1. 6. Prove that if an angle is not an odd multiple of a right angle its sine is between — 1 and + 1 ; and conversely. For what angles is sin a = + 1 ; sin a = — 1 ; cos a = + 1 ? X, §68] FUNCTIONS OF ANY ANGLE 93 7. Show that tan a = sin cc/cos a for all values of a, if cos a ^ 0. 8. Show that tan a and ctn a may have any values whatever, 9. Show that vers a and hav a are always positive or zero, 10. If an angle a starts at 0^ and gradually increases to 360°, show that the behavior of sin a and cos a will be as indicated in this table : a 0° 0°<-<90° 90° 90°-axis above and below. The angles XOH, XOK; XOL, XOM) XON.XOS', and XOT, are the angles mentioned above, and XOP is the angle 0. Denote the coordinates of the point P by (a, h) ; then because the triangle OAH is congruent to the triangle OOP the coordinates of the point H are (6, a), and in the same way the coordinates of the points K, i, M, N, S, T are easily seen to be as indicated in the figure. We are now able to read off the values of the trigonometric func- tions of the various angles from the figure, in terms of a, 6, and r ; thus sin = 6/r, cos (90° -0)== h/r, cos (90° + 6>) = - h/r, sin (180° -0)= h/r, sin (180° + (9) = - h/r, cos (270° - ^) = - 6/r, cos (270° + (9) = 6/r, sin (360° - ^)= sin (- 6>)= - h/r. Hence we have cos (90° - ^) = sin cos (90° + ^) = - sin ^ sin (180°- 0) = sin 6 sin (180° + ^) = - sin ^ cos (270° -f ^) = sin cos (270° - /) = - sin sin (360° - ^) = sin (- ^) = - sin 6 X, § 69] FUNCTIONS OF ANY ANGLE 95 Similarly we obtain from the figure cos = a/r, sin (90° -6)= a/r, sin (90° + ^) = a/r, cos (180°- 6>) = - a/r, cos (180° + ^) = - a/r, sin (270° -6) = - a/r, sin (270° + (9) = - a/r, cos (360° - ^)= cos (- e)=a/r. Whence, sin (90° -0)= cos (9 cos (180° - ^) = - cos ^ sin (90° + 6)= cos 6 cos (180° + ^) = - cos ^ cos (860° - ^) = cos ( - 6) sin (270° _ ^) = _ cos d = cos ^ sin (270° + (9) = - cos ^ These formulas together with the fact mentioned in § 66, that a function of an angle a has the same value as the same function of any angle that differs from cc by a multiple of 360°, are sufficient to enable one to find the value of any one of the functions of any angle from the tables.* Example 1. Find the sine of 793° 22'. The' angle 73° 22' differs from the given angle by 720°, which is a multiple of 360° ; hence the required value is the same as sin 73° 22'. From the tables this value is found to be .95816 69. Solution of Trigonometric Equations. We are now able to give the general solutions of the equations sin = c and cos = G where c is any number lying between + 1 and — 1. In the first place, it is clear that there are two and only two angles between 0° and 360° which will satisfy either of these equations. For in the above figure there are only two points of the circle for which x has a given value between + r and — r, and likewise, only two points for which y has a given value between + r and — r ; a radius drawn to either of these two points will be the terminal side of an angle between 0° and 360°, satisfying the first equation if y is chosen so that y/r = c and satisfying the second if x is chosen so that x/r = c. To obtain the general solution we add or subtract any whole * The proofs given above are for the case in which is an acute positive angle. The formulas, however, are true for any value of whatever. 96 PLANE TRIGONOMETRY [X, § 70 multiple of 360° to either of the solutions just found. The solutions which lie between 0° and 360° can be found from the tables by means of the formulas given above. 70. Illustrative Examples on Composition and Resolution of Forces. Example 1. Find the components, Rj.t -Ry » of the resultant of two forces, the first of 12 lb. acting at an angle of 30° with the horizontal, the second of 20 lb. acting at an angle 60° with the horizontal. Solution. To solve this we make use of the principle that the projec- tion on any line of the resultant of any number of forces is the algebraic sum of the projections of the component forces. By equations (1), § 65, the horizontal component of the first is 12 cos 30°, and of the second, 20 cos 60° : hence B^ = 12 cos 30° 4- 20 cos 60° = 10.392 + 10.000 = 20.392 In a similar manner we find By = 12 sin 30° + 20 sin 60° = 6.000 + 17.320 = 23.320 We can easily find the magnitude of the resultant from the equation B2 = E2^ + B\ = (20.392)2 + (23.320)2 = 959.665 Hence 12 =\/(959.665)= 30.979 The direction of the resultant is given by the equation tan^ = By-^ B^ = 23.320 -- 20.392 = 1.1436 Hence ^ = 48°50^ Example 2. Find the magnitude and the direction of the resultant of the two forces F = (17, 128°), G =(24, 213°). [Note. The notation (24, 213°) means a force of magnitude 24 acting at an angle of 213° with the positive ic-axis.] The method of solution is the same as in Example 1 ; we find F^ = 17 cos 128° = - 17 sin 38° (by § 68). G^ = 24 cos 213° = - 24 cos 33° (by § 68) . TTpticp i?, = - 17 sin 38° - 24 cos 33° = - 10.466 - 20. 128 = - 30.594 Similarly we obtain By = 1 7 cos 38° -24 sin 33° = 13.396 - 13.071 = .325 B = V(B\ + B\)=S0.6n = arctan (.325/- 30.594) = arctan (- .01062) = 180°- 36^4 = 179° 23'.6 X, § 70] FUNCTIONS OF ANY ANGLE 97 EXERCISES XXV. — READING OF TABLES — REDUCTION TO FUNCTIONS OF ACUTE ANGLES 1. Express the following as functions of acute angles not greater than 45°. Make use of congruent angles whenever advantageous : (a) sin 150° 21'. (6) cos 125° 15'. ( c ) tan 283° 45'. (d) ctn(--36°16'). (e) sec460^ (/) esc (- 210° 20'). (g) sin(-943''24'). (/t) cos55P23'. (i) tan (- 546° 28'). 2. From the tables find the values of the following logarithms-, (a) log (- cos 161° 11'). (6) log sin 161° 11'. (c) log (-sin 217° 17'). (d) log (- cos 252° 480- [Note that the numbers in parentheses in (a), (c), and (d) are posi- tive ; if the minus sign were absent, each of them would be negative. Negative numbers have no real logarithms. ] 3. Compute the values of the following expressions by logarithms : (a) 2.35 sin 148° 23'. (6) 24.8 cos 160° 40'. (c) 16.2 cos 320° 45'. 4. Solve the following trigonometric equations : (a) cos2 1 — sin2 t = smt. Solution. In this equation cos2 t may be replaced by its equal 1 — sin2 i ; the equation then becomes a quadratic in sini, viz.: 2sm'^t + sint— 1 = 0. This equation is equivalent to the given one; i.e. every solution of either is a solution of the other. The solutions may now be found by factoring : (2sini- l)(sini4- 1) = 0. Hence we have either sin i + 1 = 0, whence sin t = — 1, and t = 270° or t = 270° + k 360°; or else 2 sin i = 1, whence sin i =1/2 and f = 30° + k 360° or t = 150° + k 360°. There are no other solutions. (6)2 sin2 X — cos X = 1. (g) sec2 x + tan x = S. (c) cos2x=sin2x. (^) 4 sec2 x + tan x = 7. (d) cos2x + 5sinx= 3. (i) tan x -f- ctn x = 2. (e) cos 2 X — sin X = 1/2. (j) sinx + 3 = cscx. (/) 5 sin X 4- 2 cos^x = 5. (k) sin2 x cos x = sinx. 5. Find the resultant (i?, 6) of three forces (100, 350°), (150, 490°), (200, 720°), where (F, a) indicates a force of magnitude F and direc- tion a. 6. Find the components on the axes of a force of magnitude 5.74 lb. which makes an angle of 215° 20' with the positive end of the x-axis. 7. Find the magnitude and the direction of a force whose components on two perpendicular axes are F^ = 25.46, Fy = 38.72 CHAPTER XI THE ADDITION FORMULAS 71. The Addition Formtilas. In the reduction of certair trigonometric expressions to simpler or more convenient forms it is sometimes desirable to express a trigonometric functioi of the sum or difference of two angles in terms of functions o: the separate angles forming the sum or difference. Withou reflection the student might think that sin (a -f p) would b( equal to sin a + sin )8 by analogy with the formula |^(a + 6 = i a + 1- 6, but a trial of one or two special cases will sho^ this is not always true; thus, sin (60° + 30°) is equal to one but sin 60° + sin 30° is equal to ^V3 -f- \^ which is greater thai one. In order to find the correct formulas for sin (a -f y8) anc cos {a + p) we make use of the theory of directed quantities as explained in §§ 26, 65, 57, 58, and Qf5, Suppose a force of magnitude A makes an angle a with th( positive a>axis, while another force of magnitude B makes ai angle a + 90° with this axis ; then the resultant R oi A anc B is represented by the diagonal OP of the rectangle of whicl A and B are two sides. The ^/-component, R^ of this re sultant is (1) ^^ = ^sina + 5sin(a + 90°) = ^ sin a 4- J5 cos a. Similarly, the a>component of R is (2) jR, = ^ cos a -f- -S cos (a -f 90°) = A cos a — B sin a. Now by § 65, Fig. 85. (3) i?, = i?cos(a4-iS), R^ = Rsm(a-\-p), where /3 is the angle between A and the resultant R. XI, § 73] ADDITION FORMULAS 99 Inserting these values in formulas (1) and (2) we find (4) R sin (a + yS) = A sin a -h J5 cos a. (5) R cos {a -\- P)= A ao^ a — B sin a. Moreover, from the hgure, -4 = jK cos ^, B = R sin fi. Substituting these values in (4) and (5) and dividing through by R we finally obtain the formulas (6) sin(a + P)= sin a cos p + cos a sin p. (7) cos (a + P)= cos a cos p — sin a sin p. It should be carefully noticed that, although in the figure the angles a and fi are acute angles, the proof does not at all depend on this fact. Formulas (6) and (7) are therefore true for all values of the angles a and p. 72. The Subtraction Formulas. It can be shown in a man- ner exactly similar to the preceding that we have also (8) sin (a — p)= sin a cos p — cqs a sin p. (9) cos (a — p)= cos a cos p + sin a sin p. It is easy to derive (8) and (9) directly from (6) and (7), how- ever. Thus, if, in (6), we replace /S by — ^ we obtain sin (a— P) = sin a cos (— /8)-f cos a sin (— )8), or, since by § 68, cos (— /3)=cos /3 and sin ( — )8) = — sin )3, sin (« — /?)= sin a cos p — cos a sin )3, which is (8). We prove (9) in a similar manner from (7). These formulas are also true for all values of the angles a and /?. They are examples of trigonometric identities involv- ing two angles. 73. Reduction of A cos o,±B sin a. Such expressions as A cos a ± ^ sin a which appeared in formulas (1) and (2) of the previous article arise in various connections ; for example, a combination of two vibrations gives rise to such a form. It is possible, and often convenient, to reduce such expres- sions to the product of a single numl>er, and the sine (or the 100 PLANE TRIGONOMETRY \X1, § 71 cosine) of the sum of two angles. The method depends oi formulas (6) and (7) and upon the fact that any two number! are proportional to the sine and the cosine of some angle. Example 1. Express 3 cos a + 4 sin a in the form k sin (a + j3). To solve this we first find an angle whose sine and cosine are propor tional to 3 and 4. We may clearly choose an angle /3 so that sin )3 = | and cos jS = |; hence we may write 3 cos Of + 4 sin a = 5(f cos a + | sin a) = 5(sin j8 cos a + cos /3 sin a). Hence by formula (6) we have 3 cos a + 4 sin a = 5 sin (/3 4- a). From the tables ^ = 36° 62'. EXERCISES XXVI. — ADDITION FORMULAS 1. Given sin a = 3/5, sin ^ = 5/13 ; find sin (a -^ ^). (a) When a and /3 are both acute ; (6) when a and jS are both obtuse 2. Find sin (45° + x), cos (45° + x), sin (30° + x), cos (30° + x) ii terms of sin x and cos x. 3. Given that x and y are both obtuse angles and that sin x = 1/2 sin 2/ = 1/ 3 ; find sin (x + y) and cos (x -\-y). 4. Use the addition formulas to express sin (90°+ a) and cos (90°+ a in terms of sin a and cos a. 5. Prove that sin (60° + x) — cos (30° + x) = sin x. 6. Express sin (ot + /3 + ^) in terms of sines and cosines of a, /9 and 0. [Hint. Let = oc + /3 and obtain sin ( + 6); then replace by it value, a + /3. ] 7. Express cos (a + /8 + ^) in terms of sines and cosines of a, /9 and e, 8. Reduce the combination of two simple harmonic motions 5 cos t - 12 sin t to the form r cos (t -\- 0). 9. Keduce 3 sin i + 4 cos t to the form rsin (t -\- 0). 10. Reduce each of the following to the product of a number and th sine or the cosine of a single angle : (a) sinx — 2cosx. ( e) \/3 cos x — sin x, (6) 3 cos 2/ — 4 sin y. (/ ) sin ?/ + .5 cos y. (c) 5 cos ^ + 12 sin 0, ( g) .7 cos — sin 0. (d) 3 sin i — 3 cos t. (h) .55667 sin c + 5 cos c. XI, § 75] ADDITION FORMULAS 101 11. Given two forces of intensities 2 and 3 that make angles of 30® and 120°, respectively, with the positive x-axLs ; find the horizontal and the vertical components of their resultant without finding the resultant itself ; find the same quantities by using the resultant. 12. Given .66 sin c + .5 cos c = — .34, find an angle ^, and a number r, such that .56 sin c + .6 cos c = r sin (c + ^), by means of § 70. Then, from r sin (c + ^) = — .34, find sin (c + ^), and therefore (from the Tables) find c -\- $. Hence find c. 74. Double Angles. Since formulas (6) and (7), § 71, are true for all angles, they hold when a = a, any angle whatever, and /3 = a, the same angle ; hence, sin (^a-\-a)= sin a cos a -\- cos a sin a, and cos (a -\- a)= cos a cos a — sin a sin a. Therefore the following formulas hold for any angle what- ever : (10) sin 2 a = 2 sin a cos a ; (11) cos 2 a = cos2 a — sin^ a ; or, since sin^ a + cos^ a = 1, (12) cos 2 a = 1 — 2 sin^ a = 2 cos2 a — 1. 75. Tangent of a Sum or of a Difference. Since formulas (6) and (7) hold for all values of a and ^, the formula sin (a + )S) __ sin a cos fi + cos a si n fi cos (a + )8) ~" cos a cos fi—aina sin fi holds good for all values of a and ^ except those which make cos (a-j-/S)=0, i,e. except when a-}- ft = 90°, or 270°, or an angle that differs from one of these by an integral number of times 360°. For example, it does not hold for a = 47**, P = 43^ Dividing both numerator and denominator by cos a cos p, we obtain the formula (13) tan(a+P)= tana-ftanP ^ ^ V Try i_tanatanp which holds for all angles a and ^ such that a, )S, and « 4- ^ have tangents. 102 PLANE TRIGONOMETRY [XI, § 75 Similarly from formulas (8) and (9), we obtain (14) tan(a-P)=i^?^LIl^, which holds for all angles a and j8 such that a, p, and a — p have tangents. From formulas (10) and (11) we find ,^^. X ft 2 tan a (15) tan 2a = , ^ ^ l-tan2a which holds for every angle a such that a and 2 a have tan- gents. The same formula ma}^ be obtained directly from (13) by putting a in place of p. 76. Applications. The formulas of this chapter are fre- quently used for reducing expressions whose values are to be calculated, to a form in which logarithms may be used. Example. Suppose the height of an object CD is to be determined and that it is not convenient to measure a base hne bearing directly toward the base G. The following method is then sometimes employed. The angle of elevation a is measured from some convenient point A\ a line AB = d is then measured at right angles to the line A G ; finally the angle of elevation, j3, is ob- served from B. The height h can then be de- termined by solving a succession of triangles. With the aid of the formulas of this chapter it Fig. 86. ^^B i^ frequently possible in such cases to reduce the calculation to a single logarithmic computation. In the case just mentioned we have BG = hctn^ AG = h ctn a, d2 = BG^ _ Jc^ = 7^2 (ctn2 ^ - ctn2 a) = h^ (ctn )3 — ctn a) (ctn /3 + ctn a) __ , 2 (sin a cos j3 — cos a sin /3) (sin a cos j8 + cos a sin /3) ^ ~ sin2 a sin2 /3 ' hence, using formulas (6) and (8), we have , _■ dsinasiujS Vsin (a — /3) sin (a+^1 Let the student show, by opening a book and studying the dihedral angle formed by two leaves, that a > )3. XI, § 76] ADDITION FORMULAS 103 EXERCISES XXVn. — SECONDARY FORMULAS — APPLICATIONS 1. Find sin 15°, cos 15°, tan 15° from the known values of sin 30^, cos 30°, tan 30°, and sin 45^, cos 45°, tan 45°. [Hint. 15° = 46° — 30°.] 2. Find tan 75°, tan 105°, sin 165°, cos 255°. [Hint. 7 5°= 45'= -|- 30°.] 3. Given sin36°52' = .0 ; find the sine, cosine, and tangent of 66° 52'; find sin 73° 44'. 4. Given tan 26° 34' = .5 ; find sine, cosine, tangent of 71° 34'; find tan 53° 8'. 6. Given sin a = 5/13 and 90° < a < 180° ; cos /3 = 8/17 and 0° < /3 < 90°; find sin (a — /3) , cos (a — /3), tan (a + /3), sin 2 a, cos 2 /3. 6. Given tan a = 15/8 and 0° < a < 90° ; cos /3 = 4/5 and 270° < /S < 360°; find sin (a - )3) , cos (/3 — a) , tan 2 a, cos 2 j8. 7. Given sin ex. = 1/3 and 90° < a < 180^ ; find sin (135° - «) and tan 2 a. 8. The angular elevation of an object from an upper window is ob- served to be a. The angular elevation from a point on the ground h feet directly beneath the window is /3. Show that the height of the object is h sin /3 cos a -T- sin (/3 — a). 9. To determine the difference in elevation of two stations, a flagstaff of known height h is held at the upper of two stations and the angles of elevation of its top and bottom are observed to be a and j8, respectively. Show that the difference in elevation of the two stations is h tan /3 — (tan a — tan j8) ; reduce this expression to a form convenient for logarithmic computation. 10. A tree leans directly toward two points of observation distant a and &, respectively, from its foot. The angles of elevation of the top of the tree from these two points are a and /3. Show that the perpendicular height of the tree is (6 — a) -r- (cot /S — cot a) ; reduce this expression to a form suitable for logarithmic computation. 11. Prove that sin Sa = sin a (3 - 4 sin^ «) = sin a (4 cos2 a - 1), and state for what values of a it holds. Use formuUis (6) and (7). 12. Prove that cos 3 a = cos a (4 cos2 a — 3) = cos « (1 — 4 sin* «), and state for what values of a it holds. Use formulas ((>) and (7). 13. Prove that tan Sa= ^ tan « - tan^ a ^^^^^ ^j^^^ ^y^^^ j^^ j^^^j^,^ f^^ 1—3 tan2 a all values of a such that « and 3 a have tangents. 14. Prove that sin (45° + a) sin (45° - «) = 1/2 cos 2 a for all values of a. 15. Prove that sin (« + /3) sin (« - ^)= sin2 « - sin2 ^ for all values of a and /3. 16. Prove that cos (a+/3) cos fi + sin (a + /3) sin /3 = cos a. 104 PLANE TRIGONOMETRY [XI, § 77 77. Functions of Half Angles. The formulas cos^ a + sin^ a = 1 and cos^ a — sin^ a = cos 2 a are true for all values of a. If we subtract one of these from the other, and if we also add them, we obtain the formulas : (16) 2sin2a=:l — cos2a, (17) 2 cos2 a = 1 + cos 2 a. These formulas are true for all values of a ; for a = a' /2 they become 2sin2(a72)=l-cosa' and 2 cos2 (a72) = 1 + cos a', or since these are true for all values of a\ we may write (18) sin (a/2) =±'^- cos a (19) . cos(a/2)=±V^^4^, which hold good for all values of a. The same formulas may be obtained from (12) by solving for sin (a'/ 2), or for cos (cc'/2), after putting a^/2 for a. Erom (18) and (19) we get by division ir^ cos a sm a (20) tan a/2 = ± \/ t— = z ^ 1 + cos a 1 + cos a sm a which hold for all values of a except when a denominator vanishes. The ambiguity of sign of the radical is determined in a given case by the fact that tan (ct/2) is positive or nega- tive according as a/2 is or is not in the first or second quadrant. The relations between an angle and its half are frequently useful in problems that relate to a chord of a circle and the angle which it subtends at the center ; this occurs, for example XI, § 77] ADDITION FORMULAS 105 in laying out railroad curves where it is convenient to make measurements along chords of the curve. This is illustrated in some of the exercises below. The relations are also useful in simplifying trigonometric expressions and in adapting for- mulas to logarithmic computation. EXERCISES XXVIII.— -HALF- ANGLE FORMULAS 1. Find the sine, the cosine, and the tangent of 22° 30' from the kfiown values of sin 45°, cos 45°, tan 45°. 2. Find the sine, cosine, and tangent of 15°. 3. Given that sin a = 4/5, and that a is an acute angle ; find sin (a/2) and tan (a/2). 4. Given tan 26° 34'= 1/2 ; find tan 13° 17'. 5. Given tan 36° 52' = 3/4 ; find sine, cosine, and tangent of 18° 26'. 6. If r denotes the radius of the circle in the accom- panying figure, c a chord, and 6 the angle which c sub- tends at the center ; show that sin (6/2) =c/(2 r) . 7. In the figure, draw the line BD tangent to the circle, and AD perpendicular to BD from the opposite end of the chord BA. Show that (a) ZABD = 6/2 ; (6) BD = AB cos (6/2) == 2 r sin (6/2) cos (6/2) = r sin 6. 8. Prove that tan (45° -f a/2) = sec a + tan a, if tan a exists. 9. Prove that tan (45° + a/2) tan (45° — a/2) = tan 45° if tan a exists. 10. Prove that tan (a/2) + 2 sin2 (a/2) ctn a = sin a, ilsma=^ 0. 11. Prove that tan (a/2) + ctn (a/2) = 2 esc a, if sin a =7^ 0. 12. Prove that [sin (a/2)+ cos (a/2)]2= 1 + sin a for all values of a. 13. Prove that [sin (a/2) — cos (a/2)]2 = 1 — sin a for all values of a. 14. In the figure, COA is a diameter of a circle of radius r ; A OP = a is any acute angle ; OCP = a/2, by geometry ; and PB is perpendicular to OA. Show that ' OB = r cos a, BP = r sin a, BA = r vers a, CB = r(l + cos a). Fig. 87. CP = ^PB^ + CB^ = rV2(l + cos a). Fig. 106 PLANE TRIGONOMETRY [XI, § 78 15. From Ex. 14, show that the functions of a/2 can be read directly from the figure in the form : sin (a/2) = ^"^^^ = Jl-cosa . rV2(l + cosa) ^ 2 cos (a/2) = ^ + ^o^<^ = 11 + cos a . V2(l + cos a) ^ 2 Vl — cos2 a / 1 — cos a 1 — cos a + /^/oN sin a \/l — cos2a ^/l tan (a/2) = = = \ ~ 1 + cos a 1 + cos a ^ 1 1 + cos a 1 + cos a ^ 1 + cos a sin a 16. If a numerical value of any function of a is given, all the other functions of a and of a/2 can be found geometrically from Ex. 14. Thus, if sin a = 4/5 is given, lay off 0P= 5, BP=4:; then OB = \/52 - 42= 3. Hence, CB = S, BA=z2; and CP = ■y/'cB'^ + BP^ = VS^ + 42= \/80. It follows that sin a = 4/5, cos a = 3/5, tan a = 4/3, sin (a/2) = 4/V80 =l/\/5 == \/5/5, cos (a/2)= 8/\/80 = 2/V5 = 2V5/5, tan (a/2) = 4/8 = 1/2. 17. Eind the remaining functions of a and those of a/2 by means of Ex. 16, if cos a = 5/13 ; if tan a = 1/3. 18. The remaining functions of (a/2) and those of a can be* found when any function of a/2 is given from the figure of Ex. 14, by dropping a perpendicular from O to CP. Do this if tan (a/2) = 3/4. 19. Since, in the figure of Ex. 14, by geometiy BP^ = CB • BA, show that (1 + cos a) vers a = sin2 a. 20. Derive trigonometric formulas from the geometric identities (Ex. 14) : __ BP'PA = AB\ BP' CP= CB^. 78. Factor Formulas. In adapting trigonometric formulas to logarithmic computation it is often desirable to express the sum (or difference) of two sines (or cosines) as the product of other functions. Example 1. Reduce sin 35° -f sin 15° to the form 2 sin 25° cos 10°. To do this, set x-\-y = 35°, x — y = 15°, and solve for x and y : x = 25°, y = 10°. Then sin (x -\- y)= sin x cos ?/ + cos x sin y, sin (x — y)=sinx cos y — cosx sin y ; whence, adding, sin (x -\- y) + sin (x - y) = 2 sin xcosy ; substituting x = 25°, y = 10°, we get sin 35° H- sin 15° = 2 sin 25° cos 10°. XI, § 78] ADDITION FORMULAS 107 Example 2. Reduce sin s — sin (s — c) to a product, where s = (a + 6 + c)/2. Let x-{-y=s, x — y = s — c; then aj = (a -f 6)/2, 2/ = c/2, and sin (x + ?/) = sin x cos 2/ + cos aj sin y^ sin (« — 2/) = sin aj cos y — cos x sin ?/ ; subtracting sin (x -\- y) — sin (x — 2/) = 2 cos x sin 2/, whence sin s — sin (s — c) = 2 cos [(a + 6)/2] sin (c/2). EXERCISES XXIX.— FACTORING 1. Reduce each of the following forms to products : (a) sin 70° - sin 10°. (6) sin 70^ + sin 50°. (c) sin 13° + sin 41°. (d) sin 34° - sin 19°. (e) cos26°--cos35°. (/) sin 43° + sin 28°. (g) cos 20° + cos 10°. (h) cos 61° - sin 11°. .. . sin 15° + cos 45° , . . sin 28° + sin 12° cos45° — sinl5° ^ cos 28° + cos 12° (k^ sii^ ^4° + sin 16° m ®^^ ^^° ~ ^^^ ^^° sin 64° - sin 16° cos 40° — cos 80° 2. Prove that cos (x + 2/) + cos (x — y)=2 cos x cos y, 3. Prove that cos (x + 2/) — cos (x — 2/) = — 2 sin x sin y. 4. Prove that cos A -\-cosB = 2 cos ^ "^ ^ cos ^""^ . 2 2 by substituting ^ = x + ?/, -B = x — y in Ex. 2. 5. Prove by means of Ex. 3 that A 4- B A cos i4 — cos 5 = — 2 sin ^ sin — 2 2 6. By the method of Example 1, § 78, show that ' sin i4 + sin 5 = 2 sin ^-^ cos ^^—-? . 2 2 7. By the method of Example 2, § 78, show that sin i4 - sin B = 2 cos ^^-^ sin ^^-^. 2 = lan — —^^iiL sin X — sin 2/ 8. Prove ?iH^±.!iM = tan^+^ctn 9. Prove cosx+cos2/^_ ctn ^±J^ ctn ^^li^. cos X — COS 2/ 2 2 10. Prove ^'"g + sin^O ^ ^^^ ^^/g). cose — cos2« 108 PLANE TRIGONOMETRY [XI, § 78 11. Prove si"(2x-3 j/ )+sin3y ^ ^^^ ^ COS (2 X — 3 y) + COS 3 2/ 12. sin (45'' + x) + sin (45° — x) = V2 cos x. 13. sin 3 X + sin 6 X = 2 sin 4 x cos x. 14. If a + 6 4- c = 2 s, show that (a) cos (6 — c)— cos a = 2 sin (s — h) sin (s (6) cos a — cos (6 + c) = 2 sin s sin (s — a) ; -c): (c) sin (8 ■ c) _ tan I c tan J (ct + 6) 15. 16. sin s + sin (s — c) tan X tan y __ sin x sin y tan X — tan y sin (x — 2/) The so-called "method of offsets" for laying out a circular track is illustrated in the adjoining figure. The track OAB is tangent at to 05', and the dis- tances OA', A'B'^ A' A, CB, are easily shown to be as marked in the figure, where a/2 = ZAOA' is half the angle at the center subtended by a 100-foot chord. In practice, the hne OA'B' is run, and A' and B' marked. Show that B'B, the distance actually to be laid off from B', is B'B = A' A + CB = 200 sin a cos (a/2).' 8lnC3«/2), CHAPTER XII GRAPHS OF TRIGONOMETRIC FUNCTIONS 79. Scales and Units. The graph of the function sin a; is a curve passing through all points whose coordinates (x, y), satisfy the equation y = sin x. The graph of any other trigo- nometric function as cos a;, tan ic, etc., is similarly determined. The radian is the unit angle commonly used in plotting the graphs and in the further study of the trigonometric functions in the Calculus and in other advanced mathematical subjects. Unless otherwise specified, the equation y = sin x is understood to mean that y is the sine of x radians ^ as explained in ^ 64. In plotting curves it is of advantage in many ways to make the horizontal and vertical scale units the same, and this should be done if not too inconvenient, f 80. Plotting Points. In Table V are given the values of the sine, cosine, and tangent of acute angles measured in radians which are very convenient for plotting the graphs of these functions on cross-section paper. 81. Graph of sinjc. Draw a pair of coordinate axes and choose the scale unit = 10 small divisions of the cross-section paper. Take from Table V the sines of the angles in the first quadrant for each tenth radian and tabulate : X .1 .2 .3 .4 .5 etc. . . . 1.6 1.57 ?/ = sin X .099 .198 .295 .389 .479 etc. . . . 1.000 * In any case, y = sin x means that y is the sine of x units of angle. The right angle, the 60° angle, the 45" angle, the degree, or any other angle might be chosen as the unit, if it were convenient. 't If we were to take the two scale units the same in plotting the curve y = sin X where the unit angle is the degree, one arch of the curve would be 180 units long and only 1 unit high. 109 110 PLANE TRIGONOMETRY p:ii,§8i Plot these points and draw a smooth curve through them as OA in the figure. |y 1 III ill 1 II II m^^^^^^ /k - - - -Jfr----""-^ s£ - ^^ ^S^ - s - - - - ^^ _ ___i^_ --- fi ^5" : ■ i^ s 'B^ ' : i '^ ^^ ::^: -S^? £— :: — -+ ^^^ ± F-4^ 5-1^- 9::::::::v::::t::±::::::::i|lv:::::±::::5fe$::::::::-,z5: 5^ ^ ^^ ^? '^ _.^t ->-- -^"■"r - ^^ - - - ,^ s / ^ !» IS ,*^- - ------- -- -__.----^-. -_- - - Fig. 90. It is readily seen by the principles of § 68 that the exten- sion of the curve through the second, third, and fourth quad- rants is as shown by AB, BCj and CD ; and that the curve extends to the left and to the right of the origin in a succes- sion of arches such as OAB, BCD, etc. The graph of sin x can be drawn without the aid of Table V as follows : Choose a convenient scale unit and lay off on the aj-axis OP = - = 1.57 approximately, and divide this segment into a convenient number of equal parts, 15 say ; the points of r> o division correspond to a:= 0, ^, -^, -^, •••, ^' Take from «jU oyj o\j Z a table of sines, such as the one printed on p. 21 for example, the sines of the angles in the first quadrant for each 6° and tabulate : X TT 30 2^ 30 37r 30 47r 30 etc. . . . IT 2 y = sin « .105 .208 .309 .407 etc. . . . 1.000 Plot these points and draw a smooth curve through them. XII, § 82] TRIGONOMETRIC GRAPHS 111 Fig. 91. The same methods may be used, with obvious modifications, to plot the graphs of cos x^ tan x^ and in fact any one of the trigonometric functions. 82. Mechanical Construction of the Graph. If an angle of X radians be laid off at the center of a unit circle (i.e. a circle whose radius is the scale unit), as AOB in Tig. 91, the numerical measure of the arc AB is the number of radians in the angle, Le. x\ the measure of CB 'v. sin a;, the measure of AD is tan x^ the measure of 00 is cos a?, and the measure of OB is sec X, These facts can be used to construct the graphs of these functions without the use of any tables whatever. If we lay off on the ic-axis a segment equal in length to the arc AB and at its end point erect a perpendicular equal to CjS, its end point will lie on the graph of sin x. It remains to show how to lay off a line segment approximately equal in length to a circular arc. If the arc AB is a known part of the quadrant AQ whose measure is 1.5708", the meas- ure of ^jB can be computed and laid off with a scale. This will be the case if B is one of the points of division which divide the quad- rant into a number of equal arcs. But even if the ratio of AB to ^Q is unknowii, provided AB| ^ ^ ^ '■*v ^X k »^ Kd-* "^ A ^ L^ .8 ^ -^ ^ s i^ s> yf- ^ •'^. / *■ N s ^ y /^ •\\ / s \ 4 u ^^ / \ \ X \ c 2 .< \ .( ) .i 5 . ~s V ~^ >^ \ v y N \ \ / :os [s c V k. \ / s :\ ==- R . y •v ^ ^ "€ Fia. 93. 1, 1.2, 1.4, 1.57, on the oj-axis and erect perpendiculars equal to the ordinates of the corresponding points on the arc. These give points on the graph of sin x. By erecting perpendiculars to the oj-axis equal to the hori- zontal distances from CQ of the corresponding points on the arc we shall get points on the graph of cos x. By drawing radiating lines from the center G of the unit circle through the points of division of the arc we can lay off the tangents of these arcs on the y-axis and construct the graph •# * The proof of this cannot be given until the student has studied Calculus. The distance ^P is greater than x, but the error is less than S)Ylx^. The greatest error, about .017, occurs when AB is an arc of about 74° 29', or when X — l.S^**^ approximately. The error for a 45° arc is .007 and for a quad- rant, .006. XII, § 82] TRIGONOMETRIC GRAPHS 113 of tan X ; and in an obvious manner (see Fig. 91) the graph of sec a; can be drawn. These graphs can be extended through the other three quadrants, and to the left of the y-Sixis, as in § 81. If the angle increases beyond 2 tt (radians) the values of all the trigonometric functions repeat themselves and the graph from x=2 tt to x = Air will be a repetition of those from a; = to oj = 2 TT. Functions which repeat themselves as x increases are called periodic functions. The period is the smallest amount of increase in x which produces the repetition of the value of the function. Thus, sin a? is a periodic function with a period of 2 TT, while the period of tan x is tt. EXERCISES XXX. — GRAPHS OF TRIGONOMETRIC FUNCTIONS 1. Plot the graphs of the following functions using Table V, and Table VI when necessary. (a) cosx (6) tana; (c) versa (d) ctnx (e) sec a; (/) cscx (gr) sin2x (h) cos2x (i) Vsinx 2. Plot the graphs of the following functions without the use of tables: (a) cosx (6) tanaj (c) secx 3 . Plot the graph of cos x by dividing the second quadrant of the unit circle into fifths of a radian (see Fig. 91) and making use of the fact that cos x = sin (7r/2 + x) . 4. Plot on the same axes the graphs of sinx, sinjx, sin2x, and 2 sin X. 5. Plot on the same axes the graphs of cosx, cosjx, cos3x, and 3 cos X. 6. Discuss the graphs of sin x/n, sin nx, and nsinx (where n Is a natural number) in view of the results of Ex. 4 and 6. 7. Plot the graph of sin x + cos x by adding the corresponding ordi- nates of the curves y = sin x and y = cos x plotted on the same axes. 8. Plot the graphs of the following functions by adding ordinates : (a) sin X — cos X ( 6 ) 2 sin x + cos x (c) tan X — 2 sin X (d) — cosx (i.e. — cosx) (e) x + sinx (/) x — cosx 9. Plot on the same axes the graphs of sin x, and sin (x — ^/6). 10. Plot on the same axes the graphs of sin x, cos x, and cos (x — Tr/2), I 114 PLANE TRIGONOMETRY [XII, § 84 83. Inverse Functions. We have seen in § 69 that the equation (1) y = sin X can be solved for a; if 2/ is any number whatever between — 1 and + 1, and that there are an infinite number of solutions. Any one of these solutions is denoted by ^ (2) X = arcsin y. If we suppose that the angle is measured in radians, (2) means that X is the number of radians in an angle (or arc) whose sine is y; it is read " arc sine y " or " an angle whose sine is 3/." Likewise arccos y denotes an angle whose cosine is y ; arc- tan y denotes an angle ivhose tangent is y. The expressions y = sin x, x = arcsin y, are two aspects of one relation, just as are the two statements " A is the uncle of B " and " B is the nephew of A " ; either one implies the other ; both mean the same thing. As we wish to study the arcsine function, and in particular to compare it with the sine function, it is convenient and customary to think of it as depending on the same variable x, and write (3) y= arcsin x, [i.e. x = sin 2/]. We note that (3) is obtained from (1) by two steps, (a) solv- ing (1) for x; and (b) interchanging x and y in (2). Two func- tions so related that each can be obtained from the other in this manner are called inverse functions ; each is the inverse of the other. In the same sense, y = cos x and y = arccos x; y=z tan x and 2/=arctan x-, y= sec x and y = arcsec x-, y = vers x and y = arcvers x ; etc., are inverse functions. 84. Graphical Representation of Inverse Functions. Since the equations (1) y = sm X and (2) x = arcsin y * The notation sin-i y also is used very frequently to denote arcsin y, it is necessary to notice carefully that sin— 1 y does not mean (sin y)-'^. XII, § 84] INVERSE FUNCTIONS 115 are equivalent, the same pairs of values of x and y which satisfy one of them satisfy the other. Hence either of these [:::±t-iT-r-i:T-4T:^d±:±::::::i:i^ 11 -III lijjjJjJjJ 11 iJ jJjjJjJ lJJ4J44+y^--^ 1 IHTtTT: ffUl y s n-xum S~ "" i - -- T ±4- 4-- --3-^ - --i . . . S^JJ . _ . ::5-::^::::::T:::::g:irr:J::rrr,:::;!^::::::::ffi::::::^:::::::::: ^,±4.. ;r5f^_. . ..4./^...[ y^\ __::i;|::?v' -"== = = ■ t^^-" ^ T-47--T-r"--ti^4--!j — i.j-i--.t>^ a.-(i-_.^ -X- ^ ^s ' 1 1 / 1 ' - - - >. " s 'ML/ ' ^v. J ' 1 1 > ' " " 1 ^s" ^ * ; 1 > T . ,1 . . • . . . _ _ i. . . . . . rv^^__4;._ J,«£ 1 ._U^ j_^ ^_; _:__. =.Jc-=?=;,4,J.,Jj^.-.a,j,^-,^,Jo-L-i---LlJJ- -_Xt 1 II L J 1 \ ' : \ • . . \ ■ \ \ \ \ \ 1 i ! 1 ! ! II 1 Fig. 94. two equivalent equations is represented graphically by the curve drawn in Fig. 94. From the manner in which equation (3) y = arcsin x is derived from (2) it follows that the graph of arcsin x is obtained from the graph of sin x by interchanging the x- and 2/-axes; or, what gives the same result, by leaving the axes fixed and rotating the curve through an angle of 180° about the line through the origin which makes an angle of 45° with the cc-axis. The result is shown in Fig. 96, p. 116. llllllllllllillllllfiiyi!i!»igi#»lil^ Fig. 95. 116 PLANE TRIGONOMETRY pen, § 84 Similarly from the graph of cos x, Eig. 95, we derive the graph of arccos x in Eig. 97 ; and in the same way the graphs of arctan aj, arcsec a?, arcctn aj, arccsc x, arcvers x, can be drawn from those of tan x, sec x, ctn x, esc x, vers x. ::::::: ::::::::::Tr :?:::-::::: ±h ::::::::::^-:-::S^-::::-:: :::::::::::::::::^:::::::::: C-0.-_WLd^R ___X _/^ X llMlfi^JMU^ :::::::::^;^^:i::#::::T:::: /^ 1 i t __:^__::___:: + :: ::::::::::: i[ J— T— 1 'i"T^T"l — J. ^ :::^r::::: :::=:=== :===:::= ^ _ 3^ 1 hrhU M II :::i::::M:::::::::::::::::: ml"^h^ M n zr SJ— ^ — 5ss"p!i ._..___. ....^__.__ rnwHmmiffliy :: ::::::::::::::::::^F :::: :::: 1 1 |\ 1 1 ::G;:::::::::::k;:: i^riin.x: M 0| :::::::::::: ::::::::2 ::::::::: ::: ::::::: :::::::2:::::::::: 1 P'^fF#-^ II :::::::::H:;?I::::::::::::: -.=17/1^ :::::::::g::::::::::::::::: -:::-.^S--:::--:;:;:; Fig. 96. Fig. 97. EXERCISES XXXI. — INVERSE FUNCTIONS 1. Draw the graph of y = arcsin » as in § 82. 2. Draw the graph oiy — arccos jc as in § 82. 3. Draw the graph oiy — arctan x. 4. Draw the graph oiy = arcsec x. XII, § 84] INVERSE FUNCTIONS 117 Fig. 98. 1 . ' M pf :::::::::l:it:fe:z^x :::::::::: -- X_ ,* \^ -- "": ::::" t" /'*:::::::"" : — : .,j: 4 .^ : 'rr^ - . J : : IJt _z^._.T _ ; '" 2L~r "• - :y-s arx : £ r : : : : : tT. .!j __ ._ __ J=iC-' '■ T L+: l-JY 4 — " I'lX" .-ii: I""!"" : "". j_. — ^. ~& — X" :: : : _ _ . -_■»(_- 1^5] p_:- ;::::;:;;||^|g:;::;;;;:;;:; IJ^teN i: -*-j- _ _: zit i^' a " " " " I m 1^^ ::-;?^:::::^::::-^ I: ■:!!!!"■;::::::: 1 1 |i||i' li' Ml IIIH^ -_\ \i\:\ll\-W^^^ Zl i :: = - = = " -■ ._ . .- _^ . ._._.- ,,« T ^!-. •:::::::::::::::: rr ~^ 17 ' R" =; ::::::::::±L:±::i: ::;,:::::::::: 1 1 i 1 1 1 1 1 1 1 M 1 1 1 1 1 1 i Fig. 99. LOGARITHMIC AND TRIGONOMETRIC TABLES LOGARITHMIC AND TRIGONOMETRIC TABLES REVISED EDITION PREPARED UNDER THE DIRECTION OP EARLE RAYMOND HEDRICK NetD gotfe THE MACMILLAN COMPANY 1921 All rights reserved Copyright, 1913 and 1920, By the MACMILLAN COMPANY Set up and electrotyped. Revised edition published August, 1920. J. 8. Gushing Co. — Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE The present edition of this book contains several tables not contained in the previous editions. The probability of the occurrence of errors has been minimized by using electrotype reproductions of the tables previ- ously included, even when changes were made. Remarkably few errors existed in the original edition ; what few have been discovered have been corrected. Minor changes only occur in the earlier pages. Care has been taken to preserve the page numbers of the principal tables up to page 114, so that older editions may be used in class-work without confusion, and texts which contain the principal tables may be used in the same class. Among the minor changes are the insertion of a condensed table of logarithms and antilogarithms (Table la, p. 20) , the insertion of a table of values of S and T for interpolation in logarithmic trigonometric functions (Table Ilia, p. 45), and the insertion on pages 1-19 of the logarithms of a few important numbers at appropriate points. The principal changes follow page 114. Tables VIII and IX (pp. 115- 122) make reasonably complete the tables of hyperbolic functions formerly represented only by Table XII (pp. 112-114): These functions are of increasing importance, notably in Electrical Engineering-. The table of haversines (Table X, pp. 123-125) will be welcomed particularly by those interested in navigation. The table of factors of composite numbers and logarithms of primes (Table XI, pp. 126-127) has obvious uses. Tables XII a, 6, c, df, e, /, pages 128-132, are intended for work in- volving compound interest, annuities, depreciation, etc. They will be useful for statistics, insurance, accounting, and the mathematics of business. The same care has been exercised to eliminate errors in the new tables that resulted in so great a degree of reliability in the original edition of these tables. E. R. HEDRICK. CONTENTS Explanation of the Tables TABLES PRINCIPALLY TO FIVE PLACES Table I. Table la Table 11. Table Ilia. Table III. Table IV. Table V. Table Va. Table VI. Table VII. Table VIII. Table IX. Table X. Table XI. Table Xlla. Table XII&. Table XIIc. Table Xlld. Table Xlle. Table XII/. Table XIII. Common Logarithms of Numbers . Condensed Logarithms and Antilogarithms Actual Values of the Trigonometric Func- tions Values of S and T for Interpolation . Common Logarithms of the Trigonometric Functions ...... Reduction of Degrees to Radians Trigonometric Functions in Radian Measure Reduction of Radians to Degrees Powers — Roots — Reciprocals Napierian or Natural Logarithms Multiples of M and of 1/M . . Values and Logarithms of Hyperbolic Functions Values and Logarithms of Haversines Factor Table — Logarithms of Primes Compound Interest ..... Compound Discount .... Amount of an Annuity .... Present Value of an Annuity Logarithms for Interest Computations American Experience Mortality Table Important Constants .... BRIEF TABLES — PRINCIPALLY TO FOUR PLACES Table XlVa. Common Logarithms 134-135 Table XIV6. Antilogarithms 136-137 Table XIVc. Values and Logarithms of Trigonometric Functions 138-142 EXPLANATION OF THE TABLES* TABLE I. FIVE-PLACE COMMON LOGARITHMS OF NUMBERS FROM 1 TO 10 000 1 . Powers of 10. Consider the following table of values of powers of 10: Column A Column £ Column A Column B 101 = 10 100 = 1. 102 = 100 10-1 = .1 103 = 1000 10-2 = .01 104 = 10000 10-3 = .001 105 =: 100000 10-4 = .0001 106 = 1000000 10-5 .00001 107 = 10000000 10-6 _ .000001 108 = 100000000 10-7 = .0000001 109 = 1000000000 10-8 = .00000001 1010 = 10000000000 10-9 = .000000001 This table may be used for multiplying or dividing powers of 10, by means of the rules 10« • 10» = 10«+^ 10« -f- 10» = 10«-^ Thus, to multiply 1000 by 100,000, add the exponent of 10 in column J. opposite 1000 to the exponent of 10 opposite 100,000 : 3+5=8; and look for the number in column B opposite 108, ^-^g. 100,000,000. Similarly 1,000,000 x .0001 = 100, since 6+ (—4) =2. To divide 1,000,000 by 100, from the exponent of 10 opposite 1,000,000 subtract the exponent of 10 opposite 100 ; 6 — 2=4; and look for the number opposite 10*, i.e. 10,000. Similarly .001 h- 1,000,000 = .000000001, since — 3 — 6 = — 9. To find the 4th power of 100, multiply the exponent of 10 opposite 100 by 4 : 4x2 = 8, and look for the number opposite 108, i.e. 100,000,000. Likewise (.001)3 = .000000001, since 3 x (- 3J =- 9. To find the cube root of 1,000,000,000, divide the exponent of 10 opposite 1,000,000,000 by 3, 9-4-3 = 3, and look for the number opposite 103. * This Explanation, written to accompany the five-place tables, may be used also for the four-place tables by omitting the last figure in each example in a manner obvious to the teacher. vii VUl EXPLANATION OF THE TABLES [§2 2. Common Logarithms. The exponent of 10 in any row of column A is called the common logarithm * of the number opposite in column B ; thus log 10 = 1, log 100 = 2, log 1000 = 3, etc.; log 1 = 0, log .1 =- 1 ; log .01 =—2, log .001 =—3, etc. In general, if 10^ = n, Z is called the common logarithm of w, and is denoted by log n. 3. Fundamental Principles. Logarithms are useful in reducing the labor of performing a series of operations of multiplication, division, raising to powers, extracting roots, as above ; they have no necessary connection with trigonometry, since all the operations could be performed without them ; but they are a great labor-saving device in arithmetical computations. They do not apply to addition and subtraction. The principles of their application are stated as follows : I. The logarithm of a product is equal to the sum of the logarithms of the factors : log ab = log a + log b. This follows from the fact that if 10^ = a and 10^ = 6, 10^+^ = a • &. In brief : to multiply, add logarithms. II. The logarithm of a fraction is equal to the difference obtained by subtracting the logarithm of the denominator from the logarithm of the numerator : log {a/b) = log a — log b. For, if 10^ = a and 10^ — b, then lOi-L _ ^ _^ ^^ jji i^rief : to divide, subtract logarithms. III. The logarithm of a power is equal to the logarithm of the base multiplied by the exponent of the power : log a^ = b log a. This follows from the fact that if 10^ = a, then W^ - ap. IV. The logarithm of a root of a number is found by dividing the loga- rithm of the number by the index of the root: log Va = (log a)/&. This follows from the fact that if W = a, then lOV^ = «!/& = \/a. Corollary of II. The logarithm of the reciprocal of a number is the negative of the logarithm of the number : log (1/a) = — log a, since log 1 = 0. 4. Characteristic and Mantissa. It is shov^n in algebra that every real positive number has a real common logarithm, and that if a and b are any two real positive numbers such that a < 6, then log a < log b. Neither zero nor any negative number has a real logarithm. An inspection of the following table, which is a restatement of a part a 1 10 100 1000 10000 100000 1000000 10000000 log a 1 2 3 4 5 6 7 * Common logarithms are exponents of the base 10 ; other systems of logarithms have bases different from 10 ; Napierian logarithms (see Table VII, p. 112) have a base denoted by e, an irrational number whose value is approximately 2.71828. When it is necessary to call attention to the base, the expression log^o n will mean common logarithm of n ; loge n will mean the Napierian logarithm, etc. ; but in this book log n denotes logjo^ unless otherwise explicitly stated. ^4] COMMON LOGARITHMS IX of the table of § 1, p. v, shows that the logarithm of every number between 1 and 10 is a proper fraction, the logarithm of every number between 10 and 100 is 1 -f a fraction, the logarithm of every number between 100 and 1000 is 2 + a fraction ; and so on. It is evident that the logarithm of every number (not an exact power of 10) consists of a whole number + a fraction (usually written as a decimal). The whole number is called the characteristic; the decimal is called the mantissa. The characteristic of the logarithm of any number greater than 1 may be determined as follows : Rule I. The characteristic of any number greater than 1 is one less than the number of digits before the decimal point. The following table, which is taken from § 1, p. v, shows that a .0000001 .000001 .00001 .0001 .001 .01 .1 1 log a -7 -6 -5 -4 ~3 — 2 - 1 the logarithm of every number between .1 and 1 is — 1 + a fraction, the logarithm of every number between .01 and .1 is — 2 + a fraction, the logarithm of every number between .001 and .01 is — 3 + a fraction ; and so on. Thus the characteristic of every number between and 1 is a negative whole number ; there is a great practical advantage, however, in comput- ing, to write these characteristics as follows : — 1 = 9 — 10, — 2 = 8 — 10, — 3 = 7 — 10, etc. E.g. the logarithm of .562 is - 1 + .74974, but this should be written 9.74974 — 10 ; and similarly for all numbers less than 1. Rule II, The characteristic of a number less than 1 is found by sub- tracting from 9 the number of ciphers between the decimal point and the first significant digits and writing — 10 after the result. Thus, the characteristic of log 845 is 2 by Rule I ; the characteristic of log 84.5 is 1 by (I) ; of log8.45 is by (I) ; of log. 845 is 9 - 10 by (II) ; of log. 0845 is 8 - 10 by (II). An important consequence of what precedes is the following : To move the decimal point in a given number one place to the right is equivalent to adding one unit to its logarithm, because this is equivalent to multiplying the given number by 10. Likewise, to move the decimal point one place to the left is equivalent to subtracting one unit from the logarithm. Hence, moving the decimal point any number of places to the right or left does not change the mantissa but only the characteristic* Thus, 5345, 5.345, 534.6, .05345, 534500 all have the same mantissa. * Another rule for finding the characteristic, based on this property, is often useful : if the decimal point were just after the first significant figure, the characteristic would be zero ; start at this point and count the digits passed over to the left or right to the actual decimal point ; the number obtained is the characteristic, except for sign ; the sign is nega- tive if the movement was to the left, positive if the movement was to the right. X EXPLANATION OF THE TABLES [§ 5 5. Use of the Table. To use logarithms in computation we need a table arranged so as to enable us to find, with as little effort and time as possible, the logarithms of given numbers and, vice versa, to find numbers when their logarithms are known. Since the characteristics may be found by means of llules I and II, p. ix, only mantissas are given. This is done in Table I. Most of the numbers in this table are irrational, and must be represented in the decimal system by approximations. A five- place table is one which gives the values correct to five places of decimals. Problem 1. To Ji7id the logarithm of a given number. First, deter- mine the characteristic, then look in the table for the mantissa. To find the mantissa in the table when the given number (neglecting the decimal point) consists of four, or less, digits (exclusive of ciphers at the beginning or end), look in the column marked iVfor the first three digits and select the column headed by the fourth digit : the mantissa will be found at the intersection of this row and this column. Thus to find the logarithm of 72050, observe first (Eule I) that the characteristic is 4. To find the mantissa, fix attention on the digits 7205 ; find 720 in column iV, and opposite it in column 5 is the desired mantissa, .85763 ; hence log 72050 = 4.85763. The mantissa of .007826 is found opposite 782 in column 6 and is .89354 ; hence log .007826 = 7.89354— 10. 6. Interpolation. If there are more than four significant figures in the given number, its mantissa is not printed in the table ; but it can be found approximately by assuming that the mantissa varies as the number varies in the small interval not tabulated ; while this assumption is not strictly correct, it is sufficiently accurate for use with this table. Thus, to find the logarithm of 72054 we observe that log 72050 = 4.85763 and that log 72060 = 4.85769. Hence a change of 10 in the number causes a change of .00006 in the mantissa ; we assume therefore that a change of 4 in the number will cause, approximately, a change of .4 x .00006 = .00002 (dropping the sixth place) in the mantissa ; and we write log 72054 = 4.85763 + .00002 = 4.85765. The difference between two successive values printed in the table is called a tabular difference (.00006, above). The proportional part of this difference to be added to one of the tabular values is called the cor- rection (.000002, above), and is found by multiplying the tabular difference by the appropriate fraction (.4, above). These proportional parts are usually written without the zeros, and are printed at the right-hand side of each page, to be used when mental multiplications seem uncertain. Example 1. Find the logarithm of .0012647. Opposite 126 in column 4 find .10175; the tabular difference is 34 (zeros dropped) ; .7 x 34 is given in the margin as 24 ; this cor- rection added gives .10199 as the mantissa of .0012647 ; hence log .0012647 = 7.10199 - 10. Example 2. Find the logarithm of 1.85643. Opposite 185 in column 6 find .26858 ; tabular difference 23 ; .43 x 23 is given in the margin as 10 ; this correction added gives .26868 as the mantissa of 1.86643 ; hence log 1.85643= 0.26868. §8] COMMON LOGARITHMS xi 7. Reverse Reading of the Table. Problem 2. To find the number when its logarithm is known.* First, fixing attention on the mantissa only, find from the table the number having this mantissa, then place the decimal point by means of the two following rules : t Rule III. If the characteristic of the logarithm is positive (in which case the mantissa is not followed by — 10), begin at the left, count digits one more than the characteristic, and place the decimal point to the right of the last digit counted. Rule IV. If the characteristic is negative (in which case the mantissa will be preceded by a number n and followed by — 10), prefix 9— n ciphers, and place the decimal point to the left of these ciphers. Example 1. Given log x = 1.22737, to find x. Since the mantissa is 22737, we look for 22 in the first column and to the right and below for 737, which we find in column 8 opposite 168. The number is therefore 1688. Since the characteristic is + 1, we begin at the left, count 2 places, and place the point ; hence X = 16.88. Example 2. Given log x = 2.24912, to find x. This mantissa is not found in the table ; in such cases we interpolate as follows : select the mantissa in the table next less than the given mantissa, and write down the corre- sponding number ; here, 1774 ; the tabular difference is 25 ; the actual difiference (found by- subtracting the mantissa of 1774 from the given mantissa) is 17 ; hence the proportionality- factor is 17/25 = .68 or .7 (to the nearest tenth). Since moving the decimal point does not affect the mantissa, it follows that the digits in the required number are 17747 (to five places). The characteristic 2 directs to count 3 places from the left ; hence x = 177.47. Rule. In general, when the given mantissa is not found in the table, write down four digits of the number corresponding to the mantissa in the table next less than the given mantissa, determine a fifth figure by dividing the actual difference by the tabular difference, and locate the decimal point by means of the characteristic. 8. Illustrations of the Use of Logarithms in Computation. Example 1. To find 832.43 X 302.43 X 16.725 X .000178. log 832.43 = 2.92034 log 302.43 = 2.48062 log 16.725 = 1.22337 log .000178 = 6.25042 - 10 (add) log X = 2.87475 whence x — 749.47. Example 2. To find 461.29 ^ 21.4. log 461.29 = 2.66397 log 21 .4 = 1.33041 (subtract) log X = 1.33356 whence x = 21.556. * The number whose logarithm is k is often called the antilogarithm oik. t Another convenient form of these rules is as follows : if the characteristic were zero, the decimal point would fall just after the first significant figure ; move the decimal point one place to the right for each positive unit in the characteristic, one place to the left for each negative unit in the characteristic. xii EXPLANATION OF THE TABLES [§8 Illustration of Cologarithms E^mpUZ. Tofindl5^25xm76XJT45. 1415.3 We might add the logarithms of the factors in the numerator and from this sum subtract the logarithm of the denominator ; but we can shorten the operation by adding the nega- tive of the logarithm of the denominator instead of subtracting the logarithm itself. The negative of the logarithm of a number (when written in convenient form for computation) is called the cologarithm of the number. We may find the negative of any number by subtracting it from zero, and it is convenient in logarithmic computation to write zero in the form 10.00000 - 10. Thus the negative of 2.17 is 7.83 - 10 ; the negative of 1.1432 - 10 is 8.8568. Remembering that the cologarithm of a number is its negative we have the follow- ing rule : To find the cologaHthm of a nuniber hegin at the left of its logarithm {including the characteristic) and subtract each digit from 9, except the last,* which subtract from 10 ; if the logarithm, has not — 10 after the mantissa^ write — 10 after the result; if the logarithm has — 10 after the mantissa, do not write — 10 after the result. By this rule the cologarithm of a number can be read directly out of the table without taking the trouble to write down the logarithm. Attention must be given not to forget the characteristic. The use of the cologarithm is governed by the principle : Adding the cologarithm is equivalent to subtracting the logarithm, Eeturning to the computation of the given problem we should write : log 48. 25 =1.68350 log 132.76= 2.12307 log .1745= 9.24180 -10 colog 1415.3 = 6.84915 - 10 (add) log x= 9.89752 - 10 whence x= .7898 Esaample 4. Find the 5th power of 7.26842 log 7.26842= 0.86144 5 (multiply) log X = 4.30720 whence x = 20286. Example 5. Find the 4th root of .007564 log .007564 =7.87875 -10. (It is convenient to have, after the division by 4s — 10 after the mantissa ; hence before the division we add 30.00000 - 30.) log .007564= 37.87875 - 40 (divide by 4), log X = 9.46969 - 10 whence x = .2949 Example^. Find the value of ! / (34.55)(- 856.7)(- 43 ]!) \ (98.75)(- 186.3) We have no logarithms of negative numbers, but an inspection of this problem shows that the result will be negative and numerically the same as though all the factors wer« positive ; hence we proceed as follows : log 34.55 =1.53845 log 856.7 = 2.93288 log 43.5 =1.63849 colog 98.75= 8.00546 - 10 colog 186.3 = 7.72979 - 10 (add) 1.84502 (divide by 3) log(- a;) = 0.61501 whence tc = - 4.121 * If the logarithm ends in one or more ciphers, the last significant digit is to be under Btood here. §9] THE SLIDE RULE Xlll 9. The Slide Rule. A slide rule consists of two pieces of the shape of a ruler, one of which slides in grooves in the other ; each is marked 3'^ 6 7 8 9 1 Mnirni IMIMltr lllllillllllllll Fig. 1 (Fig. 1) in divisions (scale A and scale B) whose distances from one end are proportional to the logarithms of the numbers marked on them. It follows that the sum of two logarithms can be obtained by simply 1 2 S'^ 4567891 A 1 1 ' ' ' ' ! 1 1 1 1 i 2 ^hiiliiili ihliiiiililililil:'^'' ' ' ' ' ' : ''ih thtr Ttltltltlltlll]] ' . : -nll|llll|llll!llll|llll|lll'!!lll|li| A L 1 2 ' 3^ 4 5 6 7 8 91 ; 2 K , , M / m IJTT It Ti- mill Wm\l ti: , , ,1, , Vi'.lW pJIIII Nil Nil lllllllllll 1 1 il.L ll II II 1 III 4|ii '\'i U lU D^ 1 [ \ 1 1 ' ' ■ ■ ^ : ■ : 1 j ■ . : .^i ■ ■ . V ^'i""'"' Fig. 2 sliding one rule along the other ; thus if (see Fig. 2) the point marked 1 on scale B is set opposite the point marked 2. 5 on scale A^ the point on scale B marked 2 will be opposite the point on scale A marked 6, since log 2.5 + log 2 = log 5. Likewise, opposite 3 (scale B) read 7.6 (scale A) \ opposite 2.5 (J5) read 6.25 (^), i.e. 2.5 x 2.6 = 6.25. Other multiplications can be performed in an analogous manner. Divi- sions can be performed by reversing the operation. Thus, if 4.5 {B) be set on 11.25 (^), then 1 {B) will be opposite 2.5 (J.), as in Fig. 2. Scales C and J) are made just twice as large as scales A and B. It fol- lows that the numbers marked on and B are the square roots of the numbers marked opposite them on scales ^ and B. For a description of more elaborate slide rules, and full directions for use, see the catalogues of instrument makers. A slide rule for practice may be made from the cut printed on one of the fly-leaves in the back of this book. xiv EXPLANATION OF THE TABLES [§ 10 la. CONDENSED LOGARITHMS AND ANTILOGARITHMS 10. Method of Computing Logarithms. This table is a rearrangement of the condensed table given by Hoiiel.* From it, the logarithm of any number whatever may be obtained to within 5 in the fifteenth place ; or to any desired degree of accuracy less than this. To illustrate the process, we shall compute log w to nine places. Tak- ing TT = 3.1415926535 8979, we divide it by 3, the first significant digit, obtaining 7r/3 = 1.04719 755 •••. We then divide this quotient by 1.04, etc., obtaining finally TT = 3(1.04) (1.006) (1.0009) (1.00001 5217225). We can obtain the logarithm of each of the first four factors from this table. The logarithm of the last factor can be obtained by multiplying its decimal part hjM= .4342944819 ; for the error made in writing log(l +x) = Mx is less than Mx'^/2. We find Mx either by using the fact that the last column in this table gives multiples of If, or (preferably) by Table VIII, page 115. Adding the five logarithms just mentioned, we find log7r= .4971498727 4, which is surely correct to within 1 in the tenth place. The correct value is .4971498726 9 .... The process may be applied to any other number in an analogous man- ner. Such high-place logarithms are occasionally needed in statistical work and in the preparation of tables. 11. Method of Computing Antilogarithms. The condensed table of antilogarithms gives eleven significant figures (ten decimal places). From it, the antilogarithm of any number can be computed to within 6 in the tenth significant digit. Thiis, to compute the antilogarithm of .4342944819 to 8 significant figures, we may write 10-4342944819 — (10-4) (lO-^^) (10-004) (10•0002^ H 0.00009) (10.0000044819) . The first five factors may be obtained directly from the table. The last factor may be calculated from the formula 10* = 1-1- {\/M)x. The error in this formula is less than 3 in the (2 A:)th decimal place if x is less than (.1)*, where A:>1. However, a much more rapid process depends on the use of Tables I and XI with this table. Thus, by Table I, 10-43429 _ 2. 718, nearly. By Table XI, log 2.718 = .43424 94524 .... Hence 10.4342944819 =^(2.718) (10-0000450295) = (2.718) (10-00004) (10-0000050296). Obtaining the second factor from this table, and the last factor from the formula 10' = 1 4- (l/if)x, by Table VHI, we find 10-4342944819=^2.718281826; while the correct value is 2.718281828 •••. This process requires only two long multiplications. * HotJEL, Becueil de Formules ei de Tables numiriques. § 12] TRIGONOMETRIC FUNCTIONS XV II. FIVE-PLACE TABLE OF THE ACTUAL VALUES OF THE TRIGONOMETRIC FUNCTIONS OF ANGLES 12. Direct Readings. This table gives the sines, cosines, tangents, and cotangents of the angles from 0° to 45° ; and by a simple device, indicated by tlie printing, the values of these functions for angles from 45° to 90° may be read directly from the same table. For angles less than 45° read down the page, the degrees being found at the top and the min- utes on the left ; for angles greater than 45° read up the page, the degrees being found at the bottom and the minutes on the right. To find a function of an angle (such as 15°27'.6, for example) v^hich does not reduce to an integral number of minutes, we employ the process of interpolation. To illustrate, let us find tan 15° 27'. 6. In the table we find tan 15° 27' = .27638 and tan 15° 28' = .27670 ; we know that tan 15° 27 '.6 lies between these two numbers. The process of interpola- tion depends on the assumption that between 15° 27' and 15° 28' the tan- gent of the angle varies directly as the angle ; while this assumption is not strictly true, it gives an approximation sufficiently accurate for a five-place table. Thus we should assume that tan 15° 27'. 5 is halfway between .27638 and .27670. We may state the problem as follows : An increase of 1' in the angle increases the tangent .00032 ; assuming that the tangent varies as the angle, an increase of 0'.6 in the angle will increase the tan- gent by .6 X .00032 = .00019 (retaining only five places); hence tan 15° 27'.6 = .27638 + .00019 = .27657. The difference between two successive values in the table is called, as in Table I, the tabular difference (.00032 above). The proportional part of the tabular difference which is used is called the correction (.00019 above), and is found by multiplying the tabular difference by the appro- priate fraction of the smallest unit given in the table. Example 1 . Find sin 63° 52 ' .8. We find sin 63*'i52 ' = . 89777 ; tabular difference = .00013 (subtracted mentally from the table), correction = .8 x .00013= .00010 (to be added). Hence sin 63" 62'.8 = .89787. ExampU 2. Find cos 65° 24'.8. cos 65° 24' = .41628 ; tabular difference = 26 ; .8 x 26 = 21 (to be subtracted because the cosine decreases as the angle increases) . Hence cos 65° 24'. 8 = .41607. Rule. To find a trigonometric function of an angle by interpolation : select the angle in the table which is next smaller than the given angle, and read its sine (cosine or tangent or cotangent as the case may be) and the tabular difference. Compute the correction as the proper proportional part of the tabular difference. In case of sines or tangents add the correc- tion ; in case of cosines or cotangents, subtract it. xvi EXPLANATION OF THE TABLES [§ 13 13. Reverse Readings. Interpolation is also used in finding the angle when one of its functions is given. Example 1. Given sin a?= .32845, to find x. Looking in the table we find the sine which is next less than the given sine to be .32832, and this belongs to 19*'10'. Subtract the value of the sine selected from the given sine to obtain the actual difi'erence= ,00013 ; note that the tabular difference = ,00027. The actual difference divided by the tabular difference gives the correction => 13/27 = .5 as the decimal of a minute (to be added). Hence x— 19** 10'. 5. Example 2. Given cos x= .28432, to find x. The cosine in the table next less than this is .28429 and belongs to 73** 29' ; the tabular difference is 28; the actual difference is 3; correction = 3/28= .1 (to be subtracted). Hence 85 = 73" 28 '.9. KuLE. To find an angle when one of its trigonometric functions is given : select from the table the same named function which is next less than the given function, noting the corresponding angle and the tabular difference ; compute the actual difference (between the selected value of the function and the given value) and divide it by the tabular difference ; this gives the correction which is to be added if the given function is sine or tangent, and to be subtracted if the given function is cosine or cotangent. in. FIYE-PLACE COMMON LOGARITHMS OF THE TRIGO]N^OMETRIC FUNCTIONS 14. Use of the Table. If it is required to find the numerical value of X = 27.85 X sin 51° 27', we may apply logarithms as follows : log27.85 = 1.44483. log sin 51° 27' = 9.89324 - 10 (add) . logx = 1.33807 x = 21.78 The only new idea here is the method of finding log sin 51° 27', which means the logarithm of the sine of 51° 27'. The most obvious way is to find in Table I, sin 51° 27' = .78206, and then to find in Table II, log. 78206 = 9.89324 — 10, but this involves consulting two tables. To avoid the necessity of doing this. Table HI gives the logarithms of the sines, cosines, tangents, and cotangents. The arrangement and the principles of interpolation are similar to those given on p. viii for Table I. The sines and cosines of all acute angles, the tangents of all acute angles less than 45° and the cotangents of all acute angles greater than 45° are proper fractions, and their logarithms end with — 10, which is not printed in the table, but which should be written down whenever such a logarithm is used. Example 1. Find log sin 68° 25'. 4. On the page having 68** at the bottom, and in the row having 25' on the right find log sin 68° 25' = 9.96843 - 10 ; the tabular difference is 5 ; .4 x 5 is given in the margin as 2 ; this is the correction to be added, giving log sin 68° 25'. 4= 9.96845 - 10. (In case of sine and tangent add the correction. In case of cosine and cotangent, sub- tract the correction.) § 15] RADIAN MEASURE xvii Example 2. Given log cos a; = 9.72581 — 10, to find x. The logarithmic cosine next less than the given one is 9.72562—10 and belongs to 57" 53' ; the actual difference is 19 ; the tabular difference is 20 ; hence the correction is 19/20= 1.0 (to the nearest tenth) ; (subtract) ; hence x= 57" 52'. 0. In finding log ctn a for any angle a, note that log ctn ct = — log tan a, since ctn a = 1 /tan a. Hence the tabular differences for log ctn are pre- cisely the same as those for log tan throughout the table, but taken in reversed order. Likewise, log sec a =— log cos a, log esc cc = — log sin a ; hence log sec a and log esc a are omitted. For angles near 0° or near 90°, the interpolations are not very accurate if the differences are large. For the calculation of sine or tangent near 0°, Table Ilia, page 45, gives the values of S = log sin A — log A' and T = log tan A — log A', where A is the given angle and A' is the number of minutes in A, for values of J. between 0° and 3°. Then log sin A = \ogA' -^ S and log tan A = log A' + T, for small angles. Moreover, since we have cos J. =: sin (90° — J.) and ctnJ. = tan(90°- J.), log cos J. = log (90° - ^)'4- ^ and log ctn J. = log (90° - A)' + T, when A is near 90°. Another method practically equivalent to the preceding is to use the approximate relations log sin A — log sin B = log A' — log B' and log tan A — log tan B = log A^ — log B', where A is the given angle and B is the nearest angle to A that is given in the table. If J. < 3° and \A — 5 1 < 1', these formulas give log sin A and log tan A to five decimal places. IV-y. RADIAN MEASURE 15. Computations in Radian Measure. The reduction of degrees to radians is facilitated by Table TV — Conversion of Degrees to Radians. Since tt radians = 180°, this table may be regarded as a table of multiples of 7r/180. The values of sinx, coscc, tana;, are stated for every angle x from 0.00 radians to 1.60 radians at intervals of .01 radian in Table V — Trigo- nometric Functions in Badian Measure. The values of any of these func- tions for larger values of x may be computed by first converting the value of the angle in radian measure to degree measure, by Table Va, and then finding the value of the function from Table II. The reduction of radians to degrees can be performed directly by Table V ; or, for greater accuracy, by the supplementary Table Va. xviii EXPLANATION OF THE TABLES [§ 16 VL POWERS — ROOTS — RECIPROCALS 16. Arrangement. This table is arranged so that the square, cube, square root, cube root, or reciprocal can be read directly to five decimal places for any number n of three significant figures. To attain this, not only n2, n^, Vn, Vn, 1/n, but also VlO n, VlOn, VlOO n are printed on every page. All values have been carefully recomputed and checked. Thus to find Vl. 17, re ad in V7i column the result: 1.08167. To find Vu.T, read in the same line, in 's/Ton col umn the r esult : 8.42053. To find Vll7, read 10 times the entry in \/n column, since Vll7 = lOVToY. Similarly, v^I.17 = 1.05373 from -y/n column ; VH-'^ = 2.27019 from the same line in y/li) n column ; \/ll7"= 4.89097 from the same line in ■yjl'd^n column. The effect of a change in the decimal point in n^, n^, and 1/n is only to shift the decimal point in the result, without altering the digits printed. VIL NAPIERIAN OR NATURAL LOGARITHMS 17. The Base e. —Natural Logarithms. The number e = 2.7182818 ... is called the natural base of logarithms. The logarithms of numbers to this base are given in Table VII at intervals of .01 from 0.01 to 10.09, and at unit intervals from 10 to 409. The fundamental relation loge n = loge 10 X logio u cuablcs us to transfer from the base 10 to the base e, or conversely ; where log^ 10 = 2.30258509. VIIL MULTIPLES OF M AND OF 1/M 18. Multiples of M and 1/ilf. This table is convenient whenever a number is to be multiplied by M or by 1/M. This occurs whenever it is desired to change from common logarithms to natural logarithms, or con- versely, since M = logio e and since we have logio X = (loge «) (logio e) = (l/M)\oge X and log, x = M logio x. Other formulas that require these multiples are logio e^ = x logio e= X- M and loge(10" . x) = log^ x + n(l/M) ; and the appropriate formulas (see §§ 10, 11, p. xiv) logio(l =tx) = =tx.if and 10^^ = \ ^(l/M)x, IX. VALUES AND LOGARITHMS OF HYPERBOLIC FUNCTIONS 19. Hyperbolic Functions. This table gives the values of e^, e-*, sinh aj, cosh cc, tanh x ; and the logarithms of 6^, sinh x, cosh x, at varying intervals from x = to x = 10. It is to be noted that log e-^= — log e* and log tanh x = log sinh x — log cosh x. The table may be extended indefinitely by means of Table VIII, since logio e* = x . if ; for this reason Table VIH may be regarded as a table of values of logio e^- § 22] VALUES AND LOGARITHMS xix X. VALUES AND LOGARITHMS OF HAVERSINES 20. Haversines. This table gives the values and the logarithms of the haversines of angles from 0° to 180° at intervals of 10'. The haversine, which means half of the versed sine, is hav^=(l/2) vers^= (1/2)(1 - cos^) ; hence its values to five places may be computed from the table of cosines. It is used extensively in navigation, and it may be used to advantage in the solution of ordinary oblique triangles. XL FACTOR TABLE — LOGARITHMS OF PRIMES 21. Factors of Composite Numbers. Logarithms of Primes. The uses of this table are evident in questions involving factoring, and for finding high-place logarithms of numbers whose prime factors are less than 2018. We shall illustrate the finding of logarithms of other numbers by finding log TT. Taking tt = 3.14159 26536, divide by 3 (the first digit), obtaining 1.0471975512 •••. Divide this quotient by 1.047 (in general, by the nearest first four digits), obtaining 1.00018 8683 .... By Table VIII, the approxi- mate formula log(l ±x) = ±x - M gives log 1.00018 8683 = .00008 1943 (Table VIII) log 3 = .47712 12547 (Table XI) log 1.047 = log 3 -f- log .349 = .01994 66817 (Table XI) logTT =.497149879 while the true value of log ir is .49714 987269, so that the error is less than 1 in the eighth place. In general, this process will give the logarithm of any number to within 6 in the eighth decimal place, and the probable error is less than 1.5 in the eighth place. For still greater accuracy, see Table la and § 10. XII. INTEREST TABLES 22. Interest Tables. Tables XII a, 6, c, d give compound interest and annuity data for various per cents up to fifty years. Aside from the obvious uses, formulas involving this data will be found in works on statistics, accounting, and the mathematics of business. Table XHe gives the logarithms of (1 -h r) to fifteen places, for all ordinary values of r from 1/2 ^o to 10%. For other values of r, log(l + r) may be computed from Table la (see § 10). The final result in interest calculations may be obtained to nine significant figures by the antilogarithms of Table la (see § 11). Table XII/ is the American Experience Mortality Table. XX EXPLANATION OF THE TABLES [§§23,24 XIV. FOUR-PLACE TABLES 23- Four-place Tables. These are duplicates of the preceding five- place tables, reduced to four places, and with larger intervals betweer the tabulations. The value of such four-place tables consists in the greater speed with which they can be Used, in case the degree of accuracy they afford is sufficient for the purpose in hand. XlVflf. Logarithms of Numbers. The only special feature of this table is that the proportional parts are printed for every tenth in every row ; hence the logarithm of any number of four significant figures can be read directly. XI V6. Antilogarithms. This table will be found to facilitate approxi- mate calculations to a marked degree. The proportional parts are stated in the right-hand margin for each row separately. This arrangement, with the corresponding one in Table XlVa, makes the tables effectively four-place each way. XI Vc. Values and Logarithms of Trigonometric Functions. In this table, the values of sin a, cos a, tan ot, ctn a, and their common loga- rithms, are stated for each 10-minute interval in a. The characteristics of the logarithms are omitted, since they can be supplied readily from the value. 24. Sources and Checks used. In arranging all of these tables, several extant tables have been used as sources ; and the proofs have been read against the standard seven-place tables of Vega, and at least one other table, or against at least two independent sources when the figures are not given by Vega. In all cases, the stereotyped plates have been proof-read five times, by three different persons. In case of apparent doubt, especially in the last place of decimals, the values have been recomputed, either by series or by the condensed fifteen- place tables of Hotiel. While errors may occur, it is believed that they must be purely typo- graphical ; in most cases such an error is revealed by the unreasonable differences it creates. Greek Alphabet Lbttbes Names Letters Names Letters Names Lbtters Names A a Alpha H^ Eta N V Nu T T Tau B)8 Beta © Theta H^ Xi Y V Upsilon ry Gamma I L Iota O Omicron 4> <^ Phi AS Delta K K Kappa n 77 Pi . Xx Chi E £ Epsilon A X Lambda Pp Rho ^ il/ Psi ZC Zeta M fJL Mu S (T S Sigma O (0 Omega LOGARITHMIC AND TRIGOTOMETRIC TABLES TABLE I COMMON LOGARITHMS OF NUMBERS FKOM 1 TO 10 000 xo FIVE DECIMAL PLACES 1 -100 K Log N Log N Log N Log N Log 20 1.30 103 40 1.60 206 60 1.77 815 80 1.90 309 1 2 3 0.00 000 0.30 103 0.47 712 21 22 23 1.32 222 1.34 242 1.36 173 41 42 43 1.61 278 1.62 325 1.63 347 61 62 63 1.78 533 1.79 239 1.79 934 81 82 83 1.90 849 1.91 381 1.91 908 4 5 6 0.60 206 0.69 897 0.77 815 24 25 26 1.38 021 1.39 794 1.41 497 44 45 46 1.64 345 1.65 321,. 1.66 276 64 65 66 1.80 618 1.81 291 1.81 954 84 85 86 1.92 428 1.92 942 1.93 450 7 8 9 10 0.84 510 0.90 309 0.95 424 27 28 29 1.43 136 1.44 716 1.46 240 47 48 49 1.67 210 1.68 124 1.69 020 67 68 69 1.82 607 1.83 251 1.83 885 87 88 89 1.93 952 1.94 448 1.94 939 1.00 000 30 1.47 712 50 1.69 897 70 1.84 510 90 1.95 424 11 12 13 1.04 139 1.07 918 1.11 394 31 32 33 1.49 136 1.50 515 1.51 851 51 52 53 1.70 757 1.71 600 1.72 428 71 72 73 1.85 126 1.85 733 1.86 332 91 92 93 1.95 90.4 1.96 379 1.96 848 14 15 16 1.14 613 1.17 609 1.20 412 34 35 36 1.53 148 1.54 407 1.55 630 54 55 56 1.73 239 1.74 036 1.74 819 74 75 76 1.86 923 1.87 506 1.88 081 94 95 96 1.9T313 1.9T772 1.98 227 17 18 19 1.23 045 1.25 527 1/27 875 37 38 39 1.56 820 1.57 978 1.59 106 57 58 59 1.75 587 1.76 343 1.77 085 77 78 79 1.88 649 1.89 209 1.89 763 97 98 99 1.98 677 1.99 123 1.^)9^564 N Log N Log N Log N Log N Log 100 — Logarithms of Numbers — 160 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 100 01 02 03 04 05 06 07 08 09 00 000 043 087 130 173 217 260 303 346 389 432 860 01284 703 02119 531 938 03 342 743 475 903 326 745 160 572 979 383 782 518 945 368 787 202 612 *019 423 822 561 988 410 828 243 653 *060 463 862 604 *030 452 870 284 694 *100 503 902 647 *072 494 912 325 735 *141 643 941 689 *115 536 963 366 776 *181 683 981 732 *157 578 996 407 816 *222 623 *021 776 *199 620 *036 449 867 *262 663 *060 817 *242 662 *078 490 898 *302 703 *100 1 2 3 4 5 6 7 8 9 44 4.4 8.8 13.2 17.6 22.0 26.4 30.8 35.2 39.6 43 4.3 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 42 4.2 8.4 12.6 16.8 21.0 25.2 29.4 33.6 37.8 110 11 12 13 14 15 16 17 18 19 04139 179 218 258 297 336 376 415 454 493 1 532 922 05 308 690 06070 446 819 07188 555 571 961 346 729 108 483 856 225 591 610 999 385 767 145 521 893 262 628 650 *038 423 805 183 558 930 298 664 689 *077 461 843 221 595 967 335 700 727 *115 600 881 258 633 *004 372 737 766 *164 638 918 296 670 *041 408 773 806 *192 576 966 333 707 *078 446 809 844 *231 614 994 371 744 *115 482 846 883 *269 652 *032 408 781 *151 618 882 1 2 3 4 5 6 7 8 9 41 4.1 8.2 12.3 16.4 20.6 24.6 28.7 32.8 36.9 40 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 39 3.9 7.8 11.7 15.6 19.5 23.4 27.3 31.2 35.1 120 918 954 990 *027 *063 *099 *135 *171 *207 *243 1 21 22 23 24 25 26 27 28 29 08279 636 991 09342 691 10037 380 721 11059 314 672 *026 377 726 072 415 755 093 350 707 *061 412 760 106 449 789 126 386 743 *096 447 795 140 483 823 160 422 778 *132 482 830 175 517 857 193 458 814 *167 517 864 209 551 890 227 493 849 *202 562 899 243 685 924 261 529 884 *237 587 934 278 619 968 294 665 920 *272 621 968 312 653 992 327 600 965 *307 656 *003 346 687 *026 361 1 2 3 4 5 6 7 8 9 38 3.8 7.6 11.4 16.2 19.0 22.8 26.6 30.4 34.2 37 3.7 7.4 11.1 14.8 18.5 22.2 26.9 29.6 33.3 36 3.6 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32.4 130 394 428 461 494 528 561 694 628 661 694 1 31 32 33 34 35 36 37 38 39 727 12057 385 710 13033 354 672 988 14301 760 090 418 743 066 386 704 *019 333 793 123 450 775 098 418 735 *051 364 826 156 483 808 130 450 767 *082 395 860 189 516 840 162 481 799 *114 426 893 222 548 872 194 513 830 *145 457 926 264 681 905 226 646 862 *176 489 959 287 613 937 258 577 893 *208 520 992 320 646 969 290 609 926 *239 551 *024 352 678 *001 322 640 956 *270 682 891 1 2 3 4 6 6 7 8 9 35 3.6 7.0 10.6 14.0 17.6 21.0 24.5 28.0 31.5 34 3.4 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 33 3.3 6.6 9.9 13.2 16.5 19.8 23.1 26.4 29.7 140 613 644 675 706 737 768 799 829 860 1 41 42 43 44 45 46 47 48 49 922 15229 534 836 16137 435 732 17026 319 953 259 564 866 167 465 761 056 348 983 290 594 897 197 495 791 085 377 *014 320 625 927 227 524 820 114 406 *045 351 655 957 256 654 850 143 435 *076 381 685 987 286 684 879 173 464 *106 412 716 *017 316 613 909 202 493 *137 442 746 *047 346 643 938 231 522 *168 473 776 *077 376 673 967 260 661 *198 503 806 *107 406 702 997 289 680 1 2 3 4 6 6 7 8 9 32 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 31 3.1 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 30 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 150 609 638 667 696 725 754 782 811 840 869 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 11 150- - Logarithms of Numbers — 200 3 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 150 17 609 638 667 696 725 754 782 811 840 869 51 898 926 955 984 *013 *041 *070 *099 *127 *156 52 18184 213 241 270 298 327 355 384 412 441 53 469 498 526 554 583 611 639 667 696 724 54 752 780 808 837 ^65 893 921 949 977 *005 55 19 033 061 089 117 145 173 201 229 257 285 56 312 340 368 396 424 451 479 507 535 562 57 590 618 645 673 700 728 756 783 811 838 58 866 893 921 948 976 *003 *030 *058 *085 *112 59 20140 167 194 222 249 276 303 330- 358 385 160 412 439 466 493 620 548 575, 602 629 656 61 683 710 737 763 790 817 844 871 898 925 29 28 27 62 952 978 *005 *032 *059 *085 *112 *139 *165 *192 1 2.9 2.8 2.7 63 21219 245 272 299 325 352 378 405 431 458 2 5.8 5.6 5.4 64 484 511 537 564 590 617 643 669 696 722 3 8.7 8.4 8.1 65 748 775 801 827 854 880 906 932 958 985 4 11.6 11.2 10.8 m 22 011 037 063 089 115 141 167 194 220 246 5 14.5 17.4 14.0 16.8 13.5 16.2 67 272 298 324 350 376 401 427 453 479 505 7 20.3 19.6 18.9 68 531 557 583 608 634 660 686 712 737 763 8 23.2 22.4 21.6 69 789 814 840 866 891 917 943 968 994 *019 9 26.1 25.2 24.3 170 23045 070 096 121 147 172 198 223 249 274 1 71 300 325 350 376 401 426 452 477 502 528 26 25 24 72 553 578 603 629 654 679 704 729 754 779 1 2.6 2.5 5.0 2.4 73 805 830 855 880 905 930 955 980 *005 *030 2 5.2 4.'8 74 24 055 080 105 130 155 180 204 229 254 279 ^ 7.8 7.5 7.2 75 304 329 353 378 403 650 428 452 477 502 527 4 10.4 10.0 9.6 76 551 576 601 625 674 699 724 748 773 5 13.0 12.5 12.0 6 15.6 15.0 14.4 77 797 822 846 871 895 920. 944 969 993 *018 7 18.2 17.5 16.8 78 25 042 066 091 115 139 164 188 212 237 261 8 20.8 20.0 19.2 79 285 310 334 358 382 406 431 455 479 503 9 23.4 22.5 21.6 180 527 551 575 600 624 648 672 696 720 744 1 81 768 792 816 840 864 888 912 935 959 983 23 22 21 82 26 007 031 055 079 102 126 150 174 198 221 1 2 2.3 4.6 2.2 4.4 2.1 4.2 83 245 269 293 316 340 364 387 411 435 458 84 482 505 529 553 576 600 623 647 670 694 3 6.9 6.6 6.3 85 717 741 764 788 811 834 858 881 905 928 4 9.2 8.8 8.4 86 951 975 998 *021 *045 *068 *091 *114 *138 *161 5 6 11.5 13.8 11.0 13.2 10.5 12.6 87 27184 207 231 254 277 300 323 346 370 393 7 16.1 15.4 14.7 88 416 439 462 485 508 531 554 577 600 623 8 18.4 17.6 16.8 89 646 669 692 715 ,738 761 784 807 830 852 9 20.7 19.8 18.9 190 875 898 921 944 967 989 *012 *035 *058 *081 91 28103 126 149 171 194 217 240 262 285 307 92 330 353 375 398 421 443 466 488 511 533 93 556 57« 601 623 646 668 691 713 735 758 94 780 803 825 847 870 892 914 937 959 981 95 29003 026 048 070 092 115 137 159 181 203 96 226 248 270 292 314 336 358 380 403 425 97 447 469 491 513 535 557 579 601 623 645 98 667 688 710 732 754 776 798 820 842 863 99 885 907 929 951 973 994 *016 *038 *060 *081 200 30103 125 146 168 190 211 233 255 276 298 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 1 200 — Logarithms of Numbers — 250 N. 8 9 Prop. Pts. 200 210 220 230 31 32 33 34 35 36 37 38 39 240 41 42 43 44 45 46 47 48 49 250 30103 125 146 168 190 211 233 255 276 298 320 535 750 963 31175 387 597 806 32 015 341 557 771 984 197 408 618 827 035 363 578 792 *006 218 429 639 848 056 384 600 814 *027 239 450 660 869 077 406 621 835 *048 260 471 681 890 098 428 643 856 *069 281 492 702 911 118 449 664 878 *091 302 613 723 931 139 471 685 899 *112 323 534 744 952 160 492 707 920 *133 345 555 765 973 181 514 728 942 *154 366 576 785 994 201 222 243 263 284 305 346 366 387 408 428 634 838 33041 244 445 646 846 34044 449 654 858 062 264 465 666 866 064 469 675 879 082 284 486 686 885 084 490 695 899 102 304 506 706 905 104 510 715 919 122 325 526 726 925 124 531 736 940 143 345 546 746 945 143 552 756 960 163 365 666 766 965 163 572 777 980 183 385 586 786 985 183 593 797 *001 203 405 606 806 *005 203 613 818 *021 224 425 626 826 *025 223 242 262 282 301 321 341 361 380 400 420 439 635 830 35025 218 411 603 793 984 459 655 850 044 238 430 622 813 *003 479 674 869 064 257 449 641 832 *021 498 694 889 083 276 468 660 851 *040 518 713 908 102 295 488 679 870 *059 537 733 928 122 315 507 698 889 *078 557 753 947 141 334 526 717 908 *097 677 772 967 im 353 545 736 927 *116 596 792 986 180 372 564 755 946 *135 616 811 *005 199 392 583 774 965 *154 36173 192 211 229 248 267' 286 305 324 342 361 549 736 922 37107 291 475 658 840 380 568 754 940 125 310 493 676 858 399 586 773 959 144 328 511 694 876 418 605 791 977 162 346 530 712 894 436 624 810 996 181 365 548 731 912 455 642 829 *014 199 383 566 749 931 474 661 847 *033 218 401 585 767 949 493 680 866 *051 236 420 603 785 967 511 698 884 *070 254 438 621 803 985 530 717 903 *088 273 457 639 822 *003 38021 039 057 075 093 112 130 148 166 184 202 382 561 739 917 39094 270 445 620 794 220 399 578 757 934 111 287 463 637 811 238 417 596 775 952 129 305 480 655 829 256 435 614 792 970 146 322 498 672 846 274 453 632 810 987 164 340 515 690 863 292 471 650 828 *005 182 358 533 707 881 310 489 668 846 *023 199 375 550 724 898 328 507 686 863 *041 217 393 568 742 915 346 525 703 881 *058 235 410 585 759 933 364 543 721 *076 252 428 602 777 950 log 2 =.30102 99566 22 21 2.2 2.1 4.4 4.2 G.a 6.3 8.8 8.4 11.0 10.5 13.2 12.6 14.7 15.4 17.6 16.8 19.8 18.9 19 18 1.9 1.8 3.8 3.6 5.7 5.4 7.6 7.2 9.5 9.0 11.4 10.8 13.3 12.6 15.2 14.4 17.1 16.2 Prop. Pts. I] 250- - Logarithms of Numbers » — 300 5 N. 1 2 3 4 3 6 7 8 9 Prop. Pts. 250 39 794 811 829 846 863 881 898 915 933 950 51 967 985 *002 *019 *037 *054 *071 *088 *106 *123 52 40140 157 175 192 209 226 243 261 278 295 53 312 329 346 364 381 398 415 432 449 466 54 483 500 518 635 552 569 586 603 620 637 55 654 671 688 705 722 739 756 773 790 807 56 824 841 858 875 892 909 926 943 960 976 57 993 *010 *027 *044 *061 *078 *095 nil *128 *145 58 41162 179 196 212 229 246 263 280 296 313 59 330 347 363 380 397 414 430 447 464 481 260 497 514 531 547 5(54 581 597 614 631 647 61 664 681 697 714 731 747 764 780 797 814 18 17 16 62 830 847 863 880 896 913 929 946 963 979 1 1.8 1.7 1.6 63 996 *012 *029 *045 *062 *078 *095 nil *127 *144 2 3.6 3.4 3.2 64 42160 177 193 210 226 243 259 275 292 308 3 4 5 6 5.4 7.2 9.0 10.8 6.1 6.8 8.5 10.2 4.8 6.4 8.0 9.6 65 325 Ml 357 374 390 406 423 439 455 472 66 488 504 521 537 553 570 586 602 619 635 67 651 667 684 700 716 732 749 765 781 797 7 12.6 11.9 11.2 68 813 830 846 862 878 894 911 927 943 959 8 14.4 13.6 12.8 69 975 991 *008 *024 *040 *056 *072 *088 *104 *120 9 16.2 15.3 14.4 270 43136 152 169 185 201 217 233 249 265 281 71 297 313 329 345 361 377 393 409 425 441 72 457 473 489 505 521 537 553 569 584 600 73 616 632 648 664 680 696 712 727 743 759 M=\ogioe 74 775 791 807 823 838 854 870 886 902 917 = logio2.718... = .4342944819 75 933 949 965 981 996 *012 *028 *044 *059 *075 76 44091 107 122 138 154 170 185 201 217 232 77 248 264 279 295 311 326 342 358 373 389 78 404 420 436 451 467 483 498 514 529 545 79 560 576 592 607 623 638 654 669 685 700 280 716 731 747 762 778 793 809 824 840 855 81 871 886 902 917 932 948 963 979 994 *010 15 14 82 45 025 040 056 071 086 102 117 133 148 163 1 2 1.5 3.0 1.4 2.8 83 179 194 209 225 240 255 271 286 301 317 84 332 347 362 378 393 408 423 439 454 469 3 4.5 4.2 85 484 500 515 530 545 561 576 591 606 621 4 6.0 5.6 86 637 652 667 682 697 712 728 743 758 773 5 6 7 7.5 9.0 10.5 7.0 8.4 9.8 87 788 803 818 834 849 864 879 894 909 924 88 939 954 969 984 *000 *015 *030 *045 *060 *075 8 12.0 11.2 89 46 090 105 120 135 150 165 180 195 210 225 9 13.5 12.6 290 240 255 270 285 300 315 330 345 359 374 91 389 404 419 434 449 464 479 494 509 523 92 538 553 568 583 598 613 627 642 657 672 93 687 702 716 731 746 -761 776 790 805 820 94 835 850 8(54 879 894 909 923 938 953 967. 95 982 997 *012 *026 *041 *056 *070 *085 *100 ni4 96 47129 144 159 173 188 202 217 232 246 261 97 276 290 305 319 334 349 363 378 392 407 98 422 436 451 465 480 494 509 524 538 553 _99 567 582 596 611 625 640 654 669 683 698 300 712 727 741 756 770 784 799 813 828 842 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 6 300- - Logarithms of Numbers -350 D N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 300 47 712 727 741 756 770 784 799 813 828 842 01 02 03 857 48 001 144 871 015 159 885 029 173 900 044 187 914 058 202 929 073 216 943 087 230 958 101 244 972 116 259 986 130 273 04 05 06 287 430 572 302 444 586 316 458 601 330 473 615 344 487 629 359 501 643 373 615 657 387 530 671 401 544 686 416 558 700 log 3 =.47712 12547 log 77= .4971498727 07 . 08 09 714 855 996 728 869 *010 742 883 *024 756 897 *038 770 911 *052 785 926 *066 799 940 *080 813 954 *094 827 968 *108 841 982 *122 310 49136 150 164 178 192 206 220 234 248 262 11 12 13 276 415 554 290 429 568 304 443 582 318 457 596 332 471 610 346 485 624 360 499 638 374 613 651 388 527 665 402 541 679 1 2 3 4 6 6 10 1.5 3.0 4.5 6.0 7.5 9.0 14 1.4 2.8 4.2 5.6 7.0 8.4 14 15 16 693 831 969 707 845 982 721 859 996 734 872 *010 748 886 *024 762 900 *037 776 914 *051 790 927 *065 803 941 *079 817 955 *092 17 18 19 50106 243 379 120 256 393 133 270 406 147 284 420 161 297 433 174 311 447 188 325 461 202 338 474 215 352 488 229 365 501 7 8 9 10.5 12.0 13.5 9.8 11.2 12.6 320 515 529 542 556 569 583 596 610 623 637 21 22 23 651 786 920 664 799 934 678 813 947 691 826 961 705 840 974 718 853 987 732 866 *001 745 880 *014 759 893 *028 772 907 *041 24 25 26 51055 188 322 068 202 335 081 215 348 095 228 362 108 242 375 121 255 388 135 268 402 148 282 415 162 295 428 175 308 441 27 28 29 455 587 720 468 601 733 481 614 746 495 627 759 508 640 772 521 654 786 534 667 799 648 680 812 561 693 825 574' 706 838 330 851 865 878 891 904 917 930 943 957 970 31 32 33 983 52114 244 996 127 257 *009 140 270 *022 153 284 *035 166 297 *048 179 310 *061 192 323 *075 205 336 *088 218 349 *101 231 362 1 2 13 1.3 2.6 12 1.2 2.4 3.6 4.8 6.0 7.2 34 35 36 375 504 634 388 517 647 401 530 660 414 543 673 427 556 686 440 569 699 453 582 711 466 595 724 479 608 737 492. 621 750 3 4 6 6 3.9 5.2 6.5 7.8 37 38 39 763 892 53020 776 905 033 789 917 046 802 930 058 815 943 071 827 956 084 840 969 097 853 982 110 866 994 122 879 *007 135 7 8 9 9.1 10.4 11.7 8.4 9.6 10.8 340 148 161 173 186 199 212 224 237 250 263 41 42 43 275 403 529 288 415 542 301 428 555 314 441 567 326 453 580 339 4(56 593 352 479 605 364 491 618 377 504 631 390 517 643 44 45 46 656 782 908 668 794 920 681 807 933 694 820 945 706 832 958 719 845 970 732 857 983 744 870 995 757 882 *008 769 895 *020 47 48 49 54033 158 283 045 170 295 058 183 307 070 195 320 083 208 332 095 220 345 108 233 357 120 245 370 133 258 382 145 270 394 350 407 419 432 444 456 469 481 494 506 518 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. I] 350- -Logarithms of Numbers — 400 7 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 350 54 407 419 432 444 456 469 481 494 506 518 51 ,52 53 531 654 777 543 667 790 555 679 802 568 691 814 580 704 827 593 716 839 605 728 851 617 741 864 630 753 876 (542 765 888 54 55 56 900 55 023 145 913 035 157 925 047 169 937 060 182 949 072 194 962 084 206 974 096 218 986 108 230 998 121 242 *011 133 255 57 58 59 360 267 388 509 279 400 522 291 413 534 303 425 546 315 437 558 328 449 570 340 461 582 352 473 594 364 485 606 376 497 618 630 642 654 6m 678 691 703 715 727 739 (51 ()2 (53 751 871 991 763 883 *003 775 895 *015 787 907 *027 799 919 *038 811 931 *050 823 943 *062 835 955 *074 847 967 *086 859 979 *098 1 2 13 1.3 2.6 12 1.2 2.4 (34 (55 m 56110 229 348 122 241 360 134 253 372 146 265 384 158 277 396 170 289 407 182 301 419 194 312 431 205 324 443 217 336 455 3 4 5 6 3.9 5.2 6.5 7 8 3.6 4.8 6.0 7 2 (57 (58 69 467 585 703 478 597 714 490 608 726 502 620 738 514 632 750 526 644 761 538 656 773 549 667 785 561 679 797 573 691 808 7 8 9 9.1 10.4 11.7 8.4 9.6 10.8 370 820 832 844 855 867 879 891 902 914 926 71 72 73 937 57 054 171 949 066 183 961 078 194 972 089 206 984 101 217 9fX) 113 229 *008 124 241 *019 136 252 *031 148 264 *043 159 276 74 75 76 287 403 519 299 415 530 310 426 542 322 438 553 334 449 565 345 4(51 576 357 473 588 368 484 600 380 4% 611 392 507 623 77 78 79 634 749 864 646 761 875 657 772 887 669 784 898 680 795 910 692 807 921 703 818 933 715 830 944 726 841 955 738 852 967 380 978 990 *001 *013 *024 *035 *047 *058 *070 *081 81 82 83 58 092 206 320 104 218 331 115 229 343 127 240 354 138 252 365 149 263 377 161 274 388 172 286 399 184 297 410 195 309 422 1 2 11 1.1 10 1.0 2.0 84 85 86 433 546 659 444 557 670 456 569 681 467 580 692 478 591 704 490 602 715 501 614 726 512 625 737 524 636 749 535 647 760 3 4 5 6 7 8 9 3.3 4.4 5.5 6.6 7.7 8.8 9.9 3.0 4.0 5.0 6.0 7.0 8.0 9.0 87 88 89 771 883 995 782 894 *006 794 906 *017 805 917 *028 816 928 *040 827 939 *051 838 950 *062 850 961 *073 861 973 *084 872 984 *095 390 59106 118 129 140 151 162 173 184 195 207 91 92 93 218 329 439 229 340 450 240 351 461 251 362 472 262 373 483 273 384 494 284 395 506 295 406 517 306 417 528 318 428 539 94 95 96 550 660 770 561 671 780 572 682 791 583 693 802 594 704 813 605 715 824 616 726 835 627 737 846 638 748 857 649 759 868 ,97 98 99 879 988 60 097 890 999 108 901 *010 119 912 *021 130 923 *032 141 934 *043 152 945 *054 163 956 *065 173 966 *076 184 977 *086 195 400 206 217 228 239 249 260 271 282 293 304 ^. 1 2 3 4 5 6 7 8 9 Prop. Pts. 400 — Logarithms of Numbers — 450 N. 1 2 3 4 6 6 7 8 9 Prop. Pts. 400 60 206 217 228 239 249 260 271 282 293 304 01 02 03 04 05 06 07 08 09 314 423 531 638 746 853 959 61066 172 325 433 541 649 756 863 970 077 183 336 444 552 660 767 874 981 087 194 347 455 563 670 778 885 991 098 204 358 466 574 681 788 895 *002 109 215 369 477 584 692 799 906 *013 119 225 379 487 595 703 810 917 *023 130 236 390 498 606 713 821 927 *034 140 247 401 509 617 724 831 938 *045 151 257 412 520 627 735 842 949 *055 162 268 410 278 289 300 310 321 331 342 352 363 374 11 12 13 14 15 16 17 18 19 384 490 595 700 805 909 62014 118 221 395 500 606 711 815 920 024 128 232 405 511 616 721 826 930 034 138 242 416 521 627 731 836 941 045 149 252 426 532 637 742 847 951 055 159 263 437 542 648 752 857 962 066 170 273 448 553 658 763 868 972 076 180 284 458 563 669 773 878 982 086 190 294 469 574 679 784 888 993 097 201 304 479 584 690 794 899 *003 107 211 315 420 325 335 346 356 366 377 387 397 408 418 21 22 23 24 25 26 27 28 29 428 531 634 737 839 941 63043 144 246 439 542 644 747 849 951 053 155 256 449 552 655 757 859 961 063 165 266 459 562 665 767 870 972 073 175 276 469 572 675 778 880 982 083 185 28(3 480 583 685 788 992 094 195 296 490 593 696 798 900 *002 104 205 306 500 603 706 808 910 *012 114 215 317 511 613 716 818 921 *022 124 225 327 521 624 726 829 931 *033 134 236 337 1 2 3 4 5 6 7 8 9 11 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 10 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 9 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 430 347 357 367 377 387 397 407 417 428 438 lo^if=log[loge] = 9.63778431 — 10 31 32 33 34 35 36 37 38 39 448 548 649 749 849 949 64048 147 246 458 558 659 759 859 959 058 157 256 468 568 669 769 869 969 068 167 266 478 579 679 779 879 979 078 177 276 488 589 689 789 889 988 088 187 286 498 599 699 799 899 998 098 197 296 508 609 709 809 909 *008 108 207 306 518 619 719 819 919 *018 118 217 316 528 629 729 829 929 *028 128 227 326 538 639 739 839 939 *038 137 237 335 440 345 355 365 375 385 395 404 414 424 434 41 42 43 44 45 46 47 48 49 444 542 640 738 836 933 65031 128 225 454 552 650 748 846 943 040 137 234 464 562 660 758 856 953 050 147 244 473 572 670 768 865 963 060 157 254 483 582 680 777 875 972 070 167 263 493 591 689 787 885 982 079 176 273 503 601 699 797 895 992 089 186 283 513 611 709 807 904 *002 099 196 292 523 621 719 816 914 *ou 108 205 302 532 631 729 826 924 *021 118 215 312 450 321 331 341 350 360 369 379 389 398 408 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. u 450- - Logarithms Of Numbers — 500 9 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 450 65 321 331 341 ;350 360 3()9 379 389 398 408 51 52 53 418 514 610 427 523 619 437 533 629 447 543 639 456 552 648 466 562 658 475 571 667 485 581 677 495 591 686 504 600 696 54 55 56 706 801 896 715 811 906 725 820 916 734 830 925 744 839 935 753 849 944 763 858 954 772 868 963 782 877 973 792 887 982 57 58 59 992 66 087 181 *001 096 191 *011 106 200 *020 115 210 *030 124 219 *039 134 229 *049 143 238 *058 153 247 *068 162 257 *077 172 266 460 276 285 295 304 314 323 332 342 351 361 61 62 63 370 464 558 380 474 567 389 483 577 398 492 586 408 502 596 417 511 605 427 521 614 436 530 624 445 539 633 455 549 642 64 65 m 652 745 839 661 755 848 671 764 857 680 773 867 689 783 876 699 792 885 708 801 894 717 811 904 727 820 913 736 829 922 67 68 69 932 67 025 117 941 034 127 950 043 136 960 052 145 969 062 154 978 071 164 987 080 173 997 089 182 *006 191 *015 108 201 470 210 219 228 237 247 256 265 274 284 293 71 72 73 302 394 486 311 403 495 321 413 504 330 422 514 339 431 523 348 440 532 357 449 541 367 459 550 376 468 560 385 477 569 1 2 10 1.0 2.0 9 0.9 1.8 8 0.8 1.6 74 75 76 578 669 761 587 679 770 596 688 779 605 697 788 614 706 797 624 715 806 633 724 815 642 733 825 651 742 834 660 7r)2 843 3 4 5 6 3.0 4.0 5.0 60 2.7 3.6 4.5 5 4 2.4 3.2 4.0 4.8 77 78 79 852 943 68 034 861 952 043 870 961 052 879 970 061 888 979 070 897 988 079 906 997 088 916 *006 097 925 *015 10(5 934 *024 115 7 8 9 7.0 8.0 9.0 6.3 7.2 8.1 5.6 6.4 7.2 480 124 133 142 151 1(30 169 178 187 196 205 81 82 83 215 305 395 224 314 404 233 323 413 242 332 422 251 341 431 260 350 440 269 359 449 278 368 458 287 377 467 296 386 476 84 85 86 485 574 664 494 583 673 502 592 681 nil 601 690 520 610 699 529 619 708 538 628 717 547 637 726 556 646 735 565 655 744 87 88 89 753 842 931 762 851 940 771 860 949 780 869 958 789 878 966 797 886 975 806 895 984 815 904 993 824 913 *002 833 922 *011 490 69 020 028 037 046 055 064 073 082 090 099 91 92 93 108 197 285 117 205 294 126 214 302 135 223 311 144 232 320 152 241 329 161 249 338 170 258 346 179 267 355 188 276 364 94 95 96 373 461 548 381 469 557 390 478 566 399 487 574 408 496 583 417 504 592 425 513 601 434 522 609 443 531 618 452 539 627 97^' 98 99 636 723 810 644 732 819 653 740 827 662 749 836 671 758 845 679 767 854 688 775 862 697 784 871 705 793 880 714 801 888 500 897 906 914 923 932 940 949 958 966 975 N. ' 1 2 3 4 5 6 7 1 8 9 Prop. Pts. 10 500- - Logarithms of Numbers -550 [I N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 500 69 897 906 914 923 932 940 949 958 966 975 log 5= .6989700043 01 02 03 984 70 070 157 992 079 165 *001 088 174 *010 096 183 *018 105 191 *027 114 200 *036 122 209 *044 131 217 *053 140 226 *062 148 234 04 05 06 243 329 415 252 338 424 260 346 432 269 355 441 278 364 449 286 372 458 295 381 467 303 389 475 312 398 484 321 406 492 07 08 09 501 586 672 509 595 680 518 689 52(7 612 697 535 621 706 544 629 714 552 638 723 661 646 731 .569 655 740 578 663 749 510 757 766 774 783 791 800 808 817 825 834 11 12 13 842 927 71012 851 935 020 859 944 029 868 952 037 876 961 046 885 969 054 893 978 063 902 986 071 910 995 079 919 *003 088 14 15 16 096 181 265 105 189 273 113 198 282 122 206 290 130 214 299 139 223 307 147 231 315 155 240 324 164 248 332 172 257 341 17 18 19 349 433 517 357 441 525 366 450 533 374 458 542 383 466 550 391 475 559 399 483 567 408 492 575 416 500 584 425 508 592 520 600 609 617 625 634 642 650 659 667 675 21 22 23 684 767 850 692 775 858 700 784 867 709 792 875 717 800 883 725 809 892 734 817 900 742 825 908 750 834 917 759 842 925 1 2 9 0.9 1.8 8 0.8 1.6 7 0.7 1.4 24 25 26 933 72016 099 941 024 107 950 032 115 958 041 123 966 049 132 975 057 140 983 066 148 991 074 156 999 082 165 *008 O^X) 173 3 4 5 6 2.7 3.6 4.5 5 4 2.4 3.2 4.0 4 8 2.1 2.8 3.5 4.2 4.9 5.6 6.3 27 28 29 181 263 346 189 272 354 198 280 362 206 288 370 214 296 378 222 304 387 230 313 395 239 321 403 247 329 411 255 337 419 7 8 9 6.3 7.2 8.1 5.6 6.4 7.2 530 428 436 444 452 460 469 477 485 493 501 31 32 33 509 591 673 518 599 681 526 607 689 534 616 697 542 624 705 550 .'532 713 558 640 722 567 648 730 575 656 738 683 665 746 34 35 36 754 835 916 762 843 925 770 852 933 779 860 941 787 868 949 795 876 957 803 884 965 811 892 973 819 900 981 827 908 989 37 38 39 997 73078 159 *006 086 167 *014 094 175 *022 102 183 *030 111 191 *038 119 199 *046 127 207 *054 135 215 *062 143 223 *070 151 231 540 239 247 255 263 272 280 288 296 304 312 41 42 43 320 400 480 328 408 488 336 416 496 344 424 504 352 432 512 360 440 520 368 448 528 376 456 536 384 464 544 392 472 552 44 45 46 560 640 719 568 648 727 576 656 735 584 664 743 592 672 751 600 679 759 608 687 767 616 695 775 624 703 783 632 711 791 47 48 49 799 878 957 807 886 965 815 894 973 823 902 981 830 910 989 838 918 997 846 926 *005 854 933 *013 862 941 *020 870 949 *028 650 74036 044 052 060 068 076 084 092 099 107 N. 1 1 2 1 3 4 5 6 7 8 9 Prop. Pts* q 550- -Logarithms of Numbers — 600 11 N= 1 2 3 4 5 6 7 8 9 Prop. Pts. 560 74 036 044 052 060 068 076 084 092 099 107 51 52 53 115 194 273 123 202 280 131 210 288 139 218 296 147 225 304 155 233 312 162 241 320 170 249 327 178 257 335 186 265 343 54 55 56 351 429 507 359 437 515 367 445 523 374 453 531 382 461 539 390 468 547 398 476 554 406 484 562 414 492 570 421 500 578 57 58 59 586 663 741 593 671 749 601 679 757 609 687 764 617 695 772 624 702 780 632 710 788 640 718 796 648 726 803 656 733 811 560 819 827 834 842 850 858 865 873 881 889 61 62 63 896 974 75 051 904 981 059 912 989 066 920 997 074 927 *005 082 935 *012 089 943 *020 097 950 *028 105 958 *035 113 966 *043 120 64 65 m 128 205 282 136 213 289 143 220 297 151 228 305 159 236 312 166 243 320 174 251 328 182 259 335 189 266 343 197 274 351 67 68 69 358 435 511 366 442 519 374 450 526 381 458 534 389 465 542 397 ,473 549 404 481 557 412 488 565 420 496 572 427 504 580 570 587 595 603 610 618 62() 633 641 648 656 71 72 73 6()4 740 815 671 747 823 679 755 831 686 762 838 694 770 846 702 778 853 709 785 861 717 793 868 724 800 876 732 808 884 1 2 8 0.8 1.6 7 0.7 1.4 74 75 76 891 967 76042 899 974 050 906 982 057 914 989 065 921 997 072 929 *005 080 937 *012 087 944 *020 095 952 *027 103 959 *035 110 3 4 5 6 2.4 3.2 4.0 4.8 2.1 2.8 3.6 4.2 77 78 79 580 118 193 268 125 200 275 133 208 283 140 215 290 148 223 298 155 2;30 305 163 238 313 170 245 320 178 253 328 185 260 335 7 8 9 6.6 6.4 7.2 4.9 5.6 6.3 343 350 358 365 373 380 388 395 403 410 81 82 83 418 492 567 425 500 574 433 507 682 440 515 589 448 522 597 455 530 604 462 537 612 470 545 619 477 552 626 485 559 634 84 85 86 641 716 790 649 723 797 656 730 805 664 738 812 671 745 819 678 753 827 686 760 834 693 768 842 701 775 849 708 782 856 87 88 89 864 938 77 012 871 945 019 879 953 026 886 960 034 893 967 041 901 975 048 908 982 0:>6 916 989 063 923 997 070 930 *004 078 590 085 093 100 107 115 122 129 137 144 151 91 92 93 159 232 305 166 240 313 173 247 320 181 254 327 188 262 335 195 269 342 203 276 349 210 283 357 217 291 364 225 298 371 94 95 96 379 452 525 386 459 532 393 466 539 401 474 546 408 481 554 415 488 561 422 495 568 430 503 576 437 510 583 444 517 690 97. 98 99 597 670 743 605 677 750 612 685 757 619 692 764 627 699 772 634 706 779 641 714 786 648 721 793 656 728 801 663 735 808 600 815 822 830 837^ 844 851 859 866 873 880 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 12 600- -Logarithms of Numbers — 650 [1 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 600 77 815 822 830 837 844 851 859 866 873 880 01 02 03 887 960 78032 895 967 039 902 974 046 909 981 053 916 988 061 924 996 068 931 *003 075 938 *010 082 945 *017 089 952 *025 097 04 05 06 104 176 247 111 183 254 118 l^X) 262 125 197 269 132 204 276 140 211 283 147 219 290 154 226 297 161 233 305 168 240 312 07 08 09 319 390 462 326 398 469 333 405 476 aio 412 483 347 419 490 355 426 497 362 433 504 369 440 512 376 447 519 383 455 526 610 533 540 547 554 561 569 576 583 590 597 11 12 13 604 675 746 611 682 753 618 689 760 625 696 767 633 704 774 640 711 781 647 718 789 654 725 796 661 732 803 668 739 810 14 15 16 817 888 958 824 895 965 831 902 972 838 909 979 845 916 986 852 923 993 859 930 *000 866 937 *007 873 944 *014 880 951 *021 17 18 19 79029 099 169 036 106 176 043 113 183 050 120 190 057 127 197 064 134 204 071 141 211 078 •148 218 085 155 225 092 162 232 620 239 246 253 260 267 274 281 288 295 302 21 22 23 309 379 449 316 386 456 323 393 463 330 400 470 337 407 477 344 414 484 351 421 491 358 428 498 365 435 505 372 442 511 1 2 8 0.8 1.6 7 0.7 1.4 6 0.6 1.2 24 25 26 518 588 657 525 595 664 532 602 671 539 609 678 546 616 685 553 623 692 560 630 699 567 637 706 574 644 713 581 650 720 3 4 5 6 2.4 3.2 4.0 48 2.1 2.8 3.5 4 2 1.8 2.4 3.0 3.6 27 28 29 727 796 865 734 803 872 741 810 879 748 817 886 754 824 893 761 831 900 768 837 906 775 844 913 782 851 920 789 858 927 7 8 9 5.6 6.4 7.2 4.9 5.6 6.3 4.2 4.8 5.4 630 934 941 948 955 962 969 975 982 989 996 31 32 33 80003 072 140 010 079 147 017 085 154 024 092 161 030 099 168 037 106 175 044 113 182 051 120 188 058 127 195 065 134 202 34 35 36 209 277 346 216 284 353 223 291 359 229 298 366 236 305 373 243 312 380 250 318 387 257 325 393 264 532 400 271 339 407 37 38 39 414 482 550 421 489 557 428 496 564 434 502 570 441 509 577 448 516 584 455 523 591 462 530 598 468 536 604 475 543 611 640 618 625 632 638 645 652 659 665 672 679 41 42 43 686 754 821 693 760 828 699 767 835 706 774 841 713 781 848 720 787 855 726 794 862 733 801 868 740 808 875 747 814 882 44 45 46 889 956 81023 895 963 030 902 969 037 909 976 043 916 983 050 922 990 057 929 996 064 936 *003 070 943 *010 077 949 *017 084 47 48 49 090 158 224 097 164 231 104 171 238 111 178 245 117 184 251 124 191 258 131 198 265 137 204 271 144 211 278 151 218 285 650 291 298 305 311 318 325 331 338 345 351 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. i] 650- -Logarithms of Numbers — 700 13 N. 1 2 3 4 5 6 7 8 9 Prop. Pt6. 650 81291 298 305 311 318 325 331 338 345 351 51 358 365 371 378 385 391 398 405 411 418 52 425 431 438 445 451 458 465 471 478 485 53 491 498 505 511 518 625 631 538 544 651 54 558 564 571 578 584 591 598 604 611 617 55 624 631 637 644 651 657 664 671 677 684 56 690 697 704 710 717 723 730 737 743 750 57 757 763 770 776 783 790 796 803 809 816 58 823 829 836 842 849 856 862 869 875 882 59 660 889 895 902 908 915 921 928 935 941 948 954 961 968 974 981 987 994 *000 *007 *014 Gl 82020 027 033 040 046 053 060 066 073 079 ()2 086 092 099 105 112 119 125 132 138 145 63 151 158 164 171 178 184 191 197 204 210 64 217 223 230 236 243 249 256 263 269 276 65 282 289 295 302 308 315 321 328 334 341 m 347 354 360 367 373 380 387 393 400 406 67 413 419 426 432 439 445 452 458 465 471 68 478 484 491 497 504 510 517 623 530 536 69 643 549 556 562 569 575 582 688 595 601 670 607 614 620 627 633 640 646 653 659 666 71 672 679 685 692 698 705 711 718 724 730 7 6 72 737 743 750 756 763 769 776 782 789 795 1 0.7 0.6 73 802 808 814 821 827 834 840 847 853. 860 2 1.4 1.2 74 866 872 879 885 892 898 905 911 918 924 3 4 5 6 2.1 2.8 3.5 4.2 1.8 2.4 3.0 3.6 75 930 937 943 950 956 963 969 975 982 988 76 995 *001 *008 *014 *020 *027 *033 *040 *046 *052 77 83059 065 072 078 085 091 097 104 110 117 7 4.9 4.2 78 123 129 136 142 149 155 161 168 174 181 8 5.6 4.8 79 187 193 200 206 213 219 225 232 238 245 9 6.3 5.4 680 81 251 257 264 270 276 283 289 296 302 308 315 321 327 334 340 347 353 359 366 372 82 378 385 391 398 404 410 417 423 429 436 83 442 448 455 461 467 474 480 487 493 499 84 506 512 518 525 531 537 544 550 556 563 85 569 575 582 588 594 601 607 613 620 626 86 632 639 645 651 658 664 670 677 683 689 87 696 702 708 715 721 727 734 740 746 753 88 759 765 771 778 784 790 797 803 809 816 89 822 828 835 841 847 853 860 866 872 879 690 885 891 897 904 910 916 923 929 935 942 91 948 954 960 967 973 979 985 992 998 *004 92 84 011 017 023 029 036 042 048 055 061 067 93 073 080 086 092 098 105 111 117 123 130 94 136 142 148 155 161 167 173 180 186 192 95 198 205 211 217 223 230 236 242 248 255 96 261 267 273 280 286 292 298 305 311 317 97 323 330 336 342 348 354 361 367 373 379 98 386 392 398 404 410 417 423 429 435 442 99 448 454 460 466 473 479 485 491 497 504 700 510 516 522 528 535 541 547 553 559 566 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 14 7 00- - Logarithms of Numbers -760 [I N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 700 84 510 516 522 528 535 541 547 553 559 566 log 7- .8450980400 01 02 03 572 634 696 578 640 702 584 646 708 590 652 714 597 658 720 603 665 726 609 671 733 615 677 739 621 683 745 628 689 751 04 05 06 757 819 880 763 825 887 770 831 893 776 837 899 782 844 905 788 850 911 794 856 917 800 862 924 807 868 930 813 874 936 07 08 09 942 85003 065 948 009 071 954 016 077 mo 022 083 967 028 089 973 034 095 979 040 101 985 046 107 991 052 114 997 058 120 710 126 132 138 144 150 156 163 169 175 181 11 12 13 187 248 309 193 254 315 199 260 321 205 266 327 211 272 333 217 278 339 224 285 345 230 291 352 236 297 358 242 303 364 14 15 16 370 431 491 376 437 497 382 443 503 388 449 509 394 455 516 400 461 522 406 467 528 412 473 534 418 479 540 425 485 546 17 18 19 720 552 612 673 558 618 679 564 ()25 685 570 631 691 576 637 697 582 643 703 588 649 709 594 655 715 600 661 721 606 667 727 733 739 745 751 757 763 769 775 781 788 21 22 23 794 854 914 800 860 920 806 866 926 812 872 962 818 878 938 824 884 944 830 890 950 836 896 956 842 ^)02 962 848 908 968 1 2 7 0.7 1.4 6 0.6 1.2 5 0.5 1.0 24 25 26 974 86034 094 980 040 100 986 046 106 902 052 112 998 058 118 *004 064 124 *010 070 130 *016 076 136 *022 082 141 *028 088 147 3 4 5 6 2.1 2.8 3.5 4*^ 1.8 2.4 3.0 36 1.5 2.0 2.5 30 27 28 29 730 153 213 273 332 159 219 2<9 165 225 285 171 231 291 177 237 297 183 243 303 189 249 308 195 255 314 201 261 320 207 267 326 7 8 9 4.9 5.6 6.3 4.2 4.8 5.4 3.5 4.0 4.5 338 344 350 356 362 368 374 380 386 31 32 33 392 451 510 398 457 516 404 463 522 410 4()9 528 415 475 534 421 481 540 427 487 546 433 493 552 439 499 558 445 604 564 34 35 36 570 629 688 576 635 694 581 641 700 587 646 705 593 652 711 599 658 717 605 6t>4 723 611 670 729 617 676 735 623 682 741 37 38 39 747 806 864 753 812 870 759 817 876 764 823 882 770 829 888 776 835 894 782 841 fX)0 788 847 906 794 853 911 800 859 917 740 923 929 935 941 947 953 958 964 970 976 41 42 43 982 87040 099 988 046 105 994 052 111 999 058 116 *005 064 122 *011 070 128 *017 075 134 *023 081 140 *029 087 146 *035 093 151 44 45 46 157 216 274 163 221 280 169 227 286 175 233 291 181 239 297 186 245 303 192 251 309 198 256 315 204 262 320 210 268 326 47 48 49 332 390 448 338 396 454 344 402 460 349 408 466 355 413 471 361 419 477 367 425 483 373 431 489 379 437 495 552 384 442 500 750 506 512 518 523 529 535 .541 547 558 N. I 2 3 4 5 6 7 8 9 Prop. Pts. I] 760- - Logarithms of Numbers — 800 15 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 750 87 506 512 518 523 529 535 541 547 552 558 51 52 53 564 622 679 570 628 685 576 633 691 581 639 697 587 645 703 593 651 708 599 656 714 604 ()62 720 610 6f)8 726 616 674 731 54 55 56 737 795 852 743 800 858 749 806 864 754 812 869 760 818 875 766 823 881 772 829 887 777 835 892 783 841 898 789 846 904 57 58 59 910 967 88 024 915 973 030 921 978 036 927 984 041 933 990 047 938 996 053 944 *001 058 950 *007 064 955 *013 070 961 *018 076 760 081 087 093 098 104 110 116 121 127 133 61 62 63 138 195 252 144 201 258 150 207 264 156 213 270 161 218 275 167 224 281 173 230 287 178 235 292 184 241 298 190 247 304 64 65 m 309 366 423 315 372 429 321 377 434 326 383 440 332 389 446 338 395 451 343 400 457 349 406 463 355 412 468 360 417 474 67 68 69 480 536 593 485 542 598 491 547 604 497 553 610 502 559 615 508 564 621 513 570 ()27 519 576 632 525 581 638 530 587 643 770 649 655 660 666 672 677 683 689 694 700 71 72 73 705 762 818 711 767 824 717 773 829 722 779 835 728 784 840 734 190 846 739 795 852 745 801 857 750 807 863 756 812 868 1 2 6 0.6 1.2 5 0.5 1.0 74 75 76 874 930 986 880 936 992 885 941 997 891 947 *003 897 953 *009 902 958 *014 908 9CA *020 913 *025 919 975 *031 925 981 *037 3 4 5 6 1.8 2.4 3.0 3.6 1.5 2.0 2.5 3 77 78 79 89042 098 154 048 104 159 053 109 165 059 115 170 064 120 176 070 126 182 076 131 187 081 137 193 087 143 198 254 092 148 204 7 8 9 4.2 4.8 5.4 3.5 4.0 4.5 780 209 215 221 226 232 237 243 248 260 81 82 83 265 321 376 271 326 382 276 332 387 282 337 393 287 343 398 293 348 4(H 298 354 409 304 360 415 310 365 421 315 371 426 84 85 86 432 487 542 437 492 548 443 498 653 448 504 559 454 509 564 459 515 570 465 520 575 470 526 581 476 531 586 481 537 592 87 88 89 597 653 708 603 658 713 609 664 719 614 669 724 620 675 730 625 680 735 631 686 741 636 691 746 642 697 752 647 702 757 790 763 768 774 779 785 790 796 801 807 812 91 92 93 818 873 927 823 878 933 829 883 938 834 889 944 840 894 949 845 900 955 851 905 960 856 911 966 862 916 971 867 922 977 94 95 96 982 90037 091 988 042 097 993 048 102 998 053 108 *004 059 113 *009 064 119 *015 069 124 *020 075 129 *026 080 135 *031 086 140 97 98 99 146 200 255 151 206 260 157 211 266 162 217 271 168 222 276 173 227 282 179 233 287 184 238 293 189 244 298 195 249 304 800 309 314 320 325 331 336 342 347 352 358 U. 1 2 3 4 5 6 7 8 9 Prop. Pts. 16 800- - Logarithms of Numbers -850 [1 N, 1 2 3 4 5 6 7 8 9 Prop. Pts. 800 90 309 314 320 325 331 336 342 347 352 358 01 02 03 363 417 472 369 423 477 374 428 482 380 434 488 385 439 493 390 445 499 396 450 504 401 455 509 407 461 515 412 466 620 04 05 06 526 580 634 531 585 639 536 590 644 542 596 650 547 601 655 553 607 660 558 612 666 563 617 671 569 623 677 574 628 682 07 08 09 687 741 795 693 747 800 698 752 806 703 757 811 709 763 816 714 768 822 720 773 827 725 779 832 730 784 838 736 789 843 810 849 854 859 865 870 875 881 886 891 897 11 12 13 902 956 91009 907 961 014 913 966 020 918 972 025 924 977 030 929 982 036 934 988 041 940 993 046 945 998 052 950 *004 057 14 15 16 062 116 169 068 121 174 073 126 180 078 132 185 084 137 190 089 142 196 094 148 201 100 153 206 105 158 212 110 164 217 17 18 19 222 275 328 228 281 334 233 286 339 238 291 344 243 297 350 249 302 355 254 307 360 259 312 365 265 318 371 270 323 376 820 381 387 392 397 403 408 413 418 424 429 21 22 23 434 487 540 440 492 545 445 498 551 450 503 656 455 508 561 461 514 566 466 519 572 471 524 577 477 529 582 482 535 587 1 2 6 0.6 1.2 5 0.5 1.0 24 25 26 593 645 698 598 651 703 603 656 709 609 661 714 614 666 719 619 672 724 624 677 730 630 682 735 635 687 740 640 693 745 3 4 5 6 1.8 2.4 3.0 3.6 1.5 2.0 2.5 3.0 27 28 29 751 803 855 756 808 861 761 814 866 766 819 871 772 824 876 777 829 882 782 834 887 787 840 892 793 845 897 798 850 903 7 8 9 4.2 4.8 6.4 3.5 4.0 4.5 830 908 913 918 924 929 934 939 944 950 955 31 32 33 960 92012 065 965 018 070 971 023 075 976 028 080 981 033 085 986 038 091 991 044 096 997 049 101 *002 054 106 *007 059 111 34 35 36 117 169 221 122 174 226 127 179 231 132 184 236 137 189 241 143 195 247 148 200 252 153 205 257 158 210 262 163 215 267 37 38 39 273 324 376 278 330 381 283 335 387 288 340 392 293 345 397 298 350 402 304 355 407 309 361 412 314 366 418 319 371 423 840 428 433 438 443 449 454 459 464 469 474 41 42 43 480 531 583 485 536 588 490 542 593 495 547 598 500 552 603 505 557 609 511 562 614 516 567 619 521 572 624 526 578 629 44 45 46 634 686 737 639 691 742 645 696 747 650 701 752 655 706 758 660 711 763 665 716 768 670 722 773 675 727 778 681 732 783 47 48 49 788 840 891 793 845 896 799 850 901 804 855 906 809 860 911 814. 865 916 819 870 921 824 875 927 829 881 932 834 886 937 850 942 947 952 957 962 967 973 978 983 988 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. . q 8 50- -Lo^ ?arit hms of Numbers — 900 17 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 850 92 942 947 952 957 962 967 973 978 983 988 51 52 53 993 93 044 095 998 049 100 *003 054 105 *008 059 110 *013 064 115 *018 069 120 *024 075 125 *029 080 13J *034 085 136 *039 090 141 54 55 56 146 197 247 151 202 252 156 207 258 161 212 263 166 217 268 171 222 273 176 227 278 181 232 283 186 237 288 192 242 293 57 58 59 298 349 399 303 354 404 308 359 409 313 364 414 318 369 420 323 374 425 328 379 430 334 384 435 339 389 440 344 394 445 860 450 455 460 465 470 475 480 485 490 495 61 62 63 500 551 601 505 556 606 510 561 611 515 560 616 520 571 621 526 576 626 531 581 631 536 586 636 541 591 641 546 596 646 64 65 66 651 702 752 656 707 757 661 712 762 666 717 767 671 722 772 676 727 777 682 732 782 687 737 787 692 742 792 697 747 797 67 68 69 802 852 902 807 857 907 812 862 912 817 867 917 822 872 922 827 877 927 832 882 932 837 887 937 842 892 942 847 897 947 997 870 952 957 962 967 972 977 982 987 992 71 72 73 94 002 052 101 007 057 106 012 062 111 017 067 116 022 072 121 027 077 126 032 082 131 037 086 136 042 091 141 047 096 146 1 2 6 0.6 1.2 5 0.5 1.0 4 0.4 0.8 74 75 76 151 201 250 156 206 255 161 211 260 166 216 265 171 221 270 176 226 275 181 231 280 186 236 285 191 240 290 196 245 295 3 4 5 1.8 2.4 3.0 3 6 1.5 2.0 2.5 30 1.2 1.6 2.0 2.4 77 78 79 300 349 399 305 354 404 310 359 409 315 364 414 320 369 419 325 374 424 330 379 429 335 384 433 340 389 438 345 394 443 7 8 9 4.2 4.8 5.4 3.5 4.0 4.5 2.8 3.2 3.6 880 448 453 458 463 468 473 478 483 488 493 81 82 83 498 547 596 503 552 601 507 557 606 512 562 611 517 567 616 522 571 621 527 576 626 532 581 630 537 586 635 542 591 640 84 85 86 645 694 743 650 699 748 655 704 753 660 709 758 665 714 763 670 719 768 675 724 773 680 729 778 685 734 783 689 738 787 87 88 89 792 841 890 797 846 895 802 851 900 807 856 f)05 812 861 910 817 866 915 822 871 919 827 876 924 832. 880 929 836 885 934 890 939 944 949 954 959 ms 968 973 978 983 91 92 93 988 95 036 085 993 041 090 998 046 095 *002 051 100 *007 056 105 *012 061 109 *017 066 114 *022 071 119 *027 075 124 *032 080 129 94 95 96 134 182 231 139 187 236 143 192 240 148 197 245 153 202 250 158 207 255 163 211 260 168 216 265 173 221 270 177 226 274 97 98 99 279 328 376 284 332 381 289 337 386 294 342 390 299 347 395 303 352 400 308 357 405 313 361 410 318 366 415 323 371 419 900 424 429 434 439 444 448 453 458 463 468 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 18 900- - Logarithms of Numbers -950 U N. 1 2 3 4 5 6 7 8 9 Prop, Pts. 900 95 424 429 434 439 444 448 453 458 463 468 01 02 03 472 521 569 477 525 574 482 530 578 487 535 583 492 540 588 497 545 593 501 550 598 506 554 602 511 559 607 516 564 612 04 05 06 617 665 713 622 670 718 626 674 722 631 679 727 636 684 732 641 689 737 646 694 742 650 698 746 655 703 751 660 708 756 07 08 09 761 809 85(i 766 813 861 770 818 866 775 823 871 780 828 875 785 832 880 789 837 885 794 842 890 799 847 895 804 852 899 910 904 909 914 918 923 928 933 938 942 947 11 12 13 952 999 96 047 957 *004 052 961 *009 057 966 *014 061 971 *019 066 976 *023 071 980 *028 076 985 *033 080 990 *038 085 995 *042 090 14 15 16 095 142 190 099 147 194 104 152 199 109 156 204 114 161 209 118 166 213 123 171 218 128 175 223 133 180 227 137 185 232 17 18 19 237 284 332 242 289 336 246 294 341 251 298 346 256 303 350 261 308 355 265 313 360 270 317 365 275 322 369 280 327 374 920 379 384 388 393 398 402 407 412 417 421 21 22 23 426 473 520 431 478 625 435 483 530 440 487 534 445 492 539 450 497 544 454 501 548 459 506 553 464 511 558 468 515 562 1 2 5 0.5 1.0 4 0.4 0.8 24 25 26 567 614 661 572 619 666 577 624 670 581 628 675 586 633 680 591 638 685 595 642 689 600 647 694 605 652 699 609 656 703 3 4 5 6 1.5 2.0 2.5 3.0 1.2 1.6 2.0 2.4 27 28 29 708 755 802 713 759 806 717 764 811 722 769 816 727 774 820 731 778 825 736 783 830 741 788 834 745 792 839 760 797 844 7 8 9 3.5 4.0 4.5 2.8 3.2 3.6 930 848 853 858 862 867 872 876 881 886 890 31 32 33 895 94- 988 900 946 993 904 951 997 909 956 *002 914 960 *007 918 965 *011 923 970 ■*016 928 974 *021 932 979 *025 937 984 *030 34 35 36 97035 081 128 039 086 132 044 090 137 049 095 142 053 100 146 058 104 151 063 109 155 067 114 160 072 118 165 077 123 169 37 38 39 174 220 267 179 225 271 183 230 276 188 234 280 192 239 285 197 243 290 202 248 294 206 253 299 211 257 304 216 262 308 940 41 42 43 313 359 405 451 317 322 327 331 336 340 345 350 354 364 410 456 368 414 460 373 419 465 377 424 470 382 428 474 387 433 479 391 437 483 396 442 488 400 447 493 44 45 46 497 543 589 502 548 594 506 552 598 511 557 603 516 562 607 520 566 612 525 571 617 529 575 621 534 580 626 539 585 630 47 48 49 635 (>81 727 640 685 731 644 690 736 649 695 740 653 699 745 658 704 749 663 708 754 667 713 759 672 717 763 676 722 768 950 772 777 782 786 791 795 800 804 809 813 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. i] 950 — Lo^arith ms of Numbers - -1000 19 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 950 97 772 777 782 786 791 795 800 804 809 813 51 52 53 818 864 909 823 868 914 827 873 918 832 877 923 836 882 928 841 886 932 845 891 937 850 896 941 855 900 946 859 905 950 54 55 56 955 98 000 046 959 005 050 964 009 055 968 014 059 973 019 064 978 023 068 982 028 073 987 032 078 991 037 082 996 041 087 57 58 59 091 137 182 096 141 186 100 146 191 105 150 195 109 155 200 114 159 204 118 164 209 123 168 214 127 173 218 132 177 223 960 227 232 236 241 245 250 254 259 263 268 61 62 63 272 318 363 277 322 367 281 327 372 286 331 376 290 336 381 295 340 385 299 345 390 304 349 394 308 354 399 313 358 403 64 65 m 408 453 498 412 457 502 417 462 507 421 466 511 426 471 516 430 475 520 435 480 525 439 484 529 444 489 534 448 493 538 67 68 69 543 588 632 547 592 637 552 597 641 556 601 646 561 605 650 565 610 655 570 614 659 574 619 664 579 623 668 583 628 673 970 677 682 686 691 695 700 704 709 713 717 71 72 73 722 767 811 726 771 816 731 776 820 735 780 825 740 784 829 744 789 834 749 793 838 753 798 843 758 802 847 762 807 851 1 2 5 0.5 1.0 4 0.4 0.8 74 75 76 856 900 945 860 905 949 865 909 954 869 914 958 874 918 963 878 923 967 883 927 972 887 932 976 892 936 981 896 941 985 3 4 5 6 1.5 2.0 2.5 3 1.2 1.6 2.0 2.4 77 78 79 989 99 034 078 994 038 083 998 043 087 *003 047 092 *007 052 096 *012 056 100 *016 061 105 *021 065 109 *025 069 114 *029 074 118 7 8 9 3.5 4.0 4.5 2.8 3.2 3.6 980 123 127 131 136 140 145 149 154 158 162 81 82 83 167 211 255 171 216 260 176 220 264 180 224 269 185 229 273 189 233 277 193 238 282 198 242 286 202 247 291 207 251 295 84 85 86 300 344 388 304 348 392 308 352 396 313 357 401 317 361 405 322 '666 410 326 370 414 330 374 419 335 379 423 339 383 427 87 88 89 432 476 520 436 480 524 441 484 528 445 489 533 449 493 537 454 498 542 458 502 546 463 506 550 467 511 555 471 515 559 990 564 568 572 577 581 585 590 594 599 603 91 92 93 607 651 695 612 656 699 616 660 704 621 664 708 625 669 712 629 673 717 634 677 721 638 682 726 642 686 730 647 691 734 94 95 96 739 782 826 743 787 8:^ 747 791 835 752 795 839 756 800 843 760 804 848 765 808 852 769 813 856 774 817 861 778 822 865 97 98 99 870 913 957 874 917 961 878 922 965 883 926 970 887 930 974 891 935 978 896 939 983 900 944 987 904 948 991 909 952 996 1000 00 000 004 009 013 .17 022 026 030 035 039 N. 1 2 3 4 5 6 7 8 9 Prop. Pts. 20 Logarithms of Important Constants [la TABLE la. LOGARITHMS OF IMPORTANT CONSTANTS i\r= Number Value of JV LOGio ^ "Vtt e = Napierian Base M=logiQe l-4-3f=logel0 180 -r- IT = degrees in 1 radian TT -7- 180 = radians in 1° TT -f- 10800 = radians in 1' TT -4- 648000 = radians in 1" sin 1" tan 1" centimeters in 1 ft. feet in 1 cm. inches in 1 m. pounds in 1 kg. kilograms in 1 lb. g (average value) weight of 1 cu. ft. of water weight of 1 cu. ft. of air cu. m. in 1 (U. S.) gallon ft. lb. per sec. in 1 H. P. kg. m. per sec. in 1 H. P. watts in 1 H. P. 3.14159265 0.31830989 9.86960440 1.77245385 2.71828183 0.43429448 2.30258509 57.2957795 0.01745329 0.0002908882 0.000004848136811095 0.000004848136811076 0.000004848136811162 30.480 0.032808 39.37 2.20462 0.453593 32.16 ft./sec./sec. = 981 cm. /sec. /sec. 62.425 lb. (max. density) 0.0807 lb. (at 32° F.) 231. 550. 76.0404 745.957 0.49714987 9.5028,5013 0.99429975 0.24857494 0.43429448 9.63778431 0.36221569 1.75812263 8.24187737 ().46372612 4.68557487 4.68557487 4.68557487 1.4840158 8.5159842 1.5951654 0.3433340 9.6566660 . 1.5073 2.9916690 1.7953586 8.907 2.3636120 2.7403627 1.8810445 2.8727135 COMMON LOGARITHMS OF THE FIRST HUNDRED PRIME NUMBERS N Logarithm N Log N Log N Log N Log 1 0000000000 71 8512583 173 2380461 281 4487063 409 6117233 2 3010299957 73 8633229 179 2528530 283 4517864 419 6222140 3 4771212547 79 8976271 181 2576786 293 4668676 421 6242821 5 6989700043 83 9190781 191 2810334 307 4871384 431 6344773 7 8450980400 89 9493900 193 2855573 311 4927604 433 6364879 11 0413926852 97 9867717 197 2944662 313 4955443 439 6424645 13 1139433523 101 0043214 199 2988531 317 5010593 443 6464037 17 2304489214 103 0128372 211 3242825 331 5198280 449 6522463 19 2787536010 107 0293838 223 3483049 337 5276299 457 6599162 23 3617278360 109 0374265 227 3560259 347 5403295 461 6637009 29 4623979979 113 0530784 229 3598355 349 5428254 463 6655810 31 4913616938 127 1038037 233 3673559 353 5477747 467 6693169 37 5682017241 131 1172713 239 3783979 359 5550944 479 6803355 41 6127838567 137 1367206 241 3820170 367 5646661 487 68752^)0 43 6334684556 139 1430148 251 3996737 373 5717088. 491 6910815 47 6720978579 149 1731863 257 4099331 379 5786392 499 6981005 53 7242758696 151 1789769 263 4199557 383 5831988 503 7015680 59 at 7708520116 TOKOOnO'JKA 157 1958997 269 071 4297523 A QonnoQ 389 rici'7 5899496 FC0C70nFC 509 K'}t 7067178 71A«^77 TABLE II ACTUAL VALUES OF THE TKIGONOMETEIC FUNCTIONS FKOM 0° TO 90° AT INTERVALS OF ONE MINUTE TO FIVE DECIMAL PLACES - III L 1 LI. I 44-^ sL u 91 41 US il L 4L 9a: i U^ n. r-_v m jz M^ fi - — ^ rit ^ '-J (2- Oj ^-r^TlM - ^ /^ Zf" ^ eroed ^./kj. / ." r ^e ^ AdT ^\ ^^t - $- T \^A f- ^' t VST V VI) -> ^^~ " "' M AV /' J -Ix ^^ ■ lA. ^o. ^ J fS^ ^^ 1- 4^'^\-i^ ^eear.-l' ^^ ^J --%% - ^.--T ¥' ^i"i- -" ^/ ^^c^V - ^^-' "^*I -/\i ^^^"/■^^^ - - _A>ir o<- ^^ -rS^^ ^ / !?»J^Vy, i,*^^ T' ^N?*? ^y - '^pv y^ T /^5S^ ^ ^ "^^ / ^-L ^. ^t ■^ ^s T 1- ^_,.,.^. ., ,. ,^ P" \-o ^ 7- - ^ / ^ .r -^f?"' X5o^ si- j^ i -_4r ^'^ . "•. ,^^ L'-c;\«^ >$J: it I'^k X ^-.^ \ it 77 antV 4,^ si-^ 5? ^0- (^-45 X-5 7::)- C^O) S >/ >^ r^. ^*y X*?!^ ^) \ > rH 'C^N / \\/ :s>r:^'' .^b?' ± 4^ S^ vUj^^ _ «6>\ 7<. c"^>>T r J ^'---^"^ ' ncliona ' ?v ^ "^ " "* -J. M QCiions L^.^^ <^-L- Tses 4/ r ^3X-- ises / 7 L 1 ^i\ nrp flin , 9 .08976 .09013 .095 696 51 ] 10 .09005 .09042 11.059 .99594 50 ' 11 034 071 11.024 591 49) 12 063 101 10.988 588 48 13 092 130 .953 586 47! 14 121 159 .918 583 46 15 .09150 .09189 10.883 .99580 45 16 179 218 .848 578 44 17 208 247 .814 675 43. 18 237 277 .780 672 42. 19 266 306 .746 570 41 20 .09295 .09335 10.712 .99567 40 21 324 365 .678 564 39 22 353 394 .645 562 38 23 382 423 .612 559 37 24 411 453 .579 556 36 25 .09440 .09482 10.546 .99553 35 26 469 511 .514 551 34 27 498 541 .481 648 33 28 527 570 .449 545 32 29 556 600 .417 642 31 30 .09585 .09629 10.385 .99540 30 31 614 658 .354 637 29 32 642 688 .322 534 28 33 671 717 .291 531 27 34 700 746 • .260 528 26 35 .09729 .09776 10.229 .99526 25 36 758 805 .199 523 24 37 787 834 .168 520 23 38 816 864 .138 517 22 39 845 893 .108 514 21 40 .09874 .09923 10.078 .99511 20 41 903 952 .048 508 19 42 932 .09981 10.019 506 18 43 961 .10011 9.9893 503 17 44 .09990 040 .9601 600 16 45 .10019 .10069 9.9310 .99497 15 46 048 099 .9021 494 14 47 077 128 .8734 491 13 48 106 158 .8448 488 12 49 135 187 .8164 485 11 50 .10164 .10216 9.7882 .99482 10 51 192 246 .7601 479 9 52 221 275 .7322 476 8 53 250 305 .7044 473 7 54 279 334 .6768 470 6 55 .10308 .10363 9.6493 .99467 5 56 337 393 .6220 464 4 57 366 422 .5949 461 3 58 395 452 .5679 458 2 59 424 481 .5411 455 1 60 .10453 .10510 9.5144 .99452 Cos Ctn Tan Sin 1 IIJ 6°— Values of Trigonometric Functions — 7° 4 5 6 7 8 9 10 11 12 13 14 15 1(J 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sin .10453 482 511 540 569 .10597 626 655 684 713 .10742 771 800 829 858 .10887 916 945 .10973 .11002 .11031 060 089 118 147 .11176 205 234 263 291 .11320 349 378 407 436 .11465 494 523 552 580 .11609 638 667 696 725 .11754 783 812 840 869 .11898 927 956 .11985 .12014 .12043 071 100 129 158 .12187 Tan Gtn Cos .10510 540 569 599 628 .10657 687 716 746 775 .10805 834 863 893 922 .10952 .10981 .11011 040 070 .11099 128 158 187 217 .11246 276 305 335 364 .11394 423 452 482 511 .11541 570 600 629 659 .11688 718 747 777 806 .11836 865 895 924 954 .11983 .12013 042 072 101 .12131 160 190 219 249 .12278 9.5144 .4878 .4614 .4352 .4090 9.3831 .3572 .3315 .30(^0 .2806 9.2553 .2302 .2052 .1803 .1555 9.1309 .1065 .0821 .0579 .0338 9.0098 8.9860 .9623 .9387 .9152 8.8919 .8686 .8455 .8225 .7996 8.7769 .7542 .7317 .7093 .6870 8.6648 .(;427 .6208 .5989 .5772 8.5555 .5340 .5126 .4913 .4701 8.4490 .4280 .4071 .3863 .3656 8.3450 .3245 .3041 .2838 .2636 8.2434 .2234 .2035 .1837 .1640 8.1443 Cos Ctn I Tan Sin ' .99452 449 446 443 440 .99437 434 431 428 424 .99421 418 415 412 409 .99406 402 399 396 393 .99390 386 383 380 377 .99374 370 367 364 360 .99357 354 351 317 344 .99341 337 334 331 327 .99324 320 317 314 310 .99307 303 300 297 293 .99290 286 283 279 276 .99272 269 265 262 258 .99255 / Sin Tan Ctn Cos .12187 .12278 8.1443 .99255 60 1 216 308 .1248 251 59 2 245 338 .1054 248 58 3 274 367 .0860 244 57 4 302 397 .0667 240 56 5 .12331 .12426 8.0476 .99237 65 6 360 456 .0285 233 54 7 389 485 8.0096 230 53 8 418 515 7.9906 226 52 9 447 544 .9718 222 51 10 .12476 .12574 7.9530 .99219 50 11 504 603 .9344 215 49 12 533 633 .9158 211 48 13 562 662 .8973 208 47 14 591 692 .8789 204 46 15 .12620 .12722 7.8606 .99200 45 16 <;49 751 .8424 197 44 17 ()78 781 .8243 193 43 18 706 810 .8062 189 42 19 735 840 .7882 186 41 20 .12764 .12869 7.7704 .99182 40 21 793 899 .7525 178 39 22 822 929 .7348 175 38 23 851 958 .7171 171 37 24 880 .12988 .6996 167 36 25 .12908 .13017 7.6821 .99163 35 26 937 047 .6647 160 34 27 9()6 076 .6473 156 33 28 .12^)95 106 .6301 152 32 29 .13024 136 .6129 148 31 30 .13053 .13165 7.5958 .99144 30 31 081 195 .5787 141 29 32 110 224 .5618 137 28 33 139 254 .5449 133 27 34 168 284 .5281 129 2(> 35 .13197 .13313 7.5113 .99125 25 36 226 343 .4947 122 24 37 254 372 .4781 118 23 38 283 402 .4615 114 22 39 312 432 .4451 110 21 40 .13341 .13461 7.4287 .99106 20 41 370 491 .4124 102 19 42 399 521 .3962 098 18 43 427 550 .3800 094 17 44 456 580 .3639 091 16 45 .13485 .13609 7.3479 .99087 15 4() 514 639 .3319 083 14 47 543 669 .3160 079 13 48 572 698 .3002 075 12 49 600 728 .2844 071 11 60 .13629 .13758 7.2687 .99067 10 51 658 787 .2531 063 9 52 687 817 .2375 059 8 53 716 846 .2220 055 7 54 744 876 .2066 051 6 55 .13773 .13906 7.1912 .99047 5 56 802 935 .1759 043 4 57 831 965 .1607 039 3 58 860 .13995 .1455 035 2 59 889 .14024 .1304 031 1 60 .13917 .14054 7.1154 .99027 Cos Ctn Tan Sin 1 26 8° — Values of Trigonometric Functions— 9° / Sin Tan Ctn Cos .13917 .14054 7.1154 .99027 60 1 946 084 .1004 023 59 2 .13975 113 .0855 019 58 3 .14004 143 .0706 015 57 4 033 173 .0558 Oil 56 5 .14061 .14202 7.0410 .99006 55 6 090 232 .0264 .99002 54 7 119 262 7.0117 .98998 53 8 148 291 6.9972 994 52 9 177 321 .9827 990 51 10 .14205 .14351 6.9682 .98986 50 11 234 381 .9538 982 49 12 263 410 .9395 978 48 13 292 440 .9252 973 47 14 320 470 .9110 969 46 15 .14349 .14499 6.8969 .98965 45 16 378 529 .8828 961 44 17 407 559 .8687 957 43 18 436 588 .8548 953 42 19 464 618 .8408 948 41 20 .14493 .14648 6.8269 .98944 40 21 522 678 .8131 940 39 22 551 707 .7994 936 38 23 580 737 .7856 931 37 24 608 767 .7720 927 36 25 .14637 .14796 6.7584 .98923 35 20 em 826 .7448 919 34 27 695 856 .7313 914 33 28 723 886 .7179 910 3*2 29 752 915 .7045 906 31 30 .14781 .14945 6.6912 .98902 30 31 810 .14975 .6779 897 29 32 838 .15005 .6646 893 28 33 867 034 .6514 889 27 34 896 064 .6383 884 26 35 .14925 .15094 6.6252 .98880 25 36 954 124 .6122 876 24 37 .14982 153 .5992 871 23 38 .15011 183 .5863 867 22 39 040 213 .5734 863 21 40 .15069 .15243 6.5606 .98858 20 41 097 272 .5478 854 19 42 126 302 .5350 849 18 43 155 332 .5223 845 17 44 184 362 .5097 841 16 45 .15212 .15391 6.4971 .98836 15 46 241 421 .4846 832 14 47 270 451 4721 827 13 48 299 481 .4596 823 12 49 327 511 .4472 818 11 50 .15356 .15540 6.4348 .98814 10 51 . 385 570 .4225 809 9 52 414 600 .4103 805 8 53 442 630 .3980 800 7 54 471 660 .3859 796 6 55 .15500 .15689 6.3737 .98791 5 66 529 719 .3617 787 4 57 557 749 .3496 782 3 58 586 779 .3376 778 2 59 615 809 .3257 773 1 60 .15643 .15838 6.3138 .98769 Cos Ctn Tan Sin 1 / Sin Tan Ctn Cos .15643 .15838 6.3138 .98769 60 1 672 868 .3019 764 59 2 701 898 .2901 760 58 3 730 928 .2783 755 57 4 758 958 .2666 751 56 6 .15787 .15988 6.2549 .98746 65 6 816 .16017 .2432 741 54 7 845 047 .2316 737 53 8 873 077 .2200 732 52 9 902 107 .2085 728 51 10 .15931 .16137 6.1970 .98723 60 11 959 167 .1856 718 49 12 .15988 196 .1742 714 48 13 .16017 226 .1628 709 47 14 046 256 .1515 704 46 15 .16074 .16286 6.1402 .98700 45 16 103 316 .1290 695 44 17 132 346 .1178 690 43 18 160 376 .1066 686 42 19 189 405 .0955 681 41 20 .16218 .16435 6.0844 .98676 40 21 246 465 .0734 671 39 22 275 495 .0624 667 38 23 304 525 .0514 662 37 24 333 555 .0405 657 36 25 .16361 .16585 6.0296 „98652 36 26 390 615 .0188 648 34 27 419 645 6.0080 643 33 28 447 674 5.9972 638 32 29 476 704 .9865 633 31 30 .16505 .16734 5.9758 .98629 30 31 533 764 .9651 624 29 32 562 794 .9545 619 28 33 591 824 .9439 614 27 34 620 854 .9333 609 26 35 .16648 .16884 5.9228 .98604 26 36 677 914 .9124 600 24 37 706 944 .9019 595 23 38 734 .16974 .8915 590 22 39 763 .17004 .8811 585 21 40 .16792 .17033 5.8708 .98580 20 41 820 063 .8605 575 19 42 849 093 .8502 570 18 43 878 123 .8400 565 17 44 906 153 .8298 561 16 45 .16935 .17183 5.8197 .98556 16 46 964 213 .8095 551 14 47 .16992 243 .7994 546 13 48 .17021 273 .7894 541 12 49 050 303 .7794 536 11 50 .17078 .17333 5.7694 .98531 10 51 107 363 .7594 526 9 52 136 393 .7495 521 8 53 164 423 .7396 516 7 54 193 453 .7297 511 6 65 .17222 .17483 5.7199 .98506 5 56 250 513 .7101 501 4 57 279 543 .7004 496 3 58 308 573 .6906 491 2 59 336 603 .6809 486 1 60 .17365 .17633 5.6713 .98481 Cos Ctn Tan Sin II] 10°— Values of Trigonometric Functions — IF 27 f Sin Tan Ctn Cos .17365 .17633 5.6713 .98481 60 1 393 663 .6617 476 59 2 422 693 -.6521 471 58 3 451 723 .6425 466 57 4 479 753 .6329 461 56 5 .17508 .17783 5.6234 .98455 55 G 537 813 .6140 450 54 7 565 843 .6045 445 53 8 594 873 .5951 440 52 9 623 903 .5857 435 51 10 .17651 .17933 5.5764 .98430 50 11 680 963 .5671 425 49 12 708 .17993 .5578 420 48 13 737 .18023 .5485 414 47 14 766 053 .5393 409 46 15 .17794 .18083 5.5301 .98404 45 16 823 113 .5209 399 44 17 852 143 .5118 394 43 18 880 173 .5026 389 42 19 909 203 .4936 383 41 20 .17937 .18233 5.4845 .98378 40 21 966 263 .4755 373 39 22 .17995 293 .4665 368 38 23 .18023 323 .4575 362 37 24 052 353 .4486 357 36 25 .18081 .18384 5.4397 .98352 35 26 109 414 .4308 347 34 27 138 444 .4219 341 33 28 166 474 .4131 336 32 29 195 504 .4043 331 31 30 .18224 .18534 5.3955 .98325 30 31 252 564 .3868 320 29 32 281 594 .3781 315 28 33 309 624 .3694 310 27 34 338 654 .3607 304 26 35 .18367 .18684 5.3521 .98299 25 36 395 714 .3435 294 24 37 424 745 .3349 288 28 38 452 775 .3263 283 22 39 481 805 .3178 277 21 40 .18509 .18835 5.3093 .98272 20 41 538 865 .3008 267 19 42 567 895 .2924 261 18 43 595 925 .2839 256 17 44 624 955 .2755 250 16 45 .18652 .18986 5.2672 .98245 15 46 681 .19016 .2588 240 14 47 710 046 .2505 234 13 48 738 076 .2422 229 12 49 767 106 .2339 223 11 50 .18795 .19136 5.2257 .98218 10 51 824 166 .2174 212 9 52 852 197 .2092 207 8 53 881 227 .2011 201 7 54 910 257 .1929 196 6 55 .18938 .19287 5.1848 .98190 5 56 ' 967 317 .1767 185 4 57 .18995 347 .1686 179 3 58 .19024 378 .1606 174 2 59 052 408 .1526 168 1 60 .19081 .19438 5.1446 .98163 Cos Ctn Tan Sin / f Sin Tan Ctn Cos .19081 .19438 5.1446 .98163 60 1 109 468 .1366 157 59 2 138 498 .1286 152 58 3 167 529 .1207 146 57 4 195 559 .1128 140 56 5 .19224 .19589 5.1049 .98135 55 6 252 619 .0970 129 54 7 281 649 .0892 124 53 8 309 680 .0814 118 52 9 338 710 .0736 112 51 10 .19366 .19740 5.0658 .98107 50 11 395 770 .0581 101 49 12 423 801 .0504 09(5 48 13 452 831 .0427 090 47 14 481 861 .0350 084 46 15 .19509 .19891 5.0273 .98079 45 16 538 921 .0197 073 44 17 566 952 .0121 067 43 18 595 .19982 5.0045 061 42 19 623 .20012 4.9969 056 41 20 .19652 .20042 4.9894 .98050 40 21 680 073 .9819 044 39 22 709 103 .9744 039 38 23 737 133 .9669 033 37 24 766 164 .9594 027 36 25 .19794 .20194 4.9520 .98021 35 26 823 224 .9446 016 34 27 851 254 .9372 010 33 28 880 285 .9298 .98004 32 29 908 315 .9225 .97998 31 30 .19937 .20345 4.91.52 .97992 30 31 965 376 .9078 987 29 32 .19994 406 .9006 981 28 33 .20022 436 .8933 975 27 34 051 466 .8860 969 2(; 35 .20079 .20497 4.8788 .97963 25 36 108 527 .8716 958 24 37 136 557 .8644 952 23 38 165 588 .8573 946 22 39 193 618 .8501 940 21 40 .20222 .20648 4.8430 .97934 20 41 250 679 .8359 928 19 42 279 709 .8288 922 18 43 307 739 .8218 916 17 44 336 770 .8147 910 16 45 .20364 .20800 4.8077 .97905 15 46 393 830 .8007 899 14 47 421 861 .7937 893 13 48 450 891 .7867 887 12 49 478 921 .7798 881 11 50 .20507 .20952 4.7729 .97875 10 51 535 .20982 .7659 869 9 52 563 .21013 .7591 863 8 53 592 043 .7522 857 7 54 620 073 .7453 851 6 55 .20649 .21104 4.7385 .97845 5 56 677 134 .7317 839 4 57 706 164 .7249 833 3 58 734 195 .7181 827 2 59 763 225 .7114 821 1 60 .20791 .21256 4.7046 .97815 Cos Ctn Tan Sin / 28 12° — Values of Trigonometi 'ic Functions — 13^ [n t Sin Tan Ctn Cos 1 Sin Tan Ctn Cos .20791 .21256 4.7046 .97815 60 .22495 .23087 4.3315 .97437 60 1 820 286 .6979 809 59 1 523 117 .3257 430 59 2 848 316 .6912 803 58 2 552 148 .3200 424 58 3 877 347 .6845 797 57 3 580 179 .3143 417 57 4 905 377 .6779 791 56 4 608 209 .3086 411 56 5 .20933 .21408 4.6712 .97784 55 5 .22637 .23240 4.3029 .97404 55 6 962 438 .6646 778 54 6 665 271 .2972 398 54 7 .20990 469 .6580 772 53 7 693 301 .2916 391 53 8 .21019 499 .6514 766 52 8 722 332 .2859 384 52 9 047 529 .6448 760 51 9 750 363 .2803 378 51 10 .21076 .21560 4.6382 .97754 50 10 .22778 .23393 4.2747 .97371 60 11 104 590 .6317 748 49 11 807 424 .2691 365 49 12 132 621 .6252 742 48 12 835 455 .2635 358 48 13 161 651 .6187 735 47 13 863 485 .2580 351 47 14 189 682 .6122 729 46 14 892 616 .2524 345 46 15 .21218 .21712 4.6057 .97723 45 15 .22920 .23547 4.2468 .97338 45 16 246 743 .5993 717 44 16 948 578 .2413 331 44 17 275 773 .5928 711 43 17 .22977 608 .2358 325 43 18 303 804 .5864 705 42 18 .23005 639 .2303 318 42 19 331 834 .5800 698 41 19 033 670 .2248 311 41 20 .21360 .21864 4.5736 .97692 40 20 .23062 .23700 4.2193 .97304 40 21 388 895 .5673 686 39 21 090 731 .2139 298 39 22 417 925 .5609 680 38 22 118 762 .2084 291 38 23 445 956 .5546 673 37 23 146 793 .2030 284 37 24 474 .21986 .5483 667 36 24 175 823 .1976 278 36 25 .21502 .22017 4.5420 .97661 35 25 .23203 .23854 4.1922 .97271 35 26 530 047 .5357 655 34 26 231 885 .1868 264 34 27 559 078 .5294 648 33 27 260 916 .1814 257 33 28 587 108 .5232 642 32 28 288 946 .1760 251 32 29 616 139 .5169 636 31 29 316 .23977 .1706 244 31 30 .21644 .22169 4.5107 .97630 30 30 .23345 .24008 4.1653 .97237 30 31 672 200 .5045 623 29 31 373 039 .1600 230 29 32 701 231 .4983 617 28 32 401 069 .1547 223 28 33 729 261 .4922 611 27 33 429 100 .1493 217 27 34 758 292 .4860 604 26 34 458 131 .1441 210 26 35 .21786 .22322 4.4799 .97598 25 35 .23486 .24162 4.1388 .97203 25 36 814 353 .4737 592 24 36 514 193 .1335 196 24 37 843 383 .4676 585 23 37 542 223 .1282 189 23 38 871 414 .4615 579 22 38 671 254 .1230 182 22 39 899 444 .4555 673 21 39 699 285 .1178 176 21 40 .21928 .22475 4.4494 .97566 20 40 .23627 .24316 4.1126 .97169 20 41 956 505 .4434 560 19 41 656 347 .1074 162 19 42 .21985 536 .4373 653 18 42 684 377 .1022 155 18 43 .22013 667 .4313 547 17 43 712 408 .0970 148 17 44 041 597 .4253 541 16 44 740 439 .0918 141 16 45 .22070 .22628 4.4194 .97534 15 45 .23769 .24470 4.0867 .97134 15 46 098 658 .4134 528 14 46 797 501 .0815 127 14 47 126 689 .4075 521 13 47 825 632 .0764 120 13 48 155 719 .4015 515 12 48 853 562 .0713 113 12 49 183 750 .3956 508 11 49 882 593 .0662 106 11 50 .22212 .22781 4.3897 .97502 10 50 .23910 .24624 4.0611 .97100 10 51 240 811 .3838 496 9 51 938 655 .0560 093 9 52 268 842 .3779 489 8 52 966 686 .0509 086 8 53 297 872 .3721 483 7 53 .23995 717 .0459 079 7 54 325 903 .3662 476 6 54 .24023 747 .0408 072 6 55 .22353 .22934 4.3604 .97470 5 55 .24051 .24778 4.0358 .97065 6 56 382 964 .3546 463 4 56 079 809 .0308 058 4 57 410 .22995 .3488 457 3 57 108 840 .0257 051 3 58 438 .23026 .3430 450 2 58 136 871 .0207 044 2 59 467 056 .3372 444 1 59 164 902 .0158 037 1 60 .22495 .23087 4.3315 .97437 60 .24192 .24933 4.0108 .97030 Cos Ctn Tan Sin 1 Cos Ctn Tan Sin f 1 iq 14°— Values of Trigononieti 'ic Functions — 15° 29 i~ Sin Tan Ctn Cos 1 Sin Tan Ctn Cos .24192 .24933 4.0108 .97030 60 .25882 .26795 3.7321 .96593 60 1 220 964 .0058 023 59 1 910 826 .7277 J 585 59 2 249 .24995 4.0009 015 58 2 938 857 .7234 578 58 3 277 .25026 3.9959 008 57 3 966 888 .7191 570 57 4 305 056 .9910 .97001 56 4 .25994 920 .7148 562 56 5 .24333 .25087 3.9861 .96994 55 5 .26022 .26951 3.7105 .96555 55 6 362 118 .9812 987 54 6 050 .26982 .7062 547 54 7 390 149 .9763 980 53 7 079 27013 .7019 540 53 8 418 180 .9714 973 52 8 107 044 .6976 632 62 9 446 211 .9665 966 51 9 135 076 .6933 624 61 10 .24474 .25242 3.9617 .96959 50 10 .26163 .27107 3.6891 .96517 50 11 503 273 .9568 952 49 11 191 138 .6848 509 49 12 531 304 .9520 945 48 12 219 169 .6806 602 48 13 559 335 .9471 937 47 13 247 201 .6764 494 47 14 587 366 .9423 930 46 14 275 232 .6722 486 46 15 .24615 .25397 3.9375 .96923 45 15 .26303 .27263 3.6680 .96479 45 16 644 428 .9327 916 44 16 331 294 .6638 471 44 17 672 459 .9279 909 43 17 359 326 .6596 463 43 18 700 490 .9232 902 42 18 387 357 .6554 456 42 19 728 521 .9184 894 41 19 415 388 .6512 448 41 20 .24756 .25552 3.9136 .96887 40 20 .26443 .27419 3.6470 .96440 40 21 784 583 .9089 880 39 21 471 451 .6429 433 39 22 813 614 .9042 873 38 22 500 482 .6387 425 38 23 841 645 .8995 866 37 23 528 513 .6346 417 37 24 869 676 .8947 858 36 24 556 545 .6305 410 36 25 .24897 .25707 3.8900 .96851 35 25 .26584 .27576 3.6264 .96402 35 26 925 738 .8854 844 34 26 612 607 .6222 394 34 27 954 769 .8807 837 33 27 640 638 .6181 386 33 28 .24982 800 .8760 829 32 28 668 670 .6140 379 32 29 .25010 831 .8714 822 31 29 696 701 .6100 371 31 30 .25038 .25862 3.8667 .96815 30 30 .26724 .27732 3.6059 .96363 30 31 066 893 .8621 807 29 31 752 764 .6018 355 29 32 094 924 .8575 800 28 32 780 795 .5978 347 28 33 122 955 .8528 793 27 33 808 826 .5937 340 27 34 151 .25986 .8482 786 26 34 836 858 .5897 332 2(i 35 .25179 .26017 3.8436 .96778 25 35 .26864 .27889 3.5856 .96324 25 36 207 048 .8391 771 24 36 892 921 .5816 316 24 37 235 079 .8345 764 23 37 920 952 .5776 308 23 38 263 110 .8299 766 22 38 948 .27983 .5736 301 22 39 291 141 .8254 749 21 39 .26976 .28015 .5696 293 21 40 .25320 .26172 3.8208 .96742 20 40 .27004 .28046 3.5656 .96285 20 41 348 203 .8163 734 19 41 032 077 .5616 277 19 42 376 235 .8118 727 18 42 060 109 .5576 269 18 43 404 266 .8073 719 17 43 088 140 .5536 261 17 44 432 297 .8028 712 16 44 116 172 .5497 253 1^ 45 .25460 .26328 3.7983 .96705 15 45 .27144 .28203 3.5457 .96246 15 46 488 359 .7938 697 14 46 172 234 .5418 238 14 47 516 390 .7893 690 13 47 200 266 .5379 230 13 48 545 421 .7848 682 12 48 228 297 .5339 222 12 49 573 452 .7804 675 11 49 256 329 .5300 214 11 50 .25601 .26483 3.7760 .96667 10 50 .27284 .28360 3.5261 .96206 10 51 629 515 .7715 660 9 51 312 391 .5222 198 9 52 657 546 .7671 653 8 52 340 423 .5183 190 8 53 685 577 .7627 645 7 53 368 454 .5144 182 7 54 713 608 .7583 638 6 54 396 486 .5105 174 6 55 .25741 .26639 3.7539 .96630 5 55 .27424 .28517 3.5067 .96166 5 56 769 670 .7495 623 4 56 452 549 .5028 158 4 57 798 701 .7451 615 3 57 480 680 .4989 150 3 58 826 733 .7408 608 2 58 508 612 .4951 142 ^ 59 854 764 .7364 600 1 59 536 643 .4912 134 1 60 .25882 .26795 3.7321 .9()593 60 .27564 .28675 3.4874 .96126 Gob Ctn Tan Sin / Cos Ctn Tan Sin f 30 16° — Values of Trigonometric Functions — 17" [" / Sin Tan Ctn Cos f Sin Tan Ctn Cos .27564 .28675 3.4874 .96126 60 .29237 .30573 3.2709 .95630 60 1 592 706 .4836 118 59 1 265 605 .2675 622 59 2 620 738 .4798 110 58 2 293 637 .2641 613 58 3 648 769 .4760 102 57 3 321 669 .2607 605 57 4 676 801 .4722 094 56 4 348 700 .2573 596 56 5 .27704 .28832 3.4684 .96086 55 5 .29376 .30732 3.2539 .95588 65 6 731 864 .4646 078 54 6 404 764 .250f3 579 54 7 759 895 .4608 070 53 7 432 79(3 .2472 571 53 8 787 927 .4570 062 52 8 460 828 .2438 562 52 9 815 958 .4533 054 51 9 487 860 .2405 554 51 10 .27843 .28990 3.4495 .96046 50 10 .29515 .30891 3.2371 .95545 60 11 871 .29021 .4458 037 49 11 543 923 .2338 536 49 12 899 053 .4420 029 48 12 571 955 .2305 528 48 13 927 084 .4383 021 47 13 599 .30987 .2272 619 47 14 955 116 .4346 013 46 14 626 .31019 .2238 511 46 15 .27983 .29147 3.4308 .9(3005 45 15 .29654 .31051 3.2205 .95502 45 16 .28011 179 .4271 .95997 44 16 682 083 .2172 493 44 17 039 210 .4234 989 43 17 710 115 .2139 485 43 18 067 242 .4197 981 42 18 737 147 .2106 476 42 19 095 274 .4160 972 41 19 765 178 .2073 467 41 20 .28123 .29305 3.4124 .95964 40 20 .29793 .31210 3.2041 .95459 40 21 150 337 .4087 956 39 21 821 242 .2008 450 39 22 178 368 .4050 948 38 22 849 274 .1975 441 38 23 206 400 .4014 940 37 23 876 306 .1943 433 37 24 234 432 .3977 931 36 24 904 338 .1910 424 36 25 .28262 .29463 3.3941 .95923 35 25 .29932 .31370 3.1878 .95415 36 26 290 495 .3904 915 34 26 960 402 .1845 407 34 27 318 526 .3868 907 33 27 .29987 434 .1813 398 33 28 346 558 .3832 898 32 28 .30015 466 .1780 389 32 29 374 590 .3796 890 31 29 043 498 .1748 380 31 30 .28402 .29621 3.3759 .95882 30 30 .30071 .31530 3.1716 .95372 30 31 429 653 .3723 874 29 31 098 562 .1684 363 29 32 457 685 .3687 865 28 32 126 594 .1652 354 28 33 485 716 .3652 857 27 33 154 626 .1620 345 27 34 513 748 .3616 849 26 34 182 658 .1588 337 26 35 .28541 .29780 3.3580 .95841 25 35 .30209 .31690 3.1556 .95328 25 36 569 811 .3544 832 24 36 237 722 .1524 319 24 37 597 843 .3509 824 23 37 265 754 .1492 310 23 38 625 875 .3473 816 22 38 '292 786 .1460 301 22 39 652 906 .3438 807 21 39 320 818 .1429 293 21 40 .28680 .29938 3.3402 .95799 20 40 .30348 .31850 3.1397 .95284 20 41 708 .29970 .3367 791 19 41 376 882 .1366 275 19 42 736 .30001 .3332 782 18 42 403 914 .1334 266 18 43 764 033 .3297 774 17 43 431 946 .1303 257 17 M 792 065 .3261 76(3 16 44 459 .31978 .1271 248 16 45 .28820 .30097 3.3226 .95757 15 45 .30486 .32010 3.1240 .95240 15 46 847 128 .3191 749 14 46 514 042 .1209 231 14 47 875 160 .3156 740 13 47 542 074 .1178 222 13 48 903 192 .3122 732 12 48 570 106 .1146 213 12 49 931 224 .3087 724 11 49 597 139 .1115 204 11 50 .28959 .30255 3.3052 .95715 10 50 .30625 .32171 3.1084 .95195 10 51 .28987 287 .3017 707 9 51 653 203 .1053 186 9 52 .29015 319 .2983 698 8 52 680 235 .1022 177 8 53 042 351 .2948 690 7 53 708 267 .0991 168 7 54 070 382 .2914 681 6 54 736 299 .0961 159 6 55 .29098 .30414 3.2879 .95673 5 55 .30763 .32331 3.0930 .95150 6 56 126 446 .2845 664 4 56 791 363 .0899 142 4 57 154 478 .2811 656 3 57 819 396 .0868 133 3 58 182 509 .2777 647 2 58 846 428 .0838 124 2 59 209 541 .2743 639 1 59 874 460 .0807 115 1 60 .29237 .30573 3.2709 .95630 60 .30902 .32492 3.0777 .95106 Cos Ctn Tan Sin 1 Cos Ctn Tan Sin / 11] 18°— Talues of Trigonometric Functions — 19° 31 / Sin Tan Ctn Cos .30902 .32492 3.0777 .95106 60 1 929 524 .0746 097 59 2 957 556 .0716 088 58 3 .30985 588 .0686 079 57 4 .31012 621 .0655 070 56 5 .31040 .32653 3.0625 .95061 55 6 068 685 .0595 052 54 7 095 717 .0565 043 53 8 123 749 .0535 033 52 9 151 782 .0505 024 51 10 .31178 .32814 3.0475 .95015 50 11 206 846 .0445 .95006 49 12 233 878 .0415 .94997 48 13 261 911 .0385 988 47 14 289 943 .0356 979 46 15 ,31316 .32975 3.0326 .94970 45 16 344 .33007 .02% 961 44 17 372 040 .0267 952 43 18 399 072 .0237 943 42 19 427 104 .0208 933 41 20 .31454 .33136 3.0178 .94924 40 21 482 169 .0149 915 39 22 510 201 .0120 906 38 23 537 233 .0090 897 37 24 565 266 .0061 888 36 25 .31593 .33298 3.0032 .94878 35 26 620 330 3.0003 869 34 27 648 363 2.9974 860 33 28 675 395 .9945 851 32 29 703 427 .9916 842 31 30 .31730 .33460 2.9887 .94832 30 31 758 492 .9858 823 29 32 786 524 .9829 814 28 33 813 557 .9800 805 27 34 841 589 .9772 795 26 35 .31868 .33621 2.9743 Mim 25 36 896 654 .9714 Til 24 37 923 686 .9686 768 23 38 951 718 .9657 758 22 39 .31979 751 .9629 749 21 40 .32006 .33783 2.9600 .94740 20 41 034 816 .9572 730 19 42 061 848 .9544 721 18 43 089 881 .9515 712 17 44 116 913 .9487 792 16 45 .32144 .33945 2.9459 .94693 15 46 171 .33978 .9431 684 14 47 199 .34010 .9403 674 13 48 227 043 .9375 665 12 49 254 075 .9347 656 11 50 .32282 .34108 2.9319 .94(346 10 51 309 140 .9291 637 9 52 337 173 .9263 627 8 53 364 205 .9235 618 7 54 392 238 .9208 609 6 55 .32419 .34270 2.9180 .94599 5 56 447 303 .9152 590 4 57, 474 335 .9125 580 3 58" 502 368 .9097 571 2 59 529 400 .9070 561 1 60 .32557 .34433 2.9042 .94552 Cos Ctn Tan Sin / 1 Sin Tan Ctn Cos .32557 .34433 2.9042 .94552 60 1 584 465 .9015 642 59 2 612 498 .8987 533 58 3 639 530 .8960 523 57 4 667 563 .8933 514 56 5 .32694 .3459(3 2.8^X)5 .94504 55 6 722 628 .8878 495 54 7 749 661 .8851 . 485 53 8 777 693 .8824 476 52 9 804 726 .8797 466 51 10 .32832 .34758 2.8770 .94457 50 11 859 791 .8743 447 49 12 887 824 .8716 438 48 13 '914 856 .8689 428 47 14 942 889 .8662 418 46 15 .32969 .34922 2.8636 .94409 45 16 .32997 954 .8(309 399 44 17 .33024 .31987 .8582 390 43 18 051 .35020 .8556 380 42 19 079 052 .8529 370 41 20 .33106 .35085 2.8502 .94361 40 21 134 118 .8476 351 39 22 161 150 .8449 342 38 23 189 183 .8423 332 37 24 216 216 .8397 322 36 25 .33244 .35248 2.8370 .94313 35 26 271 281 .8344 303 34 27 298 314 .8318 293 33 28 326 346 .8291 284 32 29 353 379 .8265 274 31 30 .33381 .35412 2.8239 .94264 30 31 408 445 .8213 254 29 32 436 477 .8187 •245 28 33 463 510 .8161 235 27 34 m) 543 .8135 225 26 35 .33518 .35576 2.8109 .94215 25 36 545 608 .8083 206 24 37 573 641 .8057 196 23 38 600 674 .8032 186 22 39 627 707 .8006 176 21 40 .33(355 .35740 2.7980 .94167 20 41 682 772 .7955 157 19 42 710 805 .7929 147 18 43 737 838 .7903 137 17 44 7(>4 871 .7878 127 16 45 .33792 .35904 2.7852 .94118 15 46 819 937 .7827 108 14 47 846 .35969 .7801 098 13 48 874 .36002 .7776 088 12 49 901 035 .7751 078 11 50 .33929 .36068 2.7725 .94068 10 51 956 101 .7700 058 9 52 .33983 134 .7675 049 8 53 .34011 167 .7650 039 7 54 038 199 .7625 029 6 55 .34065 .36232 2.7600 .94019 5 56 093 265 .7575 .94009 4 57 120 298 .7550 .93999 3 58 147 331 .7525 989 2 59 175 364 .7500 979 1 60 .34202 .36397 2.7475 .93969 Cos Ctn Tan Sin / wto nap 32 20° — Values of Trigonometric Functions — 21* / Sin Tan Ctn Cos .34202 .36397 2.7475 .93969 60 1 229 430 .7450 959 69 2 257 463 .7425 949 58 3 284 496 .7400 939 57 4 311 529 .7376 929 66 5 .34339 .36562 2.7351 .93919 55 6 366 595 ■ 7326 909 54 7 393 628 .7302 899 53 8 421 661 .7277 889 52 9 448 694 .7253 879 51 10 .34476 .36727 2.7228 .93869 50 11 603 760 .7204 859 49 12 630 793 .7179 849 48 13 657 826 .7155 839 47 14 684 859 .7130 829 46 15 .34612 .36892 2.7m) .93819 45 16 639 925 .7082 809 44 17 666 958 .7058 799 43 18 694 .36991 .7034 789 42 19 721 .37024 .7009 779 41 20 .34748 .37057 2.6985 .93769 40 21 775 090 .6961 759 39 22 803 123 .6937 748 38 23 830 157 .6913 738 37 24 857 190 .6889 728 36 25 .34884 .37223 2.6865 .93718 35 26 912 266 .6841 708 34 27 939 289 .6818 698 33 28 966 322 .6794 688 32 29 .34993 355 .6770 677 31 30 .35021 .37388 2.6746 .93667 30 31 048 422 .6723 657 29 32 676 455 .6699 647 28 33 102 488 .6675 637 27 34 130 521 .6652 626 26 35 • .35157 .37554 2.6628 .93616 25 36 184 588 .6605 606 24 37 211 621 .6581 596 23 38 239 654 .6558 585 22 39 266 687 .65:34 575 21 40 .35293 .37720 2.6511 .93565 20 41 320 754 .6488 655 19 42 347 787 .6464 644 18 43 375 820 .6441 534 17 44 402 853 .6418 524 16 45 .35429 .37887 2.6395 .93514 15 46 456 920 .6371 503 14 47 484 953 .6:348 493 13 48 511 .37986 .6325 483 12 49 538 .38020 .6302 472 11 50 .35565 .38053 2.6279 .93462 10 61 592 086 .6256 452 9 62 619 120 .6233 441 8 63 647 153 .6210 431 7 54 674 186 .6187 420 6 55 .35701 .38220 2.6165 .93410 5 56 728 253 .6142 400 4 57 756 286 .6119 389 3 58 782 320 .6096 379 2 59 810 353 .6074 368 1 60 .36837 .38386 2 6051 .93358 Cos Ctn Tan Sin / 1 Sin Tan Ctn Cos ~~0 .35837 .38386 2.6051 .93358 60 1 864 420 .6028 348 59 2 891 453 .6006 337 58 3 918 487 .6983 327 67 4 945 520 .5961 316 56 5 .35973 .38553 2.6938 .93306 55 6 .36000 687 .5916 295 64 7 027 620 .5893 285 53 8 • 054 664 .6871 274 52 9 081 687 .5848 264 51 10 .36108 .38721 2.5826 .93253 50 11 135 754 .5804 243 49 12 162 787 .5782 232 48 13 l^X) 821 .5759 222 47 14 217 854 .5737 211 46 15 .36244 .38888 2.5715 .93201 45 16 271 921 .5693 190 44 17 298 955 .5671 180 43 18 325 .38988 .5649 169 42 19 352 .39022 .5627 159 41 20 .36379 .39055 2.5605 .93148 40 21 406 089 .5583 137 39 22 434 122 .5561 127 38 23 461 156 .5539 116 37 24 488 190 .6617 106 36 25 .36515 .39223 2.5495 .93095 35 26 542 257 .6473 084 34 27 669 290 .6452 074 33 28 696 324 .5430 063 32 29 623 357 .5408 052 31 30 .36650 .39391 2.5386 .93042 30 31 677 425 .5365 031 29 32 704 458 .5343 020 28 33 731 492 .5322 .93010 27 34 758 526 .5300 .92999 26 35 .36785 .39559 2.5279 .92988 25 36 812 693 .5257 978 24 37 839 626 .5236 mi 23 38 867 660 .5214 956 22 39 894 694 .6193 945 21 40 .36921 .39727 2.5172 .92935 20 41 948 761 .5160 924 19 42 .36975 796 .5129 913 18 43 .37002 829 .5108 902 17 44 029 862 .5086 892 16 45 .37056 .39896 2.5065 .92881 15 46 083 930 .5044 870 14 47 110 963 .5023 859 13 48 137 .39997 .5002 849 12 49 164 .40031 .4981 838 11 50 .37191 .40065 2.4960 .92827 10 51 218 098 .4939 816 9 62 245 132 .4918 805 8 53 272 166 .4897 794 7 64 299 200 .4876 784 6 55 .37326 .40234 2.4865 .92773 5 56 353 267 .4834 762 4 57 380 301 .4813 751 3 58 407 335 .4792 740 2 69 434 369 .4772 729 1 60 .37461 .40403 2.4751 .92718 Cos Ctn Tan Sin 1 aQ9 11] 2 2°— Values of Trigonometric Functions — 23° 33 / Sin Tan Ctn Cos f Sin 1 Tan Ctn Cos .37461 .40403 2.4751 .92718 60 .39073 .42447 2.3559 .92050 60 1 488 436 .4730 707 59 1 100 482 .3539 039 59 2 515 470 .4709 697 58 2 127 616 .3520 028 58 3 542 504 .4689 686 57 3 153 551 .3501 016 57 4 569 538 .4668 675 56 4 180 585 .3483 .92005 56 5 .37595 .40572 2.4648 .92664 55 5 .39207 .42619 2.3464 .91994 55 6 622 606 .4627 653 54 6 234 (354 .3445 982 54 7 649 640 .4606 642 53 7 260 688 .^426 971 53 8 676 674 .4586 631 52 8 287 722 .3407 959 52 9 703 707 .4566 620 51 9 314 757 .3388 948 51 10 ,37730 .40741 2.4545 .92609 50 10 .39341 .42791 2.3369 .91936 50 11 757 775 /.4525 598 49 11 3()7 826 .3351 925 49 12 784 809 .4504 587 48 12 394 860 .3332 914 48 13 811 843 .4484 576 47 13 421 894 .3313 902 47 14 838 877 .4464 565 46 14 448 929 .3294 891 46 15 .37865 .40911 2.4443 .92554 45 15 .39474 .42963 2.3276 .91879 45 16 892 945 .4423 543 44 16 501 .42998 .3257 868 44 17 919 .40979 .4403 532 43 17 528 .43032 .3238 856 43 18 946 .41013 .4383 521 42 18 555 067 .3220 845 42 19 973 047 .4362 510 41 19 581 101 .3201 833 41 20 .37999 .41081 2.4342 .92499 40 20 .39608 .43136 2.3183 .91822 40 21 .38026 115 .4322 488 39 21 635 170 .3164 810 39 22 053 149 .4302 477 38 22 661 205 .3146 799 38 23 080 183 .•4282 466 37 23 688 239 .3127 787 37 24 107 217 .4262 455 36 24 715 274 .3109 775 36 25 .38134 .41251 2.4242 .92444 35 25 .39741 .43308 2.3090 .91764 35 26 161 285 .4222 432 34 26 768 343 .3072 752 34 27 188 319 !4202 421 33 27 795 378 .3053 741 33 28 215 353 .4182 410 32 28 822 412 .3035 729 32 29 241 387 .4162 399 31 29 848 447 .3017 718 31 30 .38268 .41421 2.4142 .92.388 30 30 .39875 .43481 2.2998 .91706 30 31 295 455 .4122 377 29 31 902 516 .2980 694 29 32 322 490 .4102 366 28 32 928 550 .2962 683 28 33 349 524 .4083 355 27 33 955 685 .2944 671 27 34 376 558 .4063 343 26 34 .39982 620 .2925 660 26 35 .38403 .41592 2.4043 .92332 25 35 .40008 .43654 2.2907 .91648 25 36 430 626 .4023 321 24 36 035 689 .2889 636 24 37 456 660 .4004 310 23 37 062 724 .2871 625 23 38 483 694 .3984 299 22 38 088 758 .2853 613 22 39 510 728 .3964 287 21 39 115 793 .2835 601 21 40 .38537 .41763 2.3945 .92276 20 40 .40141 .43828 2.2817 .91590 20 41 564 797 .3925 265 19 41 168 862 .2799 578 19 42 591 831 .390\'5 254 18 42 195 897 .2781 566 18 43 617 865 .3886 243 17 43 221 932 .2763 555 17 44 644 899 .3867 231 16 44 248 .43<)66 .2745 543 \^ 45 .38671 .41933 2.3847 .92220 15 45 .40275 .44001 2.2727 .91531 15 46 698 .419()8 .3828 209 14 46 301 036 .2709 519 14 47 725 .42002 .3808 198 13 47 328 071 .2691 508 13 48 752 036 .3789 186 12 48 355 105 .2673 496 12 49 778 070 .3770 175 11 49 381 140 .2655 484 11 50 .38805 .42105 2.3750 .92164 10 50 .40408 .44175 2.2637 .91472 10 51 832 139 .3731 152 9 51 434 210 .2620 461 9 52 859 173 .3712 141 8 52 461 244 .2602 449 8 53 886 207 .3693 130 7 53 488 279 .2584 437 7 54 912 242 .3673 119 6 54 514 314 .2566 425 6 55 .38939 .42276 2.3654 .92107 5 55 .40541 .44349 2.2549 .91414 5 56 966 310 .3635 096 4 56 567 384 .2531 402 4 57 .38993 345 .3616 085 3 57 594 418 .2513 390 3 58 .39020 379 .3597 073 2 58 621 453 .2496 378 2 59 046 413 .3578 062 1 59 647 488 .2478 366 1 60 .39073 .42447 2.3559 .92050 60 .40674 .44523 2.2460 .91355 Cos Gtn Tan Sin / Cos Ctn Tan Sin 1 34 2r — Values of Trigonometric Functions -^ 25^ Pl / Sin Tan Ctn Cos 1 Sin Tan Ctn Cos .40674 .44523 2.2460 .91355 60 .42262 .46631 2.1445 .90631 60 1 700 558 .2443 343 69 1 288 666 .1429 618 59 2 727 593 .2425 331 68 2 316 702 .1413 606 58 3 753 627 .2408 319 67 3 341 737 .1396 594 67 4 780 662 .2390 307 56 4 367 772 .1380 582 56 5 .40806 .44697 2.2373 .91295 55 5 .42394 .46808 2.1364 .90569 55 6 833 732 .2356 283 54 6 420 843 .1348 557 54 7 860 767 .2338 272 53 7 446 879 .1332 645 53 8 886 802 .2320 260 62 8 473 914 .1316 532 52 9 913 837 .2303 248 61 9 499 950 .1299 620 51 10 .40939 .44872 2.2286 .91236 50 10 .42626 .46986 2.1283 .90507 60 11 966 907 .2268 224 49 11 652 .47021 .1267 495 49 12 .40992 942 .2251 212 48 12 678 056 .1251 483 48 13 .41019 .44977 .2234 200 47 13 604 092 .1235 470 47 14 045 .45012 .2216 188 46 14 631 128 .1219 458 46 15 .41072 .45047 2.2199 .91176 45 15 .42657 .47163 2.1203 .90446 45 16 098 082 .2182 164 44 16 683 199 .1187 433 44 17 125 117 .2165 152 43 17 709 234 .1171 421 43 18 151 152 .2148 140 42 18 736 270 .1155 408 42 19 178 187 .2130 128 41 19 762 305 .1139 396 41 20 .41204 .45222 2.2113 .91116 40 20 .42788 .47341 2.1123 .90383 40 21 231 257 .2096 104 39 21 815 377 .1107 371 39 22 257 292 .2079 092 38 22 841 412 .1092 358 38 23 284 327 .2062 080 37 23 867 448 .1076 346 37 24 310 362 .2046 068 36 24 894 483 .1060 334 36 25 .41337 .45397 2.2028 .91056 35 25 .42920 .47519 2.1044 .90321 35 26 363 432 .2011 044 34 26 946 555 .1028 309 34 27 390 467 .1994 032 33 27 972 590 .1013 296 33 28 416 502 .1977 020 32 28 .42999 626 .0997 284 32 29 443 538 .1960 .91008 31 29 .43025 662 .0981 271 31 30 .41469 .46573 2.1943 .90996 30 30 .43051 .47698 2.0965 .90259 30 31 496 608 .1926 984 29 31 077 733 .0950 246 29 32 522 643 .1909 972 28 32 104 769 .0934 233 28 33 549 678 .1892 960 27 33 130 805 .0918 221 27 34 675 713 .1876 948 26 34 156 840 .0903 208 26 35 .41602 .45748 2.1859 .90936 25 35 .43182 .47876 2.0887 .90196 25 36 628 784 .1842 924 24 36 209 912 .0872 183 24 37 655 819 .1825 911 23 37 236 948 .0866 171 23 38 681 864 .1808 899 22 38 261 .47984 .0840 158 22 39 707 889 .1792 887 21 39 287 .48019 .0826 146 21 40 .41734 .46924 2.1775 .90875 20 40 .43313 .48055 2.0809 .90133 20 41 760 960 .1758 863 19 41 340 091 .0794 120 19 42 787 .46995 .1742 851 18 42 366 127 .0778 108 18 43 813 .460130 .1725 839 17 43 392 163 .0763 095 17 44 840 065 .1708 826 16 44 418 198 .0748 082 16 45 .41866 .46101 2.1692 .90814 15 45 .43445 .48234 2.0732 .90070 15 46 892 136 .1675 802 14 46 471 270 .0717 057 14 47 919 171 .1659 790 13 47 497 306 .0701 045 13 48 945 206 .1642 778 12 48 623 342 .0686 032 12 49 972 242 .1625 766 11 49 649 378 .0671 019 11 50 .41998 .46277 2.1609 .90753 10 50 .43575 .48414 2.0655 .90007 10 51 .42024 312 .1592 741 9 61 602 450 .0640 .89994 9 62 051 348 .1576 729 8 52 628 486 .0625 981 8 53 077 383 .1560 717 7 63 654 621 .0609 968 7 54 104 418 .1543 704 6 64 680 567 .0594 956 6 55 .42130 .46454 2.1527 .90692 5 55 .43706 .48593 2.0679 .89943 5 56 166 489 .1510 680 4 56 733 629 .0564 930 4 57 183 525 .1494 668 3 57 759 665 .0549 918 3 58 209 560 .1478 665 2 68 785 701 .0633 905 2 59 235 595 .1461 643 1 59 811 737 .0518 892 1 60 .42262 .46631 2.1445 .90631 60 .43837 .48773 2 0503 .89879 Cos Ctn Tan Sin 1 Cos Ctn Tan Sin ; 26° — Values of Trigonometric Functions — 27° 35 / Sin Tan Ctn Cos / Sin Tan Ctn Cos .43837 .48773 2.0503 .89879 60 .45399 .50953 1.9626 .89101 60 1 863 809 .0^88 867 59 1 425 .50989 .9612 087 59 2 889 845 .0473 854 58 2 451 .51026 .9598 074 58 3 916 881 .0458 841 57 3 477 063 .9584 061 57 4 942 917 .0443 828 56 4 503 099 .9570 048 56 5 .43968 .48953 2.0428 .89816 55 5 .45529 .51136 1.9556 .89035 55 6 .43994 .48989 .0413 803 54 6 554 173 .9542 021 54 7 .44020 .49026 .0398 790 53 7 580 209 .9528 .89008 53 8 046 062 .0383 777 52 8 606 246 .9514 .88995 52 9 072 098 .0368 764 51 9 632 283 .9500 981 51 10 .44098 .49134 2.0353 .89752 50 10 .45658 .51319 1.9486 .88968 50 11 124 170 .0338 739 4<) 11 684 356 .9472 955 49 12 151 206 .0323 726 48 12 710 393 .9458 942 48 13 177 242 .0308 713 47 13 736 430 .9444 928 47 14 203 278 .0293 700 46 14 762 467 .9430 915 46 15 .44229 .49315 2.0278 .89687 45 15 .45787 .51503 1.9416 .88902 45 16 255 351 .0263 674 44 16 813 540 .9402 888 44 17 281 387 .0248 662 43 17 839 577 .9388 875 43 18 307 423 .0233 649 42 18 865 614 .9375 862 42 19 333 459 .0219 636 41 19 891 651 .9361 848 41 20 .44359 .49495 2.0204 .89623 40 20 .45917 .51688 1.9347 .88835 40 21 385 532 .0189 610 39 21 942 724 .9333 822 39 22 411 568 .0174 597 38 22 968 761 .9319 808 38 23 437 604 .0160 584 37 23 .45994 798 .9306 795 37 24 464 640 .0145 571 36 24 .46020 835 .9292 782 36 25 .44490 .49677 2.0130 .89558 35 25 .46046 .51872 1.9278 .88768 85 26 516 713 .0115 545 34 26 072 909 .9265 755 34 27 542 749 .0101 532 33 27 097 946 .9251 741 33 28 568 786 .0086 519 32 28 123 .51983 .9237 728 32 29 594 822 .0072 506 31 29 149 .52020 .9223 715 31 30 .44620 .49858 2.0057 .89493 30 30 .46175 .52057 1.9210 .88701 30 31 646 894 .0042 480 29 31 201 094 .9196 688 29 32 672 931 .0028 467 28 32 226 131 .9183 674 28 33 698 .49967 2.0013 454 27 33 252 168 .9169 661 27 34 724 .50004 1.9999 441 26 34 278 205 .9155 647 26 35 .44750 .50040 1.9984 .89428 25 35 .46304 .52242 1.9142 .88634 25 36 776 076 .9970 415 24 36 330 279 .9128 620 24 37 802 113 .9955 402 23 37 355 316 .9115 607 23 38 828 149 .9941 389 22 38 381 353 .9101 693 22 39 854 185 .9926 376 21 39 407 390 .9088 580 21 40 .44880 .50222 1.9912 .89363 20 40 .46433 .52427 1.9074 .88566 20 41 906 258 .9897 350 19 41 458 464 .9061 553 19 42 932 295 .9883 337 18 42 484 501 .9047 539 18 43 958 331 .9868 324 17 43 510 538 .9034 526 17 44 .44984 368 .9854 311 16 44 536 575 .9020 612 16 45 .45010 .50404 1.9840 .89298 15 45 .46561 .52613 1.9007 .88499 15 46 036 441 .9825 285 14 46 587 650 .8993 485 14 47 062 477 .9811 272 13 47 613 687 .8980 472 13 48 088 514 .9797 259 12 48 639 724 .8967 458 12 49 114 550 .9782 245 11 49 664 761 .8953 445 11 50 .45140 .50587 1.9768 .89232 10 50 .46690 .52798 1.8940 .88431 10 51 166 623 .9754 219 9 51 716 836 .8927 417 9 52 192 660 .9740 206 8 52 742 873 .8913 404 8 53 218 696 .9725 193 7 53 767 910 .8900 390 7 54 243 733 .9711 180 6 54 793 947 .8887 377 6 55 .45269 .50769 1.9697 .89167 5 55 .46819 .52985 1.8873 .88363 5 56 295 806 .9683 153 4 56 844 .53022 .8860 349 4 •'57 321 843 .9669 140 3 57 870 059 .8847 336 3 58 347 879 .9654 127 2 58 896 096 .8834 322 2 59 373 916 .9640 114 1 59 921 134 .8820 308 1 60 .45399 .50953 1.9626 .89101 60 .46947 .53171 1.8807 .88295 Cos Ctn Tan Sin / Cos Ctn Tan Sin / A0»o 36 28° — Values of Trigonometric Functions — 29° m r Sin Tan Ctn Cos / Sin Tan Ctn Cos .46947 .53171 1.8807 .88295 60 .48481 .55431 1.8040 .87462 60 1 973 208 .8794 281 59 1 506 469 .8028 448 59 2 .46999 246 .8781 267 58 2 532 507 .8016 434 58 3 .47024 283 .8768 254 57 3 557 545 .8003 420 57 4 050 320 .8755 240 56 4 583 583 .7991 406 56 5 .47076 .53358 1.8741 .88226 55 5 .48608 .55621 1.7979 .87391 55 6 101 395 .8728 213 54 6 634 659 .7966 377 54 7 127 432 .8715 199 53 7 659 697 .7954 363 53 8 153 470 .8702 185 52 8 684 736 .7942 349 52 9 178 507 .8689 172 51 9 710 774 .7930 335 51 10 .47204 .53545 1.8676 .88158 50 10 .48735 .55812 1.7917 .87321 50 11 229 582 .8663 144 49 11 761 850 .7905 306 49 12 255 620 .8650 130 48 12 786 888 .7893 292 48 13 281 657 .8637 117 47 13 811 926 .7881 278 47 14 306 694 .8624 103 46 14 837 .55964 .7868 264 46 15 .47332 .53732 1.8611 .88089 45 16 .48862 .56003 1.7856 .87250 45 16 358 769 .8598 075 44 16 888 041 .7844 235 44 17 383 807 .8585 062 43 17 913 079 .7832 221 43 18 409 844 .8572 048 42 18 938 117 .7820 207 42 19 434 882 .8559 034 41 19 964 156 .7808 193 41 20 .47460 .53920 1.8546 .88020 40 20 .48989 .56194 1.7796 .87178 40 21 486 957 .8533 .88006 39 21 .49014 232 .7783 164 39 22 511 .53995 .8520 .87993 38 22 040 270 .7771 150 38 23 537 .54032 .8507 979 37 23 065 309 .7759 136 37 24 562 070 .8495 965 36 24 090 347 .7747 121 36 25 .47588 .54107 1.8482 .87951 35 25 .49116 .56385 1.7735 .87107 35 26 614 145 .8469 937 34 26 141 424 .7723 093 34 27 639 183 .8456 923 33 27 166 462 .7711 079 33 28 6a5 220 .8443 909 32 28 192 501 .7699 064 32 29 690 258 .8430 896 31 29 217 539 .7687 050 31 30 .47716 .54296 1.8418 .87882 30 30 .49242 .56577 1.7675 .87036 30 31 741 333 .8405 868 29 31 268 616 .7663 021 29 32 767 371 .8392 854 28 32 293 654 .7651 .87007 28 33 793 409 .8379 840 27 33 318 693 .7639 .86993 27 34 818 446 .8367 826 26 34 344 731 .7627 978 26 35 .47844 .54484 1.8354 .87812 25 35 .49369 .56769 1.7615 .86964 25 36 869 522 .8341 798 24 36 394 808 .7603 949 24 37 895 560 .8329 784 23 37 419 846 .7591 935 23 38 920 597 .8316 770 22 38 445 885 .7579 921 22 39 946 635 .8303 756 21 39 470 923 .7567 906 21 40 .47971 .54673 1.8291 .87743 20 40 .49195 .56962 1.7556 .86892 20 41 .47997 711 .8278 729 19 41 521 .57000 .7544 878 19 42 .48022 748 .8265 715 18 42 546 039 .7532 863 18 43 048 786 .8253 701 17. 43 571 078 .7520 849 17 44 073 824 .8240 687 16 44 596 116 .7508 834 16 45 .48099 .54862 1.8228 .87673 15 45 .49622 .57155 1.7496 .86820 15 46 124 900 .8215 659 14 46 647 193 .7485 805 14 47 150 938 .8202 645 13 47 672 232 .7473 791 13 48 175 .54975 .8190 631 12 48 697 271 .7461 777 12 49 201 .55013 .8177 617 11 49 723 309 .7449 762 11 50 .48226 .55051 1.8165 .87603 10 50 .49748 .57348 1.7437 .86748 10 51 252 089 .8152 589 9 51 773 386 .7426 733 9 52 277 127 .8140 575 8 52 798 425 .7414 719 8 53 303 165 .8127 561 7 53 824 464 .7402 704 7 54 328 203 .8115 546 6 54 849 503 .7391 690 6 55 .48354 .55241 1.8103 .87532 5 55 .49874 .57541 1.7379 .86675 5 56 379 279 .8090 518 4 56 899 580 .7367 661 4 57 405 317 .8078 504 3 57 924 619 .7355 646 3 58 430 355 .8065 490 2 58 950 657 .7344 632 2 59 456 393 .8053 476 1 59 .49975 696 .7332 617 1 60 .48481 .55431 1.8040 .87462 60 .50000 .57735 1.7321 .86603 Cos Ctn Tan Sin / Cos Ctn Tan Sin / fir fiO° II] 30"— Values of Trigouometi •ic Fuuctions — 3r 37 1 Sin Tan Ctn Cos ! Sin Tan Ctn Cos .50000 .57735 1.7321 .86603 60 .51504 .60086 1.6643 .85717 60 1 025 774 .7309 588 59 1 529 126 .6632 702 59 2 050 813 .7297 573 58 2 554 165 .6(321 687 58 S 076 851 .7286 559 57 3 579 205 .6610 672 57 4 101 890 .7274 544 56 4 604 245 .6599 657 56 5 .50126 .57929 1.7262 .86530 55 5 .51628 .60284 1.6588 .85642 55 (> 151 .57i)68 .7251 515 54 6 653 324 .6577 627 54 7 176 .58007 .7239 501 53 7 678 364 .65(36 612 53 8 201 046 .7228 486 52 8 703 403 .6555 597 52 9 227 085 .7216 471 51 9 728 443 .6545 582 51 10 .50252 .58124 1.7205 .86457 50 10 .51753 .60483 1.6534 .85567 50 11 277 162 .7193 442 49 11 778 522 .6523 551 49 12 302 201 .7182 427 48 12 803 562 .6512 536 48 13 327 240 .7170 413 47 13 828 602 .6501 521 47 14 352 279 .7159 398 4<3 14 852 642 .6490 506 46 15 .50377 .58318 1.7147 .86384 45 15 .51877 .60681 1.6479 .85491 45 Ifi 403 357 .71:36 369 44 16 902 721 .6469 476 44 17 428 396 .7124 354 43 17 927 761 .6458 461 43 18 453 435 .7113 'MQ 42 18 952 801 .6447 446 42 19 478 474 .7102 325 41 19 .51977 841 .6436 431 41 20 .50503 .58513 1.7090 .86310 40 20 .52002 .60881 1.6426 .85416 40 21 528 552 .7079 295 39 21 026 921 .6415 401 39 22 653 591 .7067 281 38 22 051 .60960 .(3404 385 (38 23 578 631 .7056 266 37 23 07(3 .61000 .6393 370 37 24 603 670 .7045 251 36 24 101 040 .6383 355 3(3 25 .50628 .58709 1.70:33 .86237 35 25 .52126 .61080 1.6372 .85340 35 26 654 748 .7022 222 34 26 151 120 .6361 325 54 27 679 787 .7011 207 33 27 175 160 .6351 310 3:3 28 704 826 .6999 192 32 28 200 200 .6340 294 32 29 729 865 .6988 178 31 29 225 240 .6329 279 31 30 .50754 .58905 1.6977 .86163 30 30 .52250 .61280 1.6319 .85204 30 31 779 944 .6965 148 29 31 275 320 .6308 249 29 32 804 .58983 .6954 133 28 32 299 360 .6297 234 28 33 829 .59022 .6^3 119 27 33 324 400 .6287 218 27 34 854 061 .6932 104 26 34 349 440 .6276 203 26 35 .50879 .59101 1.6920 .86089 25 35 .52374 .61480 1.6265 .85188 25 3t3 904 140 .6909 074 24 36 399 520 .6255 173 24 37 929 179 .6898 059 23 37 423 561 .6244 157 23 38 954 218 .6887 045 22 38 448 601 .6234 142 22 39 .50979 258 .6875 0:30 21 39 473 641 .6223 127 21 40 .51004 .59297 1.6864 .8()015 20 40 .52498 .61681 1.6212 .85112 20 41 029 336 .6853 .8(3000 19 41 522 721 .6202 096 19 42 054 376 .6J^2 .85985 18 42 547 761 .6191 081 18 43 079 415 .6831 970 17 43 572 801 .6181 066 17 44 104 454 .6820 956 16 44 597 842 .6170 051 16 45 .51129 .59494 1.6808 .85941 15 45 .52621 .61882 1.6160 .85035 15 4(5 154 533 .6797 926 14 46 646 922 .6149 020 14 47 179 573 .6786 911 13 47 671 .61962 .6139 .85005 13 48 204 612 .6775 896 12 48 696 .62003 .6128 .84989 12 49 229 651 .6764 881 11 49 720 043 .6118 974 11 50 .51254 .59691 1.6753 .85866 10 50 .52745 .62083 1.6107 .84959 10 51 279 730 .6742 851 9 51 770 124 .6097 943 9 52 304 770 .6731 836 8 52 794 164 .6087 928 8 53 329 809 .6720 821 7 53 819 204 .6076 913 7 54 354 849 .6709 806 6 54 844 245 .6066 897 6 55 .51379 .59888 1.6698 .85792 5 55 .52869 .62285 1.6055 .84882 5 56 404 928 .6687 777 4 56 893 325 .6045 866 4 57 429 .59967 .6676 762 3 57 918 366 .6034 851 3 58 454 .60007 .6665 747 2 58 943 406 .6024 836 2 59 479 046 .6654 732 1 59 967 446 .6014 820 1 60 .51504 .60086 1.6643 .85717 60 .52992 .62487 1.6003 .84805 Cos Ctn ! Tan Sin / Cos Ctn Tan Sin 1 59° 68° 38 32° — Values of Trigonometric Functions — 33° [ir / Sin Tan Ctn Cos / Sin Tan Ctn Cos .52992 .62487 1.6003 .84805 60 .54464 .64941 1.5399 .83867 60 1 .53017 527 .5993 789 59 1 488 .64982 .5389 851 59 2 041 568 .5983 774 58 2 513 .65024 .5379 835 58 3 066 608 .5972 759 57 3 537 065 .5369 819 57 4 091 649 .5962 743 56 4 561 106 .5359 804 56 5 .53115 .62689 1.5952 .84728 55 5 .54586 .65148 1.5350 .83788 55 6 140 730 .5941 712 54 6 610 189 .5340 772 54 7 164 770 .5931 697 53 7 635 231 .5330 756 53 8 189 811 .5921 681 52 8 659 272 .5320 740 52 9 214 852 .5911 666 51 9 683 314 .5311 724 51 10 .53238 .62892 1.5900 .84650 50 10 .54708 .65355 1.5301 .83708 50 11 263 933 .5890 635 49 11 732 397 .5291 692 49 12 288 .62973 .5880 619 48 12 756 438 .5282 676 48 13 312 .63014 .5869 604 47 13 781 480 .5272 660 47 14 337 055 .5859 588 46 14 805 521 .5262 645 46 15 .53361 .63095 1.5849 .84573 45 15 .54829 .65563 1.5253 .83629 45 16 386 136 .5839 557 44 16 854 604 .5243 613 44 17 411 177 .5829 542 43 17 878 646 .5233 597 43 18 435 217 .5818 526 42 18 902 688 .5224 581 42 19 460 258 .5808 511 41 19 927 729 .5214 565 41 20 .53484 .63299 1.5798 .84495 40 20 .54951 .65771 1.5204 .83549 40 21 509 340 .5788 480 39 21 975 813 .5195 533 39 22 534 380 .5778 464 38 22 .54999 854 .5185 617 38 23 558 421 .5768 448 37 23 .55024 896 .5175 501 37 24 683 462 .5757 433 36 24 048 938 .5166 485 36 25 .53607 .63503 1.5747 .84417 35 25 .55072 .65980 1.5156 .83469 35 26 632 544 .5737 402 34 26 097 .66021 .5147 453 34 27 656 584 .5727 386 33 27 121 063 .5137 437 33 28 681 625 .5717 370 32 28 145 105 .5127 421 32 29 705 666 .5707 355 31 29 169 147 .5118 405 31 30 .53730 .63707 1.5697 .84339 30 30 .55194 .66189 1.5108 .83389 30 31 754 748 .5687 324 29 31 218 230 .5099 373 29 32 779 789 .5677 308 28 32 242 272 .5089 356 28 33 804 830 .5667 292 27 33 266 314 .5080 340 27 34 828 871 .5657 277 26 34 291 356 .5070 324 26 35 .53853 .63912 1.5647 .84261 25 35 .55315 .66398 1.5061 .83308 25 36 877 953 .5637 245 24 36 339 440 .5051 292 24 37 902 .63994 .5627 230 23 37 363 482 .5042 276 23 38 926 .64035 .5617 214 22 38 388 524 .5032 260 22' 39 951 076 .5607 198 21 39 412 566 .5023 244 21 40 .53975 .64117 1.5597 .84182 20 40 .55436 .66608 1.5013 .83228 20 41 .54000 158 .5587 167 19 41 460 650 .5004 212 19 42 024 199 .5577 151 18 42 484 692 .4994 195 18 43 049 240 .5567 135 17 43 509 734 .4985 179 17 44 073 281 .5557 120 16 44 533 776 .4975 163 16 45 .54097 .64322 1.5547 .84104 15 45 .55557 .66818 1.49(36 .83147 15 46 122 363 .5537 088 14 46 581 860 .4957 131 14 47 146 404 .5527 072 13 47 605 902 .4947 115 13 48 171 446 .5517 057 12 48 630 944 .4938 098 12 49 195 487 .5507 041 11 49 654 .66986 .4928 082 11 50 .54220 .64528 1.5497 .84025 10 50 .55678 .67028 1.4919 .83066 10 51 244 569 .5487 .84009 9 51 702 071 .4910 050 9 52 269 610 .5477 .83994 8 52 726 113 .4900 034 8 53 293 652 .5468 978 7 53 750 155 .4891 017 7 54 317 693 .5458 962 6 54 775 197 .4882 .83001 6 55 .54342 .64734 1.5448 .83946 5 55 .55799 .67239 1.4872 .82985 5 56 366 775 .5438 930 4 56 823 282 .4863 969 4 57 391 817 .5428 915 3 57 847 324 .4854 953 3 58 415 858 .5418 899 2 58 871 366 .4844 936 2 59 440 899 .5408 883 1 59 895 409 .4835 920 1 60 .54464 .64941 1.5399 .83867 60 .55919 .67451 1.4826 .82904 Cos Ctn Tan Sin / Cos Ctn Tan Sin / II] 34°— Values of Trigonometric Functions — 35° 39 / Sin Tan Gtn Cos .55919 .67451 1.4826 .82904 60 1 943 493 .4816 887 59 2 968 536 .4807 871 58 3 .55992 5'78 .4798 855 57 4 .56016 620 .4788 839 56 5 .56040 .67663 1.4779 .82822 55 () 064 705 .4770 806 54 7 088 748 .4761 790 53 8 112 790 .4751 773 52 9 136 832 .4742 757 51 10 .56160 .67875 1.4733 .82741 50 11 184 917 .4724 724 49 12 208 .67960 .4715 708 48 13 232 .68002 .4705 692 47 14 256 045 .4696 675 46 15 .56280 .68088 1.4687 .82659 45 16 305 130 .4678 643 44 17 329 173 .4669 626 43 18 353 215 .4659 610 42 19 377 258 .4650 593 41 20 .56401 .68301 1.4641 .82577 40 21 425 ;^3 .4632 561 39 22 449 386 .4623 544 38 23 473 429 .4614 528 37 24 497 471 .4605 511 36 25 .56521 .68514 1.4596 .82495 35 26 545 557 .4586 478 34 27 569 600 .4577 462 33 28 593 642 •4568 446 32 29 617 685 .4559 429 31 30 .56641 .68728 1.4550 .82413 30 31 . 665 771 .4541 396 29 32 689 814 .4532 380 28 33 713 857 .4523 363 27 34 736 900 .4514 347 26 35 .56760 .68942 1.4505 .82330 25 36 784 .68985 .4496 314 24 37 808 .69028 .4487 297 23 38 832 071 .4478 281 22 39 856 114 .4469 264 21 40 .56880 .69157 1.4460 .82248 20 41 904 200 .4451 231 19 42 928 243 .4442 214 18 43 952 286 .4433 198 17 44 .56976 329 .4424 181 16 45 .57000 .69372 1.4415 .82165 15 46 024 416 .4406 148 14 47 047 459 .4397 132 13 48 071 502 .4388 115 12 49 095 545 .4379 098 11 50 .57119 .69588 1.4370 .82082 10 51 143 631 .4361 065 9 52 167 675 .4352 048 8 53 191 718 .4344 032 7 54 215 761 .4335 .82015 6 55 .57238 .69804 1.4326 .81999 5 .'56 262 847 .4317 982 4 57 286 891 .4308 965 3 58 310 934 .4299 949 2 59 334 .69977 .4290 932 1 60 .57358 .70021 1.4281 .81915 Cos Ctn Tan Sin / Sin Tan Ctn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 .58779 Cos .57358 381 405 429 453 .57477 501 524 548 572 .57596 619 643 667 691 .57715 738 762 786 810 .57833 857 881 904 928 .57952 976 .57999 .58023 047 .58070 094 118 141 165 .58189 212 236 260 283 .58307 330 354 378 401 .58425 449 472 496 519 .58543 567 590 614 637 .58661 684 708 731 755 .70021 0()4 107 151 194 .70238 281 325 368 412 .70455 499 542 586 629 .70673 717 760 804 848 .70891 935 .70979 .71023 066 .71110 154 198 242 285 .71329 373 417 461 505 .71549 593 637 681 725 .71769 813 857 901 946 .71990 .72034 078 122 167 .72211 255 299 344 388 .72432 477 521 565 610 .72654 1.4281 .4273 .4264 .4255 .4246 1.4237 .4229 .4220 .4211 .4202 1.4193 .4185 .4176 .4167 .4158 1.4150 .4141 .4132 .4124 .4115 1.4106 .4097 .4089 .4080 .4071 1.4063 .4054 .4045 .4037 .4028 1.4019 .4011 .4002 .3994 .3985 1.3976 .39(38 .3959 .3951 .3942 1.3934 .3925 .3916 .3908 .3899 1.3891 .3882 .3874 .3865 .3857 1.3848 .3840 .3831 .3823 .3814 1.3806 .3798 .3789 .3781 .3772 1.3764 Cos Ctn Tan .81915 899 882 865 848 .81832 815 798 782 765 .81748 731 714 698 681 .81664 647 631 614 597 .81580 563 546 530 513 .81496 479 462 445 428 .81412 395 378 361 344 .81327 310 293 276 259 .81242 225 208 191 174 .81157 140 123 106 089 .81072 055 038 021 .81004 .80987 970 953 936 919 .80902 Sin 40 36° — Values of Trigonometric Functions — 37° 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Sin Tan Gtn .58779 802 826 849 873 .58896 920 943 967 .58990 .59014 037 061 084 108 .59131 154 178 201 225 .59248 272 295 318 342 .59365 389 412 436 459 .59482 506 529 552 576 .59599 622 646 669 693 .59716 739 763 786 809 .59832 856 879 902 926 .59949 972 .59995 .60019 042 .60065 089 112 135 158 .60182 .72654 699 743 788 832 .72877 921 .72966 .73010 055 .73100 144 189 234 278 .73323 368 413 457 502 .73547 592 637 681 726 .73771 816 861 906 951 .73996 .74041 086 131 176 .74221 267 312 357 402 .74447 492 538 583 628 .74674 719 764 810 855 .74900 946 .74991 .75037 082 .75128 173 219 264 310 .75355 Cos Ctn Tan 1.3764 .3755 .3747 .3739 .3730 1.3722 .3713 .3705 .3697 .3688 1.3680 .3672 .:3663 .3655 .3647 1.3638 .3630 .3(322 .3613 .3605 1.3.597 .3588 .3580 .3572 .3564 1.3555 .3547 .3539 .3531 .3522 1.3514 .3506 .3498 .3490 .3481 1.3473 .3465 .3457 .3449 .3440 1.3432 .3424 .3416 .3408 .3400 1.3392 .3384 .3375 .3367 .3359 1.3351 .3343 .3335 .3327 .3319 1.3311 .3303 .3295 .3287 .3278 1.3270 Cos .80902 885 867 850 833 .80816 799 782 765 748 .80730 713 50 49 48 679 47 662 46 .80644 627 610 593 576 .80558 541 524 507 489 .80472 455 438 420 403 .80386 368 351 334 316 .80299 282 264 247 230 .80212 195 178 160 143 .80125 108 091 073 056 .80038 021 .80003 .79986 968 .79951 934 916 899 881 .79864 Sin / Sin Tan Ctn Cos .60182 .75355 1.3270 .79864 60 1 205 401 .3262 846 59 2 228 447 .3254 829 58 3 251 492 .3246 811 57 4 274 538 .3238 793 56 5 .60298 .75584 1.3230 .79776 55 6 321 629 .3222 758 54 7 344 675 .3214 741 53 8 367 721 .3206 723 52 9 390 767 .3198 706 51 10 .60414 .75812 1.3190 .79688 50 11 437 858 .3182 671 49 12 460 904 .3175 653 48 13 483 950 .3167 635 47 14 506 .75996 .3159 618 46 15 .60529 .76042 1.3151 .79600 45 16 553 088 .3143 583 44 17 576 134 .3135 565 43 18 599 180 .3127 547 42 19 622 226 .3119 630 41 20 .60645 .76272 1.3111 .79512 40 21 668 318 .3103 494 39 22 691 364 .3095 477 38 23 714 410 .3087 459 37 24 738 456 .3079 441 36 25 .60761 .76502 1.3072 .79424 85 26 784 548 .3064 406 34 27 807 594 .3056 388 33 28 830 640 .3048 371 32 29 853 686 .3040 353 31 30 .60876 .76733 1.3032 .79335 30 31 899 779 .3024 318 29 32 922 825 .3017 300 28 33 945 871 .3009 282 27 34 968 918 .3001 264 26 35 .60991 .76f)64 1.2993 .79247 25 36 .61015 .77010 .2985 229 24 37 038 057 .2977 211 23 38 061 103 .2970 193 22 39 084 149 .2962 17() 21 40 .61107 .77196 1.2954 .79158 20 41 130 242 .2946 140 19 42 153 289 .2938 122 18 43 176 335 .2931 105 17 44 199 382 .2923 087 16 45 .61222 .77428 1.2915 .79069 15 46 245 475 .2907 051 14 47 268 521 .2900 033 13 48 291 568 .2892 .79016 12 49 314 615 .2884 .78993 11 60 .61.337 .77661 1.2876 .78980 10 51 360 708 .2869 962 9 52 383 754 .2861 944 8 53 406 801 .2853 926 7 54 429 848 .2846 908 6 55 .61451 .77895 1.2838 .78891 5 56 474 941 .2830 873 4 57 497 .77988 .2822 855 3 58 520 .78035 .2815 837 2 59 543 082 .2807 819 1 60 .61566 .78129 1.2799 .78801 Cos Ctn Tan Sin 1 n] 38°— Values of Trigonometric Functions — 39° 41 / Sin Tan Ctn Cos f Sin Tan Ctn Cos .61566 .78129 1.279f) .78801 60 .62932 .80978 1.2349 .77715 60 1 589 175 .2792 783 59 1 955 .81027 .2342 696 59 2 612 222 .2784 765 58 o .62977 075 .2334 678 58 3 635 269 .2776 747 57 3 .63000 123 .2327 660 57 4 658 316 .2769 729 56 4 022 171 .2320 641 56 5 .61681 .78363 1.2761 .78711 55 5 .63045 .81220 1.2312 .77623 55 6 704 410 .2753 694 54 6 068 268 .2305 605 54 7 726 457 .2746 676 53 7 090 316 .2298 586 53 S 749 504 .2738 658 52 8 113 364 .2290 568 52 9 772 551 .2731 640 51 9 135 413 .2283 550 51 10 .61795 .78598 1.2723 .78622 50 10 .63158 .81461 1.2276 .77531 50 11 818 645 .2715 604 49 11 180 510 .2268 513 49 12 841 692 .2708 586 48 12 203 558 .2261 494 48 13 864 739 .2700 568 47 13 225 606 .2254 476 47 14 887 786 .2693 550 46 14 248 655 .2247 458 46 15 .61909 .78834 1.2685 .78532 45 15 .63271 .81703 1.2239 .77439 45 16 932 881 .2677 514 44 16 293 752 .2232 421 44 17 955 928 .2670 496 43 17 316 800 .2225 402 43 18 .61978 .78975 .2662 478 42 18 338 849 .2218 384 42 19 .62001 .79022 .2655 460 41 19 361 898 .2210 366 41 20 .62024 .79070 1.2647 .78442 40 20 .63383 .81946 1.2203 .77347 40 21 046 117 .2640 424 39 21 406 .81995 .2196 329 39 22 069 164 .2632 405 38 22 428 .82044 .2189 310 38 23 092 212 .2624 387 37 23 451 092 .2181 292 37 24 115 259 .2617 369 36 24 473 141 .2174 273 36 25 .62138 .79306 1.2609 .78351 35 25 .63496 .82190 1.2167 .77255 35 26 160 354 .2602 333 34 26 518 238 .2160 236 34 27 183 401 .2594 315 33 27 540 287 .2153 218 33 28 206 449 .2587 297 32 28 563 336 .2145 199 32 29 229 496 .2579 279 31 29 585 385 .2138 181 31 30 .62251 .79544 1.2572 .78261 30 30 .63608 .82434 1.2131 .77162 30 31 274 591 .2564 243 29 31 630 483 .2124 144 29 32 297 639 .2557 225 28 32 653 531 .2117 125 28 33 320 686 .2549 206 27 33 675 680 .2109 107 27 34 342 734 .2542 188 26 34 698 629 .2102 088 26 35 .62365 .79781 1.25:34 .78170 25 35 .63720 .82678 1.2095 .77070 25 36 388 829 .2527 152 24 36 742 727 .2088 051 24 37 411 877 .2519 134 23 37 765 776 .2081 033 23 38 433 924 .2512 116 22 38 787 825 .2074 .77014 22 39 456 .79972 .2504 098 21 39 810 874 .2066 .76996 21 40 .62479 .80020 1.2497 .78079 20 40 .63832 .82923 1.2059 .76977 20 41 502 067 .2489 061 19 41 854 .82972 .2052 959 19 42 524 115 .2482 043 18 42 877 .83022 .2045 940 18 43 547 163 .2475 025 17 43 899 071 .2038 921 17 44 570 211 .2467 .78007 16 44 922 120 .2031 903 16 45 .62592 .80258 1.2460 .77988 15 45 .63944 .83169 1.2024 .76884 15 46 615 306 .2452 970 14 46 966 218 .2017 866 14 47 638 354 .2445 952 13 47 .63989 268 .2009 847 13 48 660 402 .2437 934 12 48 .64011 317 .2002 828 12 49 683 450 .2430 916 11 49 033 366 .1995 810 11 50 .62706 .80498 1.2423 .77897 10 50 .64056 .83415 1.1988 .76791 10 51 728 546 .2415 879 9 51 078 465 .1981 772 9 52 751 594 .2408 861 8 52 100 514 .1974 754 8 53 774 642 .2401 843 7 53 123 564 .1967 735 7 54 796 690 .2393 824 6 54 145 613 .1960 717 6 55 .62819 .80738 1.2386 .77806 5 55 .64167 .83662 1.1953 .76698 5 56 842 786 .2378 788 4 56 190 712 .1946 679 4 57 864 834 .2371 769 3 57 212 761 .1939 661 3 58 887 882 .2364 751 2 58 234 811 .1932 642 2 59 909 930 .2356 733 1 59 256 860 .1925 623 1 60 .62932 .80978 1.2349 .77715 60 .64279 .83910 1.1918 .76604 Cos Ctn Tan Sin ' Cos Ctn Tan Sin / ^V ^II0 42 40° — Values of Trigonometric Functions — 41° [n / Sin Tan Ctn Cos / Sin Tan Ctn Cos .64279 .83910 1.1918 .76604 60 .65606 .86929 1.1504 .75471 60 1 301 .83960 .1910 586 59 1 628 .86980 .1497 452 59 2 323 .84009 .1903 667 58 2 650 .87031 .1490 433 58 3 346 059 .1896 648 57 3 672 082 ,1483 414 57 4 368 108 .1889 530 56 4 694 1.33 .1477 395 56 5 .64390 .84158 1.1882 .76511 55 5 .65716 .87184 1.1470 .75375 55 6 412 208 .1875 492 54 6 738 236 .14(J3 356 54 7 435 258 .1868 473 53 7 759 287 .1456 337 53 8 457 307 .1861 455 52 8 781 338 .1450 318 52 9 479 357 .1854 436 51 9 803 389 .1443 299 51 10 .64501 .84407 1.1847 .76417 50 10 .65825 .87441 1.1436 .75280 60 11 624 457 .1840 398 49 11 847 492 .1430 261 49 12 646 507 .1833 380 48 12 869 543 .1423 241 48 13 568 556 .1826 361 47 13 891 595 .1416 222 47 14 590 606 .1819 342 46 14 913 646 .1410 203 46 15 .64612 .84656 1.1812 .T6323 45 15 .65935 .87698 1.1403 .75184 45 16 635 706 .1806 304 44 16 956 749 .1396 . 165 44 17 657 756 .1799 286 43 17 .65978 801 .1389 146 43 18 679 806 .1792 267 42 18 .66000 852 .1383 126 42 19 701 856 .1785 248 41 19 022 904 .1376 107 41 20 .64723 .84906 1.1778 .76229 40 20 .66044 .87955 1.1369 .75088 40 21 746 .84956 .1771 210 39 21 066 .88007 .1363 069 39 22 768 .85006 .1764 192 38 22 088 059 .1356 050 38 23 790 057 .1757 173 37 23 109 110- .1349 030 37 24 812 107 .1750 154 36 24 131 162 .1343 .75011 36 25 .64834 .85157 1.1743 .76135 35 25 .66153 .88214 1.1336 .74992 35 26 856 207 .1736 116 34 26 175 265 .1329 973 34 27 878 257 .1729 097 33 27 197 317 .1323 953 33 28 901 308 .1722 078 32 28 218 369 .1316 934 32 29 923 358 .1715 059 31 29 240 421 .1310 915 31 30 .64945 .85408 1.1708 .76041 30 30 .66262 .88473 1.1303 .74896 30 31 967 458 .1702 022 29 31 284 524 .1296 876 29 32 .()4989 509 .1695 .76003 28 32 306 576 .1290 857 28 33 .65011 559 .1688 .75984 27 33 327 628 .1283 838 27 34 033 609 .1681 9()5 26 34 349 680 .1276 818 26 35 .65055 .85660 1.1674 .75946 25 35 .66371 .88732 1.1270 .74799 25 36 077 710 .1667 927 24 36 393 784 .1263 780 24 37 100 761 .1660 908 23 37 414 836 .1257 7(50 23 38 122 811 .1653 889 22 38 436 888 .1250 741 22 39 144 862 .1647 870 21 39 458 940 .1243 722 21 40 .65166 .85912 1.1640 .75851 20 40 .66480 .88992 1.1237 .74703 20 41 188 .85963 .1633 832 19 41 501 .89045 .1230 683 19 42 210 .86014 .1626 813 18 42 623 097 .1224 664 18 43 232 064 .1619 794 17 43 545 149 .1217 644 17 44 254 115 .1612 775 16 44 566 201 .1211 625 16 45 .65276 .86166 1.1606 .75756 15 45 .66588 .89253 1.1204 .74606 15 46 298 216 .1599 738 14 46 610 306 .1197 586 14 47 320 267 .1592 719 13 47 632 358 .1191 567 13 48 342 318 .1585 700 12 48 653 410 .1184 548 12 49 364 368 .1578 680 11 49 675 463 .1178 628 11 50 .65386 .86419 1.1571 .75661 10 50 .66697 .89516 1.1171 .74509 10 51 408 470 .1565 642 9 51 718 567 .1166 489 9 52 430 521 .1558 623 8 52 740 620 .1158 470 8 53 452 572 .1551 604 7 53 762 672 .1152 451 7 54 474 623 .1544 585 6 54 783 725 .1145 431 6 55 .65496 .86674 1.1538 .75566 5 55 .66805 .89777 1.1139 .74412 5 56 518 725 .1531 547 4 56 827 830 .1132 392 4 57 540 776 .1524 528 3 57 848 883 .1126 373 3 58 562 827 .1517 509 2 58 870 935 .1119 353 2 59 584 878 .1510 490 1 59 891 .89988 .1113 334 1 60 .65606 .86929 1.1504 .75471 60 .66913 •90010 1.1106 .74314 Cos Ctn Tan Sin / Cos Ctn Tan Sin t 49° 48° 42° — Values of Trigonometric Functions — 43° 43 Sin 1 Tan Ctn Cos f Sin Tan Ctn Cos .(5t)9i;J .1KM)40 1.1106 .74314 60 .68200 .93252 1.0724 .73135 60 1 9;35 093 .1100 295 59 1 221 306 .0717 116 59 2 956 146 ,1093 276 58 2 242 360 .0711 096 58 3 978 199 .1087 256 57 3 264 415 .0705 076 57 4 .66999 251 .1080 237 56 4 285 469 .0699 056 56 5 .67021 .90304 1.1074 .74217 55 5 .68306 .93524 1.0(592 .73036 55 () 043 357 .1067 198 54 () 327 578 .0686 .73016 54 7 064 410 .1061 178 53 7 349 633 .0680 .72996 53 8 08() 463 .1054 159 52 8 370 688 .0674 976 52 9 107 516 .1048 139 51 9 391 742 .0668 957 51 10 .67129 .90569 1.1041 .74120 50 10 .68412 .93797 1.0661 .72937 50 11 151 621 .ioa5 100 49 11 434 852 .0655 917 49 12 172 674 .1028 080 48 12 455 906 .0649 897 48 13 194 727 .1022 061 47 13 476 .93961 .0(543 877 47 14 215 781 .1016 041 46 . 14 497 .94016 .0637 857 46 15 .672.37 .90834 1.1009 .74022 45 15 .68518 .94071 1.0630 .72837 45 1(> 258 887 .1003 .74002 44 16 539 125 .0624 817 44 17 280 940 .0996 .73983 43 17 561 180 .0(518 797 43 18 301 .90993 .0990 963 42 18 582 235 .0612 777 42 19 323 .91046 .0983 944 41 19 603 290 .0606 757 41 20 .67344 .91099 1.0977 .73924 40 20 .68624 .94345 1.0599 .72737 40 21 366 153 .0971 904 39 21 645 400 .0593 717 39 22 387 206 .0964 885 38 22 666 455 .0587 697 38 23 409 259 .0958 865 37 23 688 510 .0581 677 37 24 430 313 .0951 846 36 24 709 565 .0575 657 36 25 .674.52 .91366 1.0945 .73826 35 25 .68730 .^620 1.0569 .72637 35 2(J 473 419 .0939 806 34 2(5 751 676 .05(52 617 34 27 495 473 .0932 787 33 27 772 731 .0556 597 33 28 516 526 .0926 767 32 28 793 786 .0550 677 32 29 538 580 .0919 747 31 29 814 841 .0544 657 31 30 .67559 .91633 1.0913 .73728 30 30 .68835 .94896 1.0538 .72537 30 31 580 687 .0907 708 29 31 857 .94952 .0532 517 29 32 (502 740 .0900 (588 28 32 878 .95007 .0526 497 28 33 623 794 .0894 669 27 33 899 062 .0519 477 27 34 (545 847 .0888 649 26 34 920 118 .0513 457 26 35 .67666 .91901 1.0881 .73629 25 35 .68941 .95173 1.0507 .72437 25 3(3 (588 .91955 .0875 610 24 36 %2 229 .0501 417 24 37 709 .92008 .0869 590 23 37 .68983 284 .0495 397 23 38 730 062 .0862 570 22 38 .69004 340 .0489 377 22 39 752 116 .0856 551 21 39 025 395 .0483 357 21 40 .67773 .92170 1.0850 .73531 20 40 .69046 .95451 1.0477 .72337 20 41 795 224 .0843 511 19 41 067 506 .0470 317 19 42 816 277 .0837 4i>l 18 42 088 662 .0464 297 18 43 837 331 .0831 472 17 43 109 618 .0458 277 17 4-1 859 385 .0824 452 16 44 130 673 .0452 257 16 45 .67880 .92439 1.0818 .7:5432 15 45 .69151 .95729 1.0446 .72236 15 4(j 901 493 .0812 413 14 46 172 785 .0440 216 14 47 923 547 .0805 393 13 47 193 841 .0434 196 13 48 944 601 .0799 373 12 48 214 897 .0428 176 12 49 965 655 .0793 353 11 49 235 .95952 .0422 156 11 50 .67987 .92709 1.0786 .73333 10 50 .69256 .96008 1.0416 .72136 10 51 .68008 763 .0780 314 9 51 277 064 .0410 116 9 52 029 817 .0774 294 8 52 298 120 .0404 095 8 53 051 872 .0768 274 7 53 319 176 .0398 075 7 54 072 926 .0761 254 6 64 340 232 .0392 056 6 55 .68093 .92980 1.0755 .73234 5 55 .69361 .96288 1.0385 .72035 5 56 115 .93034 .0749 215 4 56 382 344 .0379 .72015 4 57 136 088 .0742 195 3 57 403 400 .0373 .71995 3 58' 157 143 .0736 175 2 58 424 457 .0367 974 2 59 179 197 .0730 155 1 59 445 513 .0361 954 1 60 .68200 .93252 1.0724 .731.35 60 .69466 .96569 1.0355 .719134 Cos Ctn Tan Sin t Cos Ctn Tan Sin ' 4.7^ J.AO 44 44° — Values of Trigonometric Functions 1 Sin Tan Ctn Cos .69466 .96569 1.0355 .71934 60 1 487 625 .0349 914 59 2 508 681 .0343 894 68 3 529 738 .0337 873 57 4 549 794 .0331 853 56 6 .69570 .96850 1.0325 .71833 55 6 591 907 .0319 813 54 7 612 .96963 .0313 792 53 8 633 .97020 .0307 772 52 9 654 076 .0301 752 51 10 .69675 .97133 1.0295 .71732 50 11 696 189 .0289 711 49 12 717 246 .0283 691 48 13 737 302 .0277 671 47 14 768 359 .0271 650 46 15 .69779 .97416 1.0265 .71630 45 16 800 472 .0259 610 44 17 821 529 .0253 590 43 18 842 586 .0247 569 42 19 862 643 .0241 549 41 20 .69883 .97700 1.0235 .71529 40 21 904 756 .0230 508 39 22 925 813 .0224 488 38 23 946 870 .0218 468 37 24 966 927 .0212 447 36 25 .69987 .97984 1.0206 .71427 35 26 .70008 .98041 .0200 407 34 27 029 098 .0194 386 33 28 049 155 .0188 366 32 29 070 213 .0182 345 31 30 .70091 .98270 1.0176 .71325 30 31 112 327 .0170 305 29 32 132 384 .0164 284 28 33 153 441 .0158 264 27 34 174 499 .0152 243 26 35 .70195 .98556 1.0147 .71223 25 36 215 613 .0141 203 24 37 236 671 .0135 182 23 38 257 728 .0129 162 22 39 277 786 .0123 141 21 40 .70298 .98843 1.0117 .71121 20 41 319 901 .0111 100 19 42 339 .98958 .0105 080 18 43 360 .99016 .0099 059 17 44 381 073 .0094 039 16 45 .70401 .99131 1.0088 .71019 15 46 422 189 .0082 .70998 14 47 443 247 .0076 978 13 48 463 304 .0070 957 12 49 484 362 .0064 937 11 50 .70505 .99420 1.0058 .70916 10 61 525 478 .0052 896 9 52 546 536 .0047 875 8 63 567 594 .0041 855 7 64 587 652 .0035 834 6 55 .70608 .99710 1.0029 .70813 5 66 628 768 .0023 793 4 67 649 826 .0017 772 3 58 670 884 .0012 752 2 59 690 .99942 .0006 731 1 60 .70711 1.0000 1.0000 .70711 Cos Ctn Tan Sin f 45° TABLE III COMMON LOGAEITHMS OF THE TEIGONOMETKIC FUNCTIONS FROM 0° TO 90° AT INTERVALS OF ONE MINUTE TO FIVE DECIMAL PLACES TABLE Ilia — AUXILIARY TABLE OF S AND T FOR A IN MINUTES S = log sin A — log A' and T= log tan A — log A' A' S + 10 0'- 13' 6.46373 14'— 42' 72 43'— 58' 71 59'— 71' 6.46370 72'- 81' 69 82'— 91' 68 92'— 99' 6.46367 100' - 107' m 108' — 115' 65 116' — 121' 6.46364 122' - 128' 63 129' - 134' 62 135' - 140' 6.46361 141' - 146' 60 147' - 151' 59 152' - 157' 6.46358 158' - 162' 57 163' -167' 56 168' - 171' 6.46355 172' - 176' 54 177' -181' 53 A' T + 10 0' — 26' 6.46373 27' - 39' 74 40' - 48' 75 49' - 56' 6.46376 57' - 63' 77 64' - 69' 78 70' - 74' 6.46379 75' - 80' 80 81' - 85' 81 86' - 89' 6.46382 90' - 94' 83 95' — 98' 84 99' -102' 6.46385 103' — 106' 86 107' -110' 87 111' -113' 6.46388 114' -117' 89 118' — 120' 90 121' — 124' 6.46391 125' — 127' 92 128' — 130' 93 A' r + 10 131' -133' 6.46394 134' — 136' 95 137' — 139' 96 140' -142' 6.46397 143' — 145' 98 146' -148' 99 149' -150' 6.46400 151' — 153' 01 154' - 156' 02 157' — 158' 6.46403 159' - 161' 04 162' — 163' 05 164' -166' 6.46406 167' - 168' 07 169' — 171' 08 172' -173' 6.46409 174' -175' 10 176' -178' 11 179' — 180' 6.46412 181' -182' 13 183' — 184' 14 For small angles : log sin A = log A' -}- S and log tan A = A' + T For angles near 90° : log cos A = log (90° — A)' + S, log ctn A = log (90°— A)' +T where A' = number of minutes in A , and (90° — A)' = number of minutes in 90° — A 45 46 0° — Logarithms of Trigonometric Functions [in / LSin d LTan cd LCtn L Cos 0.00 000 0.00 000 60 59 1 6.46 373 30103 17609 12494 9691 6.46 373 30103 17609 12494 9691 7918 6694 5800 5115 4576 4139 3779 3476 3219 2996 3.53 627 2 6.76 476 6.76 476 3.23 524 0.00 000 58 3 6.94 085 6.94 085 3.05 915 0.00 000 57 4 7.06 579 7.06 579 2.93 421 0.00 000 56 5 7.16 270 7918 6694- 5800 5115 4576 4139 3779 3476 3218 2997 7.16 270 2.83 730 0.00 000 55 6 7.24 188 7.24 188 2.75 812 0.00 000 54 CM a:) •+3 7 7.30 882 7.30 882 2.69 118 0.00 000 53 o 8 7.36 682 7.36 682 2.63 318 0.00 000 52 Kfl H 9 10 7.41 797 7.46 373 7.41 797 7.46 373 2.58 203 2.53 627 0.00 000 0.00 000 51 50 4J 4-3 CD 11 12 7.50 512 7.54 291 7.50 512 7.54 291 2.49 488 2.45 709 0.00 000 0.00 000 49 48 1 CQ 02 13 7.57 767 7.57 767 2.42 233 0.00 000 47 io r^ 14 7.60 985 7.60 986 2.39 014 0.00 000 46 p-i -* r^ ^ Q 15 7.63 982 2802 2633 2483 2348 2227 7.63 982 2803 2633 2482 2348 2228 2.36 018 0.00 000 45 o ^ "B -. 29 7.92 612 7.92 613 2.07 387 9.99 998 31 tM O g • 1—1 5 30 7.94 084 1424 7.94 086 1424 2 05 914 9.99 998 30 OQ ^ c^ o P 31 7.95 508 1379 1336 1297 1259 7.95 510 1379 1336 1297 1259 2.04 490 9.99 998 29 4^ CD bJO ^3 '""' C^ *^ 32 33 7.96 887 7 98 223 7.96 889 7.98 225 2.03 111 2.01 775 9.99 998 9.99 998 28 27 u * 50 8.16 268 860 8.16 273 860 1.83 727 9.99 995 10 o a; >■ ;h % 51 8.17 128 843 8.17 133 843 1.82 867 9.99 995 9 ph r^ o o u 52 8.17 971 827 8.17 976 828 1.82 024 9.99 995 8 02 53 8.18 798 812 797 8.18 804 812 797 1.81 196 9 99 995 7 § u 54 8.19 610 8.19 616 1.80 384 9.99 995 6 O 55 8.20 407 782 8.20 413 782 1.79 587 9.99 994 5 56 8.21 189 769 8.21 195 769 1.78 805 9.99 994 4 57 8.21 958 755 8.21 964 756 742 730 1.78 036 9.99 994 3 58 8.22 713 743 730 8.22 720 1.77 280 9.99 994 2 59 8.23 456 8.23 462 1.76 538 9,99 994 1 60 8.24 186 8.24192 1.75 808 9.99 993 LCos d LCtn Cd L Tan L Sin / S9° — TiOe^ariflims of Tricftnoiiifttrip, FimHions Ill] 1° — Logarithms of Trigonometric Functions 47 / LSin 1 d LTan cd LCtn LCos Prop. Pts. 8.24 186 717 706 8.24 192 718 706 1.75 808 9.99 993 60 1 8.24 ^X)3 8.24 910 1.75 0^)0 9.99 993 59 720 710 690 680 670 2 8.25 609 8.25 616 1.74 384 9.99 993 58 2 144 142 138 1.36 134 3 4 8.26 304 8.26 988 695 684 673 '663 8.26 312 8.26 996 696 684 673 663 1.73688 1.73 004 9.99 993 9.99 992 57 56 3 4 5 216 288 360 213 284 355 207 276 345 204 201 272 268 340 335 5 8.27 661 8.27 669 1.72 331 9.99 992 55 6 7 432 504 426 497 414 483 408 402 476 469 (i 8.28 324 653 644 8.28 332 654 643 1.71 668 9.99 992 54 8 576 568 552 544 536 7 8.28 977 8.28 986 1.71 014 9.99 992 53 9 648 639 621 612 603 8 8.29 621 8.29 629 1.70 371 9.99 992 52 9 8.30 255 634 8.30 263 634 1.69 737 9.99 991 51 660 650 640 630 620 624 625 2 132 130 128 126 124 10 8.;^ 879 616 608 599 8.30 888 617 607 599 1.69112 9.99 991 50 3 198 195 192 189 186 11 12 8.31 495 8.32 103 8.31 505 8.32 112 1.68 495 1.67 888 9.99 991 9 99 9^)0 49 48 4 5 6 264 330 396 260 325 390 256 320 384 252 248 315 310 378 372 13 8.32 702 8.32 711 1.67 289 9.99 990 47 7 462 455 448 441 434 14 8.33 292 590 583 8.33 302 591 584 1.66 698 9.99 990 4(5 8 9 528 594 520 585 512 576 504 496 567 558 15 8.33 875 575 568 560 553 547 539 8.33 886 575 568 1.66 114 9.99990 45 16 8.34 450 8.34 461 1 .65 539 9 99 989 44 610 600 590 580 570 17 8.e35 018 8.35 029 1.64 971 9.99 989 43 2 122 120 118 116 114 18 19 8.35 578 8.36 131 8.35 590 8.36 143 561 553 546 540 1.64 410 1.63 857 9.99 989 9.99 989 42 41 3 4 5 183 244 305 180 240 300 177 236 295 174 171 232 228 290 285 20 8.36 678 8.36 689 1.63 311 9.99 988 40 6 7 366 427 360 420 354 413 348 342 406 399 21 8.37 217 533 526 8.37 229 533 527 1.62 771 9.99 988 39 8 488 480 472 464 456 22 8.37 750 8.37 762 1.62 238 9.99 988 38 9 549 540 531 522 513 23 8.38 276 8.38 289 1.61711 9*99 987 37 24 8.38 796 520 8.38 809 520 1.61191 9.99 987 36 660 550 540 530 520 514 514 2 112 110 108 106 104 25 8.39 310 508 502 496 8.39 323 509 502 496 1.60 677 9.99 987 35 3 168 165 162 159 156 26 27 8.39 818 8.40 320 8.39 832 8.40 334 1.60168 1.59 6(^6 9.99 986 9.99 986 34 33 4 5 6 224 280 336 220 275 330 216 270 324 212 208 265 260 318 312 28 8.40 816 491 485 8.40 830 491 486 1.59170 9.99 986 32 7 392 385 378 371 364 29 8.41 307 8.41 321 1.58 679 9.99 985 31 8 9 448 504 440 495 432 486 424 416 477 468 30 8.41 792 480 8.41 807 480 475 470 464 460 455 1.58193 9.99 985 30 31 8.42 272 474 470 464 459 455 8.42 287 1.57 713 9.99 985 29 510 500 490 480 470 32 8.42 746 8.42 762 1.57 238 9.99 984 28 2 102 100 98 96 94 33 34 8.43 216 8.43 680 8.43 232 8.43 696 1.56 768 1.56 304 9.99 984 9.99 984 27 26 3 4 5 153 204 255 150 200 250 i4;r 196 245 144 141 192 188 240 235 35 8.44 139 8.44 156 1.55 844 9.99 983 25 6 7 306 357 300 350 294 343 288 282 336 329 36 8.44 594 450 445 441 436 8.44 611 450 446 441 437 1.55 389 9.99 983 24 8 408 400 392 384 376 37 8.45 044 8.45 061 1.54 939 9.99 983 23 9 459 450 441 432 423 38 8.45 489 8.45 507 1.54 493 9.99 982 22 39 8.45 930 8.45 948 1.54 052 9.99 982 21 460 450 90 135 440 88 132 430 420 86 84 129 126 40 8.46 366 433 8.46 385 432 1.53 615 9.99 982 20 2 3 138 41 42 8.46 799 8.47 226 427 424 8.46 817 8.47 245 428 424 1.53 183 1.52 755 9.99 981 9.99 981 19 18 4 5 6 184 230 276 180 225 270 176 220 264 172 168 215 210 258 252 43 8.47 650 419 416 8.47 689 420 416 1.52 331 9.99 981 17 7 322 315 308 301 294 44 8.48 069 8.48 089 1.51911 9.99 980 16 8 9 368 414 360 405 352 396 344 336 387 378 45 8.48 485 411 8.48 505 412 1.51 495 9.99 980 15 4() 8.48 896 408 8.48 917 408 1.51083 9.99 979 14 410 400 395 390 385 47 8.49 304 404 400 8.49 325 404 401 1.50 675 9.99 979 13 2 82 80 79.0 78 77.0 48 8.49 708 8.49 729 1.50 271 9.99 979 12 3 4 123 164 120 118.5 160 158.0 117 115.5 156 154.0 49 8.50 108 396 8.50 130 397 1.49 870 9.99 978 11 5 205 200 197.5 195 192.5 50 8.50 504 393 8.50 527 393 1.49 473 9.99 978 10 6 7 246 287 240 237.0 280 276.5 234 231.0 273 269.5 51 8.50 897 390 8.50 920 390 1.49080 9.99 977 9 8 328 320 316.0 312 308.0 52 8.51 287 386 8.51 310 386 1.48 690 9.99 977 8 9 369 360 355.5 351 346.5 53 54 8.51 673 8.52 055 382 379 8.51 696 8.52 079 383 380 1.48 304 1.47 921 9.99 977 9.99 976 7 6 2 380 76 375 75.0 370 74 365 360 73.0 72 55 8.52 434 376 8.52 459 376 1.47 541 9.99 976 5 3 114 112.5 111 109.5 108 56 57 8.52 810 8.53 183 373 369 8.52 835 8.53 208 373 370 1.47 165 1.46 792 9.99 975 9.99 975 4 3 4 5 6 152 190 228 150.0 187.5 225.0 148 185 222 146.0 144 182.5 180 219.0 216 58 8.53 552 367 363 8.53 578 367 363 1.46 422 9.99 974 2 7 266 262.5 259 255.5 252 59 8.53 919 8.53 945 1.46 055 9.99 974 1 8 9 304 342 300.0 337.5 296 333 292.0 288 328.5 324 60 8.54 282 8.54 308 1.45 692 9.99 974 LGos d LCtn Cd LTan LSin 1 Prop . Pts. 1 88° — Logarithms of Trigonometric Functions 2° — Logarithms of Trigonometric Functions [in r LSin d LTan cd LCtn LGos Prop. Pts. 8.54 282 8.54 308 1.45 692 9.99 974 60 1 8.54 642 360 8.54 669 361 1.45 331 9.99 973 59 2 8.54 999 357 8.55 027 358 1.44 973 9.99 973 58 3 8.55 354 355 8.55 382 355 1.44 618 9.99 972 67 4 8.55 705 351 349 8.55 734 352 349 1.44 266 9.99 972 66 5 8.56054 8.56 083 1.43 917 9.99 971 55 6 8.56400 346 8.56 429 346 1.43 571 9.99 971 54 360 355 350 345 7 8.56 743 343 8.56 773 344 1.43 227 9.99 970 63 2 72 71.0 70 69.0 8 8.57 084 341 337 336 332 8.57114 341 1.42 886 9.99 970 52 3 4 108 106.5 105 103.5 144 142.0 140 138 9 8.57 421 8.57 452 338 1.42 548 9.99 969 51 5 180 177.5 175 172^5 10 8.57 757 8.57 788 336 1.42 212 9.99 969 50 6 7 216 213.0 210 207.0 252 248.5 245 241.5 11 8.58 089 8.58 121 333 1.41 879 9.99 968 49 8 288 284.0 280 276.0 12 8.58 419 330 8.58 451 330 1.41 549 9.99 968 48 9 324 319.5 315 310.5 13 8.58 747 328 8.58 779 328 1.41221 9.99 967 47 14 8.59 072 325 8.59 105 326 1.40 895 9.99 967 46 15 8.59 395 323 320 8.59428 323 321 1.40 572 9.99 967 45 2 340 335 330 325 68 67 66 ft^ 16 8.59 715 8.59 749 1.40 251 9.99 966 44 3 102 100.5 99 97.5 17 8.60 033 318 8.60068 319 1.39 932 9.99 966 43 4 136 134.0 132 130.0 18 8.60 349 316 8.60 384 316 1.39616 9.99 965 42 5 6 170 167.5 165 162.5 204 201.0 198 195 19 8.60 662 313 8.60 698 314 1.39 302 9.99 964 41 7 238 234.5 231 227.5 20 8.60 973 311 309 8.61 009 311 1.38 991 9.99 964 40 8 9 272 268.0 264 260.0 306 301.5 297 292.5 21 8.61 282 8.61 319 310 1.38 681 9.99 963 39 22 8.61 589 307 8.61 626 307 1.38 374 9.99 963 38 23 8.61 894 305 8.61 931 305 1.38 069 9.99 962 37 320 315 310 305 24 8.62 196 302 8.62234 303 1.37 766 9.99 962 36 2 64 63.0 62 61.0 25 8.62 497 301 298 8.62 535 301 299 1.37 465 9.99 961 35 3 4 96 94.5 93 91.5 128 126.0 124 122 26 8.62 795 8.62 834 1.37 166 9.99 961 34 5 160 157.5 155 152.5 27 28 8.63 091 8.63 385 296 294 293 290 288 8.63 131 8.63426 297 295 1.36 869 1.36 574 9.99 960 9.99 960 33 32 6 7 8 192 189.0 186 183.0 224 220.5 217 213.5 256 252.0 248 244.0 29 8.63 678 8.63 718 292 1.36 282 9.99 959 31 9 288 283.5 279 274.5 30 8.63 968 8.64009 291 1.35 991 9.99 959 30 31 8.64256 8.64 298 289 1.35 702 9.99 958 29 32 8.64 543 287 8.64 585 287 1.35 415 9.99 958 28 300 295 290 285 33 8.64 827 284 283 8.64 870 285 1.35 130 9.99 957 27 2 3 60 59.0 58 57.0 90 88 5 87 S'l "i 34 8.65 110 8.65 154 284 1.34 846 9.99 956 26 4 i7\J 00.«_F 553 137 136 8.96 602 8.96 739 138 137 138 1.03 398 1.03 261 9.99 815 9.99 814 43 42 2 3 4 28.4 42.6 56.8 28.2 28.0 27.8 42.3 42.0 41.7 56.4 56.0 55.6 19 8.9(5 689 136 8.96 877 1.03 123 9.99 813 41 5 71.0 70.5 70.0 69.5 20 8.96 825 136 8.97 013 136 1.02 987 9.99 812 40 6 7 85.2 99.4 84.6 84.0 83.4 98.7 98.0 97.3 21 8.<)6 960 135 8.97 150 137 1.02 850 9.99 810 39 8 113.6 112.8 112.0 111.2 22 8.97 095 135 8.97 285 135 1.02 715 9.99 809 38 9 127.8 126.9 126.0 125.1 23 24 8.97 229 8.97 363 134 134 133 8.97 421 8.97 556 136 135 135 1.02 579 1.02 444 9.99 808 9.99 807 37 36 2 138 27.6 137 136 135 27.4 27.2 27.0 25 8.97 496 8.97 691 1.02 309 9.99 806 35 3 41.4 41.1 40.8 40.5 26 8.97 629 133 133 8.97 825 134 134 1.02 175 9.99 804 34 4 5 55.2 69.0 54.8 54.4 54.0 68.5 68.0 67 5 27 8.97 762 8.97 959 1.02041 9.99 803 33 6 82.8 82.2 81.6 81.0 28 29 8.97 894 8.98026 132 132 131 8.98092 8.98 225 133 133 133 1.01908 1.01 775 9.99 802 9.99 801 32 31 7 8 9 96.6 110.4 124.2 95.9 95.2 94.5 109.6 108.8 108.0 123.3 122.4 121.5 30 8.98 157 8.98 358 132 132 1.01 642 9.99 800 30 31 8,98 288 131 8.98 4^)0 1.01510 9.99 798 29 134 133 132 131 32 8.98 419 131 8.98 622 1.01 378 9.99 797 28 2 26.8 26.6 26.4 26.2 33 8.98 549 130 8.98 753 131 131 1.01 247 9.99 796 27 3 4 40.2 53.6 39.9 39.6 39.3 53.2 52 8 52 4 34 8.98 679 130 8.98 884 1.01116 9.99 795 26 5 67.0 66.5 66.0 65.5 35 8.98 808 129 8.99 015 131 130 1.00 985 9.99 793 25 6 7 80.4 93.8 79.8 79.2 78.6 93.1 92.4 91.7 36 8.98 937 129 8.99 145 1.00 855 9S}9 792 24 8 107.2 106.4 105.6 104.8 37 8.99 066 129 8.99 275 130 1.00 725 9.99 791 23 9 120.6 119.7 118.8 117.9 38 8.99194 128 8.99 405 130 1.00 595 9.99 790 22 ,-,« 129 128 127 25.8 25.6 25.4 39 8.99 322 128 128 8.99 534 129 128 1.00 466 9.99 788 21 2 26.0 40 8.99 450 8.99 662 1.00 338 9.99 787 20 3 39.0 38.7 38.4 38.1 41 42 8.99 577 8.99 704 127 127 8.99 791 8.99 919 129 128 1.00 209 1.00 081 9.99 786 9.99 785 19 18 4 5 6 52.0 65.0 78.0 51.6 51.2 50.8 64.5 64.0 63.5 77.4 76.8 76.2 43 8.99 830 126 9.00 046 127 0.99 954 9.99 783 17 7 91.0 90.3 89.6 88.9 44 8.99 956 126 126 9.00 174 128 127 0.99 826 9.99 782 16 8 9 104.0 117.0 103.2 102.4 101.6 116.1 115.2 114.3 45 9.00 082 9.00 301 126 0.99 699 9.99 781 15 46 9.00 207 125 9.00 427 0.99 573 9.99 780 14 126 125 124 123 47 9.00 332 125 9.00 553 126 0.99 447 9.99 778 13 2 25.2 25.0 24.8 24.6 48 9.00 456 124 125 9.00 679 126 126 0.99 321 9.99 777 12 3 4 37.8 50 4 37.5 37.2 36.9 50.0 49.6 49.2 49 9.00 581 9.00 805 0.99 195 9.99 776 11 5 63.0 62.5 62.0 61.5 50 9.00 704 123 124 9.00 930 125 125 0.99070 9.99 775 10 6 7 75.6 88.2 75.0 74.4 73.8 87.5 86.8 86.1 51 9.00 828 9.01 055 0.98 945 9.99 773 9 8 100.8 100.0 99.2 98.4 52 9.00 951 123 9.01 179 124 0.98 821 9.99 772 8 9 113.4 112.5 111.6 110.7 53 9.01074 123 9.01 303 124 0.98 697 9.99 771 7 54 9.01 196 122 9.01 427 124 0.98 573 9.99 769 6 Izz izx xzu 1 122 123 2 24.4 24.2 24.0 55 9.01 318 9.01 550 0.98 450 9.99 768 5 3 36.6 36.3 36.0 56 9.01 440 122 9.01 673 123 123 122 0.98 327 9.99 767 4 4 5 6 48.8 48.4 48.0 61.0 60.5 60.0 73.2 72.6 72.0 57 9.01 561 121 121 9.01 796 0.98 204 9.99 765 3 58 9.01 682 9.01 918 0.98 082 9.99 764 2 7 85.4 84.7 84.0 59 9.01 803 121 120 9.02 040 122 122 0.97 960 9.99 763 1 8 9 97.6 96.8 96.0 109.8 108.9 108.0 60 9.01 923 9.02 162 0.97 838 9.99 761 LCos d LCtn Cd LTan LSin ' Prop. Pts. 84°— Logarithms of Trigonometric Functions 62 6° — Logarithms of Trigonometric Functions [III / LSin d LTan cd LCtn LCos Prop. Pts. 1 9.01 923 9.02 162 0.97 838 9.99761 60 1 9.02 043 120 9.02 283 121 0.97 717 9.99 760 69 2 9.02 163 120 9.02 404 121 0.97 596 9.99 759 68 3 9.02 283 120 9.02 525 121 0.97 475 9.99 757 67 4 9.02 402 119 118 119 118 9.02 645 120 121 119 120 0.97 355 9.99 756 66 5 9.02 520 9.02 766 0.97 234 9.99 756 55 121 120 119 118 6 9.02 639 9.02 885 0.97 115 9.99 753 64 2 24 2 24 23 8 23 6 7 9.02 757 9.03 005 0.96 995 9.99 752 53 3 36.3 36.0 35.7 35'a 8 9 9.02 874 9.02 992 117 118 117 117 9.03 124 9.03 242 119 118 0.96 876 0.96 758 9.99 751 9.99 749 62 51 4 5 6 48.4 60.5 72.6 48.0 47.6 60.0 59.5 72.0 71.4 47.2 59.0 70.8 10 9.03 109 9.03 361 119 118 0.96 639 9.99 748 50 7 8 84.7 96.8 84.0 83.3 96 95 2 82.6 94 4 11 9.03 226 9.03 479 0.96 521 9.99 747 49 9 108.9 108.0 107.1 10612 12 9.03 342 116 9.03 597 118 0.96 403 9.99 745 48 13 9.03458 116 9.03 714 117 0.96 286 9.99 744 47 117 116 116 114 14 9.03 574 116 116 115 9.03 832 118 116 117 0.96 168 9.99 742 46 2 23.4 23.2 23.0 22.8 15 16 9.03690 9.03 805 9.03 948 9.04 065 0.96 052 0.95 935 9.99 741 9.99 740 45 44 3 4 5 35.1 46.8 58.5 34.8 34.5 46.4 46.0 58.0 57.5 34.2 45.6 57.0 17 9.03 920 115 9.04181 116 0.95 819 9.99 738 43 6 70.2 69.6 69.0 68.4 18 9.04034 114 115 113 114 114 9.04 297 116 0.95 703 9.99 737 42 7 8 81.9 93.6 81.2 80.5 92 8 92 79.8 91 2 19 9.04 149 9.04413 116 0.95 587 9.99 736 41 9 105.3 104.4 103.5 102.6 20 9.04 262 9.04 528 115 0.95 472 9.99 734 40 21 9.04 376 9.04 643 115 0.95 357 9.99 733 39 113 112 111 110 22 9.04 490 9.04 758 115 0.95 242 9.99 731 38 2 22.6 22.4 22.2 22.0 23 9.04 603 113 112 9.04 873 115 0.95 127 9.99 730 37 3 4 33.9 45 2 33.6 33.3 44.8 44 4 33.0 44 24 9.04 715 9.04 987 114 0.95 013 9.99 728 36 5 56!5 56^0 55.5 55^0 25 9.04 828 113 112 9.05 101 114 0.94 899 9.99 727 35 6 7 67.8 79.1 67.2 66.6 78.4 77.7 66.0 77.0 26 9.04 940 9.05 214 113 0.94 786 9.99726 34 8 90.4 89.6 88.8 88.0 27 9.05 052 112 112 9.05 328 114 0.94 672 9.99 724 33 9 101.7 100.8 99.9 99.0 28 9.05 164 9.05 441 113 0.94 659 9.99 723 32 29 9.05 275 111 9.05 553 112 0.94 447 9.99 721 31 109 108 107 106 30 9.05 386 111 111 9.05 666 113 112 0.94 334 9.99 720 30 2 3 21.8 32.7 21.6 21.4 32.4 32.1 21.2 31 8 31 9.05 497 9.05 778 0.94 222 9.99 718 29 4 43.6 43.2 42.8 42.4 32 33 9.05 607 9.05 717 110 110 110 9.05 890 9.06 002 112 112 0.94 110 0.93 998 9.99 717 9.99 716 28 27 5 6 7 54.5 65.4 76.3 64.0 53.5 64.8 64.2 75.6 74.9 53.0 63.6 74.2 34 9.05 827 9.06 113 111 0.93 887 9.99 714 26 8 87.2 86.4 85.6 84.8 110 111 9 Q8.1 97.2 96.3 95 4 35 9.05 937 109 109 9.06 224 0.93 776 9.99 713 25 36 9.06 046 9.06 335 111 0.93665 9.99 711 24 37 9.06 155 9.06 445 110 0.93 655 9.99 710 23 38 9.06 264 109 9.06 556 111 0.93 444 9.99 708 22 39 9.06 372 108 109 9.06 666 110 109 0.93 334 9.99 707 21 40 9.06 481 9.06 775 0.93 225 9.99 705 20 41 9.06 589 108 9.06 885 110 0.93 115 9.99 704 19 From the top: 42 43 9.06 696 9.06 804 107 108 9.06 994 9.07 103 109 109 0.93 006 0.92 897 9.99 702 9.99 701 18 17 For 6°+ or 186°+, 44 9.06 911 107 107 106 107 9.07 211 108 0.92 789 9.99699 16 read as printed ; for 45 9.07 018 9.07 320 109 0.92 680 9.99 698 15 96°+ or 276°+, read 46 9.07 124 9.07 428 108 0.92 572 9.99 696 14 co-function. 47 9.07 231 9.07 536 108 0.92 464 9.99 695 13 48 9.07 337 106 9.07 643 107 0.92 357 9.99 693 12 49 9.07 442 105 106 105 105 9.07 751 108 107 106 107 0.92 249 9.99692 11 From trie oottom •* 50 9.07 548 9.07 858 0.92 142 9.99690 10 For 83°+ or 263°+. 1 51 52 9.07 653 9.07 758 9.07 964 9.08 071 0.92 036 0.91 929 9.99 689 9.99 687 9 8 read as printed ; for 53 9.07 863 105 9.08 177 106 0.91 823 9.99 686 7 173°+ or 353"+, read 64 9.07 968 105 104 104 9.08 283 106 106 0.91 717 9.99684 6 co-function. 55 9.08 072 9.08 389 0.91 611 9.99683 5 56 9.08 176 9.08 495 106 0.91 505 9.99 681 4 57 9.08 280 104 9.08 600 105 0.91 400 9.99680 3 58 9.08 383 103 9.08 705 105 0.91 295 9.99 678 2 59 9.08486 103 103 9.08 810 105 104 0.91 190 9.99677 1 60 9.08 589 9.08 914 0.91 086 9.99 676 LGos d LCtn Cd LTan L Sin / Prop. Pts. 1 83° — Logarithms of Trigonometric Functions Ill] T — Logarithms of Trigonometric Functions 53 L Sin LTan cd LCtn LGos Prop. Pts. 9.08 589 1 9.08 692 2 9.08 795 3 9.08 897 4 9.08 999 5 9.09 101 () 9.09 202 7 9.09 304 8 9.09 405 9 9.09506 10 9.09 606 11 9.09 707 12 9.09 807 13 9.09 907 14 9.10 006 15 9.10106 16 9.10 205 17 9.10 304 18 9.10402 19 9.10 501 20 9.10 599 21 9.10 697 oo 9.10 795 23 9.10 893 24 9.10 990 25 9.11 087 26 9.11 184 27 9.11 281 28 9.11 377 29 9.11474 30 9,11 570 31 9.11 666 32 9.11 761 33 9.11 857 34 9.11 952 35 9.12 047 36 9.12 142 37 9.12 236 38 9.12 331 39 9.12 425 40 9.12 519 41 9.12 612 42 9.12 706 43 9.12 799 44 9.12 892 45 9.12 985 46 9.13 078 47 9.13171 48 9.13 263 49 9.13 355 50 9.13447 51 9.13 539 62 9.13 630 53 9.13 722 54 9.13 813 55 9.13 904 56 9.13 994 57 9.14085 58 9.14 175 59 9.14 266 60 9.14 356 103 103 102 102 102 101 102 101 101 100 101 100 100 99 100 99 99 9.08 914 9.09 019 9.09 123 9.09 227 9.09 330 9.09 434 9.09 537 9.09 640 9.09 742 9.09 845 9.09 947 9.10049 9.10 150 9.10 252 9.10 353 9.10 454 9.10 555 9.10 656 9.10 756 9.10 856 9.10 956 11056 11 155 11254 11353 11452 11551 11649 11747 11845 Ml 943 1.12 040 f.12138 1.12 235 1.12 332 M2 428 1.12 525 1.12 621 M2 717 >.12 813 ).12 909 1.13 004 1.13099 M3 194 1.13 289 9.13 384 9.13 478 9.13 573 9.13 667 9.13 761 9.13 854 9.13 948 9.14 041 9.14134 9.14 227 9.14 320 9.14 412 9.14 504 9.14 597 9.14 688 9.14 780 0.91 086 0.90 981 0.90 877 0.90 773 0.90 670 0.90 566 0.90 463 0.90 360 0.90 258 0.90 155 0.90 053 0.89 951 0.89 850 0.89 748 0.89647 0.89 546 0.89 445 0.89 344 0.89 244 0.89 144 0.89 044 0.88 944 0.88 845 0.88 746 0.88647 0.88 548 0.88 449 0.88 351 0.88 253 0.88 155 0.88057 0.87 960 0.87 862 0.87 765 0.87 668 0.87 572 0.87 475 0.87 379 0.87 283 0.87 187 0.87 091 0.86 996 0.86 901 0.86 806 0.86 711 0.86 616 0.86 522 0.86 427 0.86 333 0.86 239 0.86 146 0.86052 85 959 0.85 866 0.85 773 0.85 680 0.85 588 0.85 496 0.85 403 0.85 312 0.85 220 9.99 675 9.99 674 9.99 672 9.99670 9.99 669 9.99 667 9.99 666 9.99664 9.99 663 9.99 661 9.99659 9.99 658 9.99 656 9.99 655 9.99 653 9.99651 9.99 650 9.99648 9.99 647 9.99 645 9.99 643 9.99 642 9.99 640 9.99 638 9.99 637 9.99 635 9.99 633 9.99 632 9.99630 9.99 629 9.99627 9.99625 9.99624 9.99622 9.99 620 9.99618 9.99 617 9.99 615 9.99 613 9.99 612 9.99 610 9.99 608 9.99 607 9.99 605 9.99 603 9.99 601 9.99 600 9.99 598 9.99 596 9.99 595 9.99 593 9.99 591 9.99 589 9.99 588 9.99 586 9.99 584 9.99 582 9.99581 9.99 579 9.99 577 9.99 575 2 105 21.0 104 20.8 103 20.6 3 31.5 31.2 30.9 4 42.0 41.6 41.2 5 52.5 52.0 51.5 6 63.0 62.4 61.8 7 73.5 72.8 72.1 8 84.0 83.2 82.4 9 94.5 93.6 92.7 101 99 98 20.2 19.8 19.6 30.3 29.7 29.4 40.4 39.6 39.2 50.5 49.5 49.0 60.6 59.4 58.8 70.7 69.3 68.6 80.8 79.2 78.4 90.9 89.1 88.2 96 95 94 19.2 19.0 18.8 28.8 28.5 28.2 38.4 38.0 37.6 48.0 47.5 47.0 57.6 57.0 56.4 67.2 66.5 65.8 76.8 76.0 75.2 86.4 85.5 84.6 20.4 30.6 40.8 51.0 61.2 71.4 81.6 91.8 97 19.4 29.1 38.8 48.5 58.2 67.9 77.6 87.3 93 18.6 27.9 37.2 46.5 55.8 65.1 74.4 83.7 92 91 2 18.4 18.2 3 27.6 27.3 4 36.8 36.4 5 46.0 45.5 6 55.2 54.6 7 64.4 63.7 8 73.6 72.8 9 82.8 81.9 90 18.0 27.0 36.0 45.0 54.0 63.0 72.0 81.0 From the top : For 7°+ or 187°+, read as printed ; for 97°+ or 277^^-, read co-function. From the bottom : For 82°+ or 262°+, read as printed ; for 172°+ or 352°+, read CQ-function. LGos LCtn c d L Tan L Sin Prop. Pts. 82°— Logarithms of Trigonometric Functions 54 8° — Logarithms of Trigonometric Functions [in L Sin L Tan c d L Ctn L Cos Prop. Pts. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.14 356 9.14 445 9.14 535 9.14 624 9.14 714 9.14 803 9.14 891 9.14 980 9cl5 069 9.15 157 9.15 245 9.15 333 9.15 421 9.15 508 9.15 596 9.15 683 9.15 770 9.15 857 9.15 944 9.16 030 9.16 116 9.16 203 9.16 289 9.16 374 9.16 460 9.16 545 9.16 631 9.16 716 9.16 801 9.16 886 9.16 970 9.17 055 9.17 139 9.17 223 9.17 307 9.17 391 9.17 474 9.17 558 9.17 641 9.17 724 9.17 807 9.17 890 9.17 973 9.18 055 9.18137 9.18 220 9.18 302 9.18 383 9.18 465 9.18 547 9.18 628 9.18 709 9.18 790 9.18 871 9.18 952 9.19033 9.19 113 9.19 193 9.19 273 9.19 353 9.19433 9.14 780 9.14 872 9.14 963 9.15 054 9.15 145 9.15 236 9.15 327 9.15 417 9.15 508 9.15 598 9.15 688 9.15 777 9.15 867 9.15 956 9.16 046 9.16 135 9.16 224 9.16 312 9.16 401 9.16 489 9.16 577 9.16 665 9.16 753 9.16 841 9.16 928 9.17 016 9.17 103 9.17 190 9.17 277 9.17 363 9.17 450 9.17 536 9.17 622 9.17 708 9.17 794 9.17 880 9.17 965 9.18 051 9-18 136 9.18 221 9.18 306 9.18 391 9.18 475 a 18 560 9.18 644 9.18 728 9.18 812 9.18 896 9.18 979 9.19063 9.19 146 9.19 229 9,19 312 9.19 395 9.19 478 9.19 561 9.19 643 9.19 725 9.19 807 9.19 889 9.19 971 0.85 220 0.85 128 0.85 037 0.84 946 0.84 855 0.84 764 0.84 673 0.84 583 0.84 492 0.84 402 0.84 312 0.84 223 0.81 133 0.84 044 0.83 954 0.83 865 0.83 776 0.83 688 0.83 599 0.83 511 0.83 423 0.83 335 0.83 247 0.83 159 0.83 072 0.82 984 0.82 897 0.82 810 0.82 723 0.82 637 0,82 550 0.82 464 0.82 378 0.82 292 0.82 206 0.82 120 0.82 035 0.81 949 0.81 864 0.81 779 0.81 694 081609 0.81 525 0.81 440 0.81 356 0.81 272 0-81 188 0.81 104 0.81 021 0.80 937 0.80 854 0.80 771 0.80 688 0.80 605 0.80 522 0.80 439 0.80 357 0.80 275 0.80 193 0.80 111 0.80 029 9.99 575 9.99 574 9.99 572 9.99 570 9.99 568 9.99 566 9.99 565 9.99 563 9.99 561 9.99 559 9.99 557 9.99 556 9.99 554 9.99 552 9.99 550 9.99 548 9.99 546 9.99 545 9.99 543 9.99 541 9.99 539 9.99 537 9.99 535 9.99 533 9.99 532 9.99 530 9.99 528 9.99 526 9.99 524 9.99 522 9.99 520 9.99 518 9.99517 9.99515 9.99 513 9.99 511 9.99 509 9.99 507 9.99 505 9.99 503 9.99 501 9.99 499 9.99497 9.99 495 9.99494 9.99 492 9.99490 9.99 488 9.99 486 9.99484 9.99 482 9.99480 9.99 478 9.99 476 9.99 474 9.99 472 9.99 470 9.99 468 9.99 466 9.99 464 9.99462 92 91 90 18.4 18.2 18.0 27.6 27.3 27.0 3(18 36.4 36.0 460 45.5 45.0 55.2 54.6 54.0 64.4 63.7 63.0 73.6 72.8 72.0 82.8 81.9 81.0 89 17.8 26.7 35.6 44.5 53.4 62.3 71.2 80.1 88 87 17.6 17.4 26.4 26.1 35.2 34.8 44.0 43.5 52.8 52.2 61.6 60.9 70.4 69.6 79.2 78.3 85 84 17.0 16.8 25.5 25.2 34.0 33.6 42.5 42.0 51.0 50.4 59.5 58.8 68.0 67.2 76.5 75.6 86 17.2 25.8 34.4 43.0 51.6 60.2 68.8 77.4 83 16.6 24.9 33.2 41.5 49.8 58.1 66.4 74.7 80 16.0 24.0 32.0 400 48.0 56.0 64.0 72.0 From the top : For 8°+ or 188°+, read as printed ; for 98°+ or 278°+, read co-function. From the bottom : For 81°+ or 261°+, read as printed ; for 171°+ or 351°+, read co-function. 82 81 2 16.4 16.2 3 24.6 24.3 4 32.8 32.4 5 41.0 40.5 6 49.2 48.6 7 57.4 66.7 8 65.6 64.8 9 73.8 72.9 LGos L Ctn c d L Tan L Sin Prop. Pts. 8F — Logarithms of Trigonometric Functions IIIJ 9° — Logarithms of Trigonometric Functions 55 LSin L Tan led L Ctn LCos Prop. Pts. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 k; 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3() 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.19 433 9.19 513 9.19 592 9.19 672 9.19 751 9.19 830 9.19 909 9.19 988 9.20 067 9.20 145 9.20 223 9.20 302 9.20 380 9.20 458 9.20 535 9.20 613 9.20 691 9.20 7(;8 9.20 845 9.20 922 9.20 9m 9.21 076 9.21 153 9.21 229 9.21 306 9.21 382 9.21 458 9.21 534 9.21 610 9.21 685 9.21 761 9.21 836 9.21 912 9.21 987 9.22 062 9.22 137 9.22 211 9.22 286 9.22 361 9.22 435 9.22 509 9.22 583 9.22 657 9.22 731 9.22 805 9.22 878 9.22 952 9.23 025 9.23 098 9.23 171 9.23 244 9.23 .317 9.23 390 9.23 462 9.23 535 9.23 607 9.23679 9.23 752 9.23 823 9.23 895 9.23 967 9.19971 9.20 0.53 9.20 lU 9.20 216 9.20 297 9.20 378 9.20 459 9.20 540 9.20 621 9.20 701 9.20 782 9.20 862 9.20 942 9.21022 9.21 102 9.21 182 9.21 261 9.21 341 9.21 420 9.21 499 9.21 578 9.21 657 9.21 7;3(> 9.21814 9.21893 9.21 971 9.22 049 9.22 127 9.22 205 9.22 283 9.22 361 9.22 438 9.22 516 9.22 593 9.22 670 9.22 747 9.22 824 9.22 901 9.22 977 9.23 054 9.23130 9.23 206 9.23 283 9.23 359 9.23435 9.23 510 9.23 586 9.23 661 9.23 737 9.23 812 9.23 887 9.23 962 9.24 037 9.24 112 9.24 186 9.24 261 9.24 335 9.24 410 9.24 484 9.24 558 9.24 632 0.80 029 0.79 4 9.36 452 9.36 509 9.36 566 9.36 624 9.36 681 9.36 738 9.36 795 9.36 852 9.36 909 9.36 966 9.37 023 9.37 080 9.37 137 9.37 193 9.37 250 9.37 306 9.37 363 9.37 419 9.37 476 9.37 532 9.37 588 9.37 644 9.37 700 9.37 756 9.37 812 9.37 868 9.37 924 9.37 980 9.38 035 9.38 091 9.38 147 9.38 202 9.38 257 9.38 313 9.38 368 9.38 423 9.38 479 9 38 534 9.38 589 9.38 (344 9.38 699 9.38 754 9.38 808 9.38 863 9.38 918 9.38 972 9.39 027 9.39 082 9.39 136 9.39 190 9.39 245 9.39 299 9.39 353 9.39407 9.39 461 9.39 515 9.39 569 9.39 623 9.39 677 0.63 6(34 0.63 606 0.63 548 0.63 491 0.63 434 0.63 376 0.63 319 0.63 262 0.63 205 0.63 148 0.63 091 0.63 034 0.62 977 0.62 920 0.62 863 0.62 807 0.62 750 0.62 694 0.62 637 0.62 581 0.62 524 0.(32 468 0.62 412 0.62 356 0.62 300 0.62 244 0.62 188 0.62 132 0.62 076 0.62 020 0.61 965 0.61 909 0.61 853 0.61 798 0.61 743 0.61 687 0.61 632 0.61 577 0.61 521 0.61 466 0.61 411 0.61 3.56 0.61 301 0.61 246 0.61 192 0.61 137 0.61 082 0.61 028 0.60 973 0.60 918 0.60 864 0.60 810 0.60 755 0.60 701 0.60 647 0.60 593 0.60 539 0.60 485 0.60 431 0.60 377 0.60 323 9.98 872 9.98 869 9.98 867 9.98 864 9.98 861 9.98 858 9.98 855 9.98 852 9.98 849 9.98 846 9.98 843 9.98 840 9.98 837 9.98 834 9.98 831 9.98 828 9.98 825 9.98 822 9.98 819 9.98 816 9.98 813 9.98 810 9.98 807 9.98 804 9.98 801 9.98 798 9.98 795 9.98 792 9.98 789 9.98 786 9.98 783 9.98 780 9.98 777 9.98 774 9.98 771 9.98 768 9.98 765 9.98 762 9.98 759 9.98 756 9.98 753 9.98 750 9.98 746 9.98 743 9.98 740 9.98 737 9.98 734 9.98 731 9.98 728 9.98 725 9.98 722 9.98 719 9.98 715 9.98 712 9.98 709 9.98 706 9.98 703 9.98 700 9.98 697 9.98694 9.98 61X) 58 57 2 11.6 11.4 3 17.4 17.1 4 23.2 22.8 5 29.0 28.5 6 34.8 34.2 7 40.6 39.9 8 46.4 45.6 9 52.2 51.3 55 54 2 11.0 10.8 3 16.5 16.2 4 22.0 21.6 5 27.5 27.0 6 33.0 32.4 7 38.5 37.8 8 44.0 43.2 9 49.5 48.6 56 11.2 16.8 22.4 28.0 33.6 39.2 44.8 50.4 53 10.6 15.9 21.2 26.5 31.8 37.1 42.4 47.7 52 2 10.4 3 15.6 4 20.8 5 26.0 6 31.2 7 36.4 8 41.6 9 46.8 4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 51 10.2 15.3 20.4 25.5 30.6 35.7 40.8 45.9 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 From the top : For 13°+ or 193^+, read as printed ; for 103°+ or 283°+, read co-function. From the bottom: For 76° or: 256°, read as printed ; for 166°+ or 346°+, read co-function. LCDS LCtn c d L Tan L Sin d f Prop. Pts. 76°— Logarithms of Trigonometric Functions 60 14° — Logarithms of Trigonometric Functions [in 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L Sin 9.38 3G8 9.38 418 9.38 469 9.38 519 9.38 570 9.38 620 9.38 670 9.38 721 9.38 771 9.38 821 9.38 871 9.38 921 9.38 971 9.39021 9.39071 9.39 121 9.39 170 9.39 220 9.39270 9.39319 9.39 369 9.39418 9.39467 9.39 517 9.39 566 9.39615 9.39 664 9.39 713 9.39 762 9.39 811 9.39 860 9.39 909 9.39 958 9.40 006 9.40 055 9.40 103 9.40 152 9.40 200 9.40 249 9.40 297 9.40 346 9.40 394 9.40442 9.40 490 9.40 538 9.40 586 9.40 634 9.40 682 9.40 730 9.40 778 9.40 825 9.40 873 9.40 921 9.40 968 9.41 016 9.41 063 9.41 111 9.41 158 9.41 205 9.41 252 9.41 300 L Tan c d L Gtn 9.39 677 9.39 731 9.39 785 9.39 838 9.39 892 9.39 945 9.39 999 9.40 052 9.40 106 9.40 159 9.40 212 9.40 266 9.40 319 9.40 372 9.40 425 9.40 478 9.40 531 9.40 584 9.40 636 9.40 689 9.40 742 9.40 795 9.40 847 9.40 900 9.40 952 9.41 005 9.41 057 9.41 109 9.41 161 9.41 214 9.41 266 9.41 318 9.41 370 9.41 422 9 41 474 9.41 526 9.41 578 9.41 629 9.41 681 9.41 733 9.41 784 9.41 836 9.41 887 9.41 939 9.41 990 9.42 041 9.42 093 9.42 144 9.42 195 9.42 246 9.42 297 9.42 348 9.42 399 9.42 450 9.42 501 9.42 552 9.42 603 9.42 653 9.42 704 9.42 755 9.42 805 0.60 323 0.60 269 0.60 215 0.60 162 0.60 108 0.60 055 0.60 001 0.59 948 0.59 894 0.59 841 0.59 788 0.59 734 0.59 681 0.59 628 0.59 575 0.59 522 0.59 469 0.59 416 0.59 364 0.59 311 0.59 258 0.59 205 0.59 153 0.59 100 0.59048 0.58 995 0.58 943 0.58 891 0.58 839 0.58 786 0.58 734 0.58 682 0.58 630 0.58 578 0.58 526 0.58 474 0.58 422 0.58 371 0.58 319 0.58 267 0.58 216 0.58 164 0.58 113 0.58 061 0.58 010 0.57 959 0.57 907 0.57 856 0.57 805 0.57 754 0.57 703 0.57 652 0.57 601 0.57 550 0.57 499 0.57 448 0.57 397 0.57 347 0.57 296 0.57 245 0.57 195 LGos 9.98 690 9.98 687 9.98 684 9.98 681 9.98 678 9.98 675 9.98 671 9.98 668 9.98 665 9.98 662 9.98 659 9.98 656 9.98 652 9.98 649 9.98 646 9.98 643 9.98 640 9.98 636 9.98 633 9.98 630 9.98 627 9.98 623 9.98 620 9.98 617 9.98 614 9.98 610 9.98 607 9.98 604 9.98 601 9.98 597 9.98 594 9.98 591 9.98 588 9.98 584 9.98 581 9.98 578 9.98 574 9.98 571 9.98 568 9.98 565 9.98 561 9.98 558 9.98 555 9.98 551 9.98 548 9.98 545 9.98 541 9.98 538 9.98 535 y.98 531 9.98 528 9.98 525 9.98 521 9.98 518 9.98 515 9.98 511 9.98 508 9.98 505 9.98 501 9.98 498 9.98 494 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 Prop. Pts. 54 53 2 10.8 10.6 3 16.2 15.9 4 21.6 21.2 5 27.0 26.5 6 32.4 31.8 7 37.8 37.1 8 43.2 42.4 9 48.6 47.7 51 50 2 10.2 10.0 3 15.3 15.0 4 20.4 20.0 6 25.5 25.0 6 30.6 30.0 7 35.7 35.0 8 40.8 40.0 9 45.9 45.0 52 10.4 15.6 20.8 26.0 31.2 36.4 41.6 46.8 49 9.8 14.7 19.6 24.5 29.4 34.3 39.2 44.1 48 2 9.6 3 14.4 4 19.2 6 24.0 6 28.8 7 33.6 8 38.4 9 43.2 4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 47 9.4 14.1 18.8 23.5 28.2 32.9 37.6 42.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 From the top : For 14°+ or 194°+, read as printed ; for 104°+ or 284°+, read co-function. ' From the bottom : For 75°+ or 255°+, read as printed; for 165°+ or 345°+, read co-function. LGos L Gtn d L Tan L Sin Prop. Pts. 75° — Logarithms of Trigonometric Functions Ill] 15° — Logarithms of Trigonometric Functions 61 L Sin LTan c d L Ctn L Cos Prop. Pts. 9.41 300 9.41 347 9.41 394 9.41 441 9-41 488 9.41 535 9.41 582 9.41 G28 9.41 675 9.41 722 9.41 768 9.41 815 9.41 861 9.41 908 9.41 954 9.42 001 9.42 047 9.42 093 9.42 140 9.42 186 9.42 232 9.42 278 9.42 324 9.42 370 9.42 416 9.42 461 9.42 507 9.42 553 9.42 599 9.42 644 9.42 690 9.42 735 9.42 781 9.42 826 9.42 872 9.42 917 9.42 962 9.43 008 9.43 053 9.43 098 9.43 143 9.43 188 9.43 233 9.43 278 9.43 323 9.43 367 9.43412 9.43 457 9.43 502 9.43 546 9.43 591 9.43 635 9.43 680 9.43 724 9.43 769 9.43 813 9.43 857 9.43 901 9.43 946 9.43 990 9 44 034 .42 805 .42 856 ,42 906 ,42 957 ,43 007 ,43 057 ,43 108 ,43 158 ,43 208 43 258 ,43 308 43 358 43 408 43 458 43 508 .43 558 .43 607 .43 657 .43 707 .43 756 .43 806 .43 855 .43 905 .43 954 .44 004 .44 053 .44 102 .44 151 .44 201 .44 250 44 299 ,44 348 ,44 397 ,44 446 ,44 495 ,44 514 ,44 592 ,44 641 ,44 690 ,44 738 ,44 787 41 836 ,44 884 ,44 933 44 981 ,45029 45 078 ,45 126 45 174 ,45 222 45 271 45 319 ,45 367 45 415 ,45 463 45 511 ,45 559 ,45 606 45 654 45 702 45 750 0.57 195 0.57 144 0.57 094 0.57 043 0.56 993 0.56 943 0.56 892 0.56 842 0.56 792 0.56 742 0.56 692 0.56 642 0.56 592 0.56 542 0.56 492 0.56 442 0.56 393 0.56 343 0.56 293 0.56 244 0.56 IM 0.56 145 0.56 095 0.56 046 0.55 996 0.55 947 0.55 898 0.55 849 0.55 799 0.55 750 0.55 701 0.55 652 0.55 603 0.55 554 0.55 505 0.55 456 0.55 408 0.55 359 0.55 310 0.55 262 0.55 213 0.55164 0.55116 0.55 067 0.55 019 0.54 971 0.54 922 0.54 874 0.54 826 0.54 778 0.54 729 0.54 681 0.54 633 0.54 585 0.54 537 0.54 489 0.54 441 0.54 394 0.54 346 0.54 298 0.54 250 9.98 494 9.98 491 9.98 488 9.98 484 9.98 481 9.98 477 9.98 474 9.98 471 9.98 467 9.98 464 9.98 460 9.98 457 9.98 453 9.98 450 9.98 447 9.98 443 9.98 440 9.98 436 9-98 433 9.98 429 9.98.426 9.98 422 9.98 419 9.98 415 9.98 412 9.98 409 9.98 405 9.98 402 9.98 398 9.98 395 9.98 391 9.98 388 9.98 384 9.98 381 9.98 377 9.98 373 9.98 370 9.98 366 9.98 363 9.98 359 9.98 356 9.98 352 9.98 349 9.98 345 9.98 342 9.98 338 9-98 334 9.98 331 9.98 327 9.98 324 9.98 320 9.98 317 9.98 313 9.98 309 9.98 306 9.98 302 9.98 299 9.98 295 9.98 291 9.98 288 9.98 284 51 50 2 10.2 10.0 3 15.3 15.0 4 20.4 20.0 5 25.5 25.0 6 30.6 30.0 7 35.7 35.0 8 40.8 40.0 9 45.9 45.0 48 47 2 9.6 . 9.4 3 14.4 14.1 4 19.2 18.8 5 24.0 23.5 () 28.8 28.2 7 33.6 32.9 8 38.4 37.6 9 43.2 42.3 49 9.8 14.7 19.6 24.5 29.4 34.3 39.2 44.1 46 9.2 13.8 18.4 23.0 27.6 32.2 3(1.8 41.4 45 2 9.0 3 13.5 4 18.0 5 22.5 6 27.0 7 31.5 8 36.0 9 40.5 4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 44 8.8 13.2 17.6 22.0 26.4 30.8 35.2 39.6 3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 From the top : For 15°+ or 195°+, read as printed ; for 105°+ or 285°+, read co-function. From the bottom : For 74°+ or 254°+, read as printed ; for 164°+ or 344°+, read co-function. L Cos LCtn c d L Tan L Sin Prop. Pts. 74° — Logarithms of Trigonometric Functions 62 16° — Logarithms of Trigonometric Functions [in L Sin L Tan c d L Ctn LGos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.44 034 9.44 078 9.44 122 9.44 166 9.44 210 9.44 253 9.44 297 9.44 341 9.44 385 9.44 428 9.44 472 9.44 516 9.44 559 9.44 602 9.44 646 9.44 689 9.44 733 9.44 776 9.44 819 9.44 862 9.44 905 9.44 948 9.44 992 9.45 035 9.45 077 9.45 120 9.45 163 9.45 206 9.45 249 9.45 292 9.45 334 9.45 377 9.45 419 9.45 462 9.45 504 9.45 547 9.45 589 9.45 632 9.45 674 9.45 716 9.45 758 9.45 801 9.45 843 9.45 885 9.45 927 9.45 969 9.46 011 9.46053 9.46 095 9.46 136 9.46 178 9.46 220 9.46 262 9.46 303 9.46 345 9.46 386 9.46 428 9.46 469 9.46 511 9.46 552 9.46 594 9.45 750 9.45 797 9.45 845 9.45 892 9.45 940 9.45 987 9.46 035 9.46 082 9.46 130 9.46 177 9.46 224 9.46 271 9.46 319 9.46 366 9.46 413 9.46 460 9.46 507 9.46 554 9.46 601 9.46 648 9.46 694 9.46 741 9.46 788' 9.46 835 9.46 881 9.46 928 9.46 975 9.47 021 9.47 068 9.47 114 9.47 160 9.47 207 9.47 253 9.47 299 9.47 346 9.47 392 9.47 438 9.47 484 9.47 530 9.47 576 9.47 622 9.47 668 9.47 714 9.47 760 9.47 806 9.47 852 9.47 897 9.47 943 9.47 989 9.48 035 9.48080 9.48 126 9.48 171 9.48 217 9.48 262 9.48 307 9.48 353 9.48 398 9.48 443 9.48 489 9.48 534 0.54 250 0.54 203 0.54 155 0.54 108 0.64 060 0.54 013 0.53 9(i5 0.53 918 0.53 870 0.53 823 0.53 776 0.53 729 0.53 681 0.53 634 0.53 587 0.53 540 0.53 493 0.53 446 0.53 399 0.53 352 0.53 306 0.53 259 0.53 212 0.53 165 0.53 119 0.53072 0.53 025 0.52 979 0.52 932 0.52 886 0.52 840 0.52 793 0.52 747 0.52 701 0.52 654 0.52 608 0.52 562 0.52 516 0.52 470 0.52 424 0.52 378 0.52 332 0.52 286 0.52 240 0.52 194 0.52 148 0.52 103 0.52 057 0.52 011 0.51 965 0.51 920 0.51 874 0.51 829 0.51 783 0.51 738 0.51693 0.51 647 0.51 602 0.51 557 0.51 511 0.51 466 9.98 284 9.98 281 9.98 277 9.98 273 9.98 270 9.98 266 9.98 262 9.98 259 9.98 255 9.98 251 9.98 248 9.98 244 9.98 240 9.98 237 9.98 233 9.98 229 9.98 226 9.98 222 9.98 218 9.98 215 9.98 211 9.98 207 9.98 204 9.98 200 9.98 196 9.98 192 9.98 189 9.98 185 9.98 181 9.98 177 9.98 174 9.98 170 9.98 166 9.98 162 9.98 159 9.98 155 9.98 151 9.98 147 9.98 144 9.98 140 9.98 136 9.98 132 9.98 129 9.98 125 9.98 121 9.98 117 9.98 113 9.98 110 9.98 106 9.98 102 9.98 098 9.98094 9.98 090 9.98 087 9.98 083 9.98 079 9.98 075 9.98071 9.98 067 9.98063 9.98 060 48 47 2 9.6 9.4 3 14.4 14.1 4 19.2 18.8 5 24.0 23.5 6 28.8 28.2 7 33.6 32.9 8 38.4 37.6 9 43.2 42.3 45 44 2 9.0 8.8 3 13.5 13.2 4 18.0 17.6 5 22.5 22.0 6 27.0 26.4 7 31.5 30.8 8 36.0 35.2 9 40.5 39.6 46 9.2 13.8 18.4 23.0 27.6 32.2 36.8 41.4 43 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 42 2 8.4 3 12.6 4 16.8 5 21.0 6 25.2 7 29.4 8 33.6 9 37.8 4 2 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 41 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 From the top : For 16°+ or 196°+, read as printed; for 106°+ or 286°+, read co-function. From the bottom : For 73°+ or 253°+, read as printed ; for 163°+ or 343°+, read co-function. LGos L Ctn led L Tan L Sin d ' Prop. Pts. 73°— Logarithms of Trigononietric Functions m] 17° — Logarithms of Trigonometric Functions 63 L Sin LTan c d L Ctn L Cos Prop. Pts. 9.46 594 9.46 635 9.4(3 676 9.46 717 9.46 758 9.46 800 9.46 841 9.46 882 9.46 923 9.46 9(54 9.47 005 9.47 045 9.47 086 9.47 127 9.47 168 9.47 209 9.47 249 9.47 290 9.47 330 9.47 371 9.47 411 9.47 452 9.47 492 9.47 533 9.47 573 9.47 613 9.47 654 9.47 694 9.47 734 9.47 774 9.47 814 9.47 854 9.47 894 9.47 934 9.47 974 9.48 014 9.48 054 9.48 094 9.48 133 9.48 173 9.48 213 9.48 252 9.48 292 9.48 332 9.48 371 9.48 411 9.48 450 9.48 490 9.48 529 9.48 568 9.48 607 9.48 (i47 9.48 686 9.48 725 9.48 764 9.48 803 9.48 842 9.48 881 9.48 920 9.48 959 9.48 998 9.48 534 9.48 579 9.48 ()24 9.48 669 9.48 714 9.48 759 9.48 804 9.48 849 9.48 894 9.48 939 9.48 984 9.49 029 9.49 073 9.49 118 9.49 163 9.49 207 9.49 252 9.49 296 9.49 341 9.49 385 9.49 430 9.49474 9.49 519 9.49 563 9.49 607 9.49 652 9.49 696 9.49 740 9.49 784 9.49 828 9.49 872 9.49 916 9.49 960 9.50 004 9.50048 9.50 092 9.50 136 9.50 180 9.50 223 9.50 267 9.50 311 9.50 355 9.50 398 9.50 442 9.50485 9.50 529 9.50 572 9.50 616 9.50 659 9.50 703 9.50 746 9.50 789 9.50 833 9.50 876 9.50 919 9.50 962 9.51 005 9.51 048 9.51 092 9.51 135 9.51 178 0.51 4(56 0.51 421 0.51 376 0.51 331 0.51 286 0.51 241 0.51 196 0.51 151 0.51 106 0.51 061 0.51 016 0.50 971 0.50 927 0.50 882 0.50 837 0.50 793 0.50 748 0.50 704 0.50 659 0.50 615 0.50 570 0.50 526 0.50 481 0.50 437 0.50 393 0.50 348 0.50 304 0.50 260 0.50 216 0.50 172 0.50 128 0.50 084 0.50 040 0.49 996 0.49 952 0.49 908 0.49 864 0.49 820 0.49 777 0.49 733 0.49 689 0.49 645 0.49 602 0.49 558 0.49 515 0.49 471 0.49 428 0.49 384 0.49 341 0.49 297 0.49 254 0.49 211 0.49 167 0.49 124 0.49 081 0.49 038 0.48 995 0.48 952 0.48 908 0.48 865 0.48 822 9.98 060 9.98 056 9.98 052 9.98 048 9.98 044 9.98 040 9.98 036 9.98 032 9.98 029 9.98 025 9.98 021 9.98 017 9.98 013 9.98 009 9.98 005 9.98 001 9.97 997 9.97 993 9.97 989 9.97 986 9.97 982 9.97 978 9.97 974 9.97 970 9.97 966 9.97 962 9.97 958 9.97 954 9.97 950 9.97 946 9.97 942 9.97 938 9.97 934 9.97 930 9.97 926 9.97 922 9.97 918 9.97 914 9.97 910 9.97 906 9.97 902 9.97 898 9.97 894 9.97 890 9.97 886 9.97 882 9.97 878 9.97 874 9.97 870 9.97 866 9.97 861 9.97 857 9.97 853 9.97 849 9.97 845 9.97 841 9.97 837 9.97 833 9.97 829 9.97.825 9.97 821 45 44 2 9.0 8.8 3 13.5 13.2 4 18.0 17.6 5 22.5 22.0 6 27.0 26.4 7 31.5 30.8 8 36.0 35.2 9 40.5 39.6 42 41 2 8.4 8.2 3 12.6 12.3 4 16.8 16.4 5 21.0 20.5 6 25.2 24.6 7 29.4 28.7 8 33.6 32.8 9 37.8 36.9 43 8.6 12.9 17.2 21.5 25.8 30.1 34.4 38.7 40 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 39 2 7.8 3 11.7 4 15.6 5 19.5 6 23.4 7 27.3 8 31.2 9 35.1 2 4 0.8 3 1.2 4 1.6 5 2.0 6 2.4 7 2.8 8 3.2 9 3.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 From the top : For 17°+ or 197^+, read as printed ; for 107°+ or 287°+, read co-function . From the bottom: For 72°+ or 252°+, read as printed ; for 162°+or 342°+, read co-function. L Cos LCtn c d L Tan L Sin d ' Prop. Pts. 72° — Logarithms of Trigonometric Functions 64 18° — Logarithms of Trigonometric Tunctions L Sin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 •39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.48 998 9.49 037 9.49076 9.49 115 9.49 153 9.49 192 9.49 231 9.49 269 9.49 308 9.49 347 9.49 385 9.49 424 9.49462 9.49 500 9.49 539 9.49 577 9.49615 9.49654 9.49 692 9.49 730 9.49 768 9.49806 9.49 844 9.49 882 9.49 920 9.49 958 9.49 996 9.50 034 9.50 072 9.50 110 9.50 148 9.50 185 9.50 223 9.50 261 9.50 298 9.50 3.36 9.50 374 9.50 411 9.50 449 9.50 486 9.50 523 9.50 561 9.50 598 9.50 635 9.50 673 9.50 710 9.50 747 9.50 784 9.50 821 9.50 858 9.50 896 9.50 933 9.50 970 9.51007 9.51 043 9.51 080 9.51 117 9.51 154 9.51 191 9.51 227 9.51 264 9.51 178 9.51 221 9.51 264 9.51 306 9.51 349 9.51 392 9.51 435 9.51 478 9.51 520 9.51 563 9.51 606 9.51 648 9.51 691 9.51 734 9.51 776 9.51 819 9.51 861 9.51 903 9.51 946 9.51 988 9.52 031 9.52 073 9.52 115 9.52 157 9.52 200 52 242 52 284 52 326 52 368 52 410 ,52 452 52 494 52 536 .52 578 52 620 52 661 52 703 52 745 52 787 ,52 829 52 870 52 912 52 953 52 995 53 037 ,53 078 53120 53 161 53 202 53 244 ,53 285 ,53 327 53 368 53 409 53 450 ,53 492 ,53 533 ,53 574 53 615 ,53 656 ,53 697 0.48 822 0.48 779 0.48 736 0.48 694 0.48 651 0.48 608 0.48 565 0.48 522 0.48 480 0.48 437 0.48 394 0.48 352 0.48 309 0.48 266 0.48 224 0.48 181 0.48 139 0.48 097 0.48 054 0.48 012 0.47 ^69 0.47 927 0.47 885 0.47 843 0.47 800 0.47 758 0.47 716 0.47 674 0.47 632 0.47 590. 0.47 548 0.47 506 0.47 464 0.47 422 0.47 380 0.47 339 0.47 297 0.47 255 0.47 213 0.47 171 0.47 130 0.47 088 0.47 047 0.47 005 0.46 963 0.46 922 0.46 880 0.46 839 0.46 798 0.46 756 0.46 715 0.46 673 0.46 632 0.46 591 0.46 550 0.46 508 0.46 467 0.46 426 0.46 385 0.46 344 0.46 303 9.97 821 9.97 817 9.97 812 9.97 808 9.97 804 9.97 800 9.97 796 9.97 792 9.97 788 9.97 784 9.97 779 9.97 775 9.97 771 9.97 767 9.97 763 9.97 759 9.97 754 9.97 750 9.97 746 9.97 742 9.97 738 9.97 734 9.97 729 9.97 725 9.97 721 9.97 717 9.97 713 9.97 708 9.97 704 9.97 700 9.97 696 9.97 691 9.97 687 9.97 683 9.97 679 9.97 674 9.97 670 9.97 666 9.97 662 9.97 657 9.97 653 9.97 649 9.97 645 9.97 ()40 9.97 636 9.97 632 9.97 628 9.97 623 9.97 619 9.97 615 9.97 610 9.97 606 9.97 602 9.97 597 9.97 593 9.97 589 9.97 584 9.97 580 9.97 576 9.97 571 9.97 567 43 42 2 8.6 8.4 3 12.9 12.6 4 17.2 16.8 5 21.5 21.0 6 25.8 25.2 7 30.1 29.4 8 34.4 33.6 9 38.7 37.8 39 38 2 7.8 7.6 3 .11.7 11.4 4 15.6 15.2 5 19.5 19.0 6 23.4 22.8 7 27.3 26.6 8 31.2 30.4 9 35.1 34.2 36 5 2 7.2 1.0 3 10.8 1.5 4 14.4 2.0 5 18.0 2.5 6 21.6 3.0 7 25.2 3.5 8 28.8 4.0 9 32.4 4.5 41 8.2 12.3 16.4 20.5 24.6 28.7 32.8 36.9 37 7.4 11.1 14.8 18.5 22.2 25.9 29.6 33.3 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 From the top : For 18°+ or 198°+, read as printed ; for 108°+ or 288°+, read co-function. From the bottom : For 71°+ or 251°+, read as printed; for 161°+ or 341°+, read co-function. LGos LCtn d L Tan L Sin Prop. Pts. 7r — Logarithms of Trigonometric Functions Ill] 19° — Logarithms of Trigonometric Functions 65 LSin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 '57 58 59 60 9.51 264 9.51 301 9.51 338 9.51 374 9.51 411 9.51 447 9.51 484 9.51 520 9.51 557 9.51 593 9.51 629 9.51 666 9.51 702 9.51 738 9.51774 9.51811 9.51 847 9.51 883 9.51 919 9.51 955 9.51 991 9.52 027 9.52 063 9.52 099 9.52 135 9.52 171 9.52 207 9.52 242 9.52 278 9.52 314 9.52 350 9.52 385 9.52 421 9.52 456 9.52 492 9.52 527 9.52 563 9.52 598 9.52 634 9.52 669 9.52 705 9.52 740 9.52 775 9.52 811 9.52 846 9.52 881 9.52 916 9.52 951 9.52 986 9.53 021 9.53 056 9.53092 9.53 126 9.53 161 9.53 196 9.53 231 9.53 266 9.53 301 9.53 336 9.53 370 9.53405 9.53 697 9.53 738 9.53 779 9.53 820 9.53 861 9.53 902 9.53 943 9.53 984 9.54 025 9.54 065 9.54 106 9.54 147 9.54 187 9.54 228 9.54 269 9.54 309 9.54 350 9.54 390 9.54 431 9.54 471 9.54 512 9.54 552 9.54 593 9.54 633 9.54 673 9.54 714 9.54 754 9.54 794 9.54 835 9.54 875 9.54 915 9.54 955 9.54 995 9.55 035 9.55 075 9.55 115 9.55 155 9.55 195 9.55 235 9.55 275 9.55 315 9.55 355 9.55 395 9.55 434 9.55 474 9.55 514 9.55 554 9.55 593 9.55 633 9.55 673 9.55 712 9.55 752 9.55 791 9.55 831 9.55 870 9.55 910 9.55 949 9.55 989 9.56 028 9.56 067 9.56 107 0.46 303 0.46 262 0.46 221 0.46 180 0.46 139 0.46 098 0.46 057 0.46 016 0.45 975 0.45 935 0.45 894 0.45 853 0.45 813 0.45 772 0.45 731 0.45 691 0.45 650 0.45 610 0.45 569 0.45 529 0.45 488 0.45 448 0.45 407 0.45 367 0.45 327 0.45 286 0.45 246 0.45 206 0.45 165 0.45 125 0.45 085 0.45 045 0.45 005 0.44 965 0.44 925 0.44 885 0.44 845 0.44 805 0.44 765 0.44 725 0.44 685 0.44 645 0.44 605 0.44 566 0.44 526 0.44 486 0.44 446 0.44 407 0.44 367 0.44 327 0.44 288 0.44 248 0.44 209 0.44 169 0.44 130 0.44090 0.44 051 0.44 011 0.43 972 0.43 933 0.43 893 9.97 567 9.97 563 9.97 558 9.97 554 9.97 550 9.97 545 9.97 541 9.97 536 9.97 532 9.97 528 9.97 523 9.97 519 9.97 515 9.97 510 9.97 506 9.97 501 9.97 497 9.97 492 9.97 488 9.97 484 9.97 479 9.97 475 9.97 470 9.97 466 9.97 461 9.97 457 9.97 453 9.97 448 9.97 444 9.97 439 9.97 435 9.97 430 9.97 426 9.97 421 9.97 417 9.97 412 9.97 408 9.97 403 9.97 399 9.97 394 9.97 390 9.97 385 9.97 381 9.97 376 9.97 372 9.97 367 9.97 363 9.97 358 9.97 353 9.97 349 9.97 344 9.97 340 9.97 335 9.97 331 9.97 326 9.97 322 9.97 317 9.97 312 9.97 308 9.97 303 9.97 299 41 40 2 8.2 8.0 3 12.3 12.0 4 16.4 16.0 5 20.5 20.0 6 24.6 24.0 7 28.7 28.0 8 32.8 32.0 9 36.9 36.0 37 36 2 7.4 7.2 3 11.1 10.8 4 14.8 14.4 5 18.5 18.0 6 22.2 21.6 7 25.9 25.2 8 29.6 28.8 9 33.3 32.4 2 34 6.8 5 1.0 3 10.2 1.5 4 13.6 2.0 5 17.0 2.5 6 20.4 3.0 7 23.8 3.5 8 27.2 4.0 9 30.6 4.5 39 7.8 11.7 15.6 19.5 23.4 27.3 31.2 35.1 35 7.0 10.5 14.0 17.5 21.0 24.5 28.0 31.5 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 From the top : For 19°+ or 199°+, read as printed ; for 109°+ or 289°+, read co-functioD. From the bottom : For 70°+ or 250°+, read as printed ; for 160°+ or 340°+, read co-function. LCos LCtn d L Tan L Sin Prop. Pts. 70°— Logarithms of Trigonometric Functions 66 20° — Logarithms of Trigonometric Functions [in LSin L Tan c d L Ctn LGos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.53 405 9.53 440 9.53 475 9.53 509 9.53 544 9.53 578 9.53 613 9.53 647 9.53 682 •9.53 716 9.53 751 9.53 785 9.53 819 9.53 854 9.53 888 9.53 922 9.53 957 9.53 991 9.54 025 9.54059 9 54 093 9.54 127 9.54 161 9.54 195 9.54 229 9.54 263 9.54 297 9.54 331 9.54 365 9.54 399 9.54 433 9.54 466 9.54 500 9.54 534 9.54 567 9.54 601 9.54 635 9.54 668 9.54 702 9.54 735 9.54 769 9.54 802 9.54 836 9.54 869 9.54 903 9.54 936 9.54 969 9.55 003 9.55 036 9.55 069 9.55 102 9.55 136 9.55 169 9.55 202 9.55 235 9.55 268 9.55 301 9.55 334 9.55 367 9.55 400 9.55 433 9.56 107 9.56 146 9.56 185 9.56 224 9.56 264 9.56 303 9.56 342 9.56 381 9.56 420 9.56 459 9.56 498 9.56 537 9.56 576 9.56 615 9.56 654 9.56 693 9.56 732 9.56 771 9.56 810 9.56 849 9.56 887 9.56 926 9.56 965 9.57 004 9.57 042 9.57 081 9.57 120 9.57 158 9.57 197 9.57 235 9.57 274 9.57 312 9.57 351 9.57 389 9.57 428 9.57 466 9.57 504 9.57 543 9.57 581 9.57 619 9.57 658 9.57 696 9.57 734 9.57 772 9.57 810 9.57 849 9.57 887 9.57 925 9.57 963 9.58 001 9.58 039 9.58 077 9.58 115 9.58 153 9.58 191 9.58 229 9.58 267 9.58 304 9.58 342 9.58 380 9.58 418 43 893 43 854 43 815 43 776 43 736 43 697 43 658 43 619 43 580 43 541 43 502 43 463 43 424 43 385 43 346 0.43 307 0.43 268 0.43 229 0.43 190 0.43 151 0.43 113 0.43 074 0.43 035 0.42 996 0.42 958 0.42 919 0.42 880 0.42 842 0.42 803 0.42 765 0.42 726 0.42 688 0.42 649 0.42 611 0.42 572 0,42 534 0.42 496 0.42 457 0.42 419 0.42 381 0.42 342 0.42 304 0.42 266 0.42 228 0.42 190 0.42 151 0.42 113 0.42 075 0.42 037 0.41 999 0.41 961 0.41 923 0.41 885 0.41 847 0.41 809 0.41 771 0.41 733 0.41 696 0.41 658 0.41 620 0.41 582 9.97 299 9.97 294 9.97 289 9.97 285 9.97 280 9.97 276 9.97 271 9.97 266 9.97 262 9.97 257 9.97 252 9.97 248 9.97 243 9.97 238 9.97 234 9.97 229 9.97 224 9.97 220 9.97 215 9.97 210 9.97 206 9.97 201 9.97 196 9.97 192 9.97 187 9.97 182 9.97 178 9.97 173 9.97 168 9.97 163 9.97 159 9.97 154 9.97 149 9.97 145 9.97 140 9.97 135 9.97 130 9.97 126 9.97 121 9.97 116 9.97 111 9.97 107 9.97 102 9.97 097 9.97 092 9.97 087 9.97 083 9.97 078 9.97 073 9.97 068 9.97 063 9.97 059 9.97 054 9.97 049 9.97 044 9.97 039 9.97 035 9.97 030 9.97 025 9.97 020 9.97 015 40 39 2 8.0 7.8 3 12.0 11.7 4 16.0 15.6 5 20.0 19.5 6 24.0 23.4 7 28.0 27.3 8 32.0 31.2 9 36.0 35.1 37 35 2 7.4 7.0 3 11.1 10.5 4 14.8 14.0 5 18.5 17.5 6 22.2 21.0 7 25.9 24.5 8 29.6 28.0 9 33.3 31.5 33 5 2 6:6 1.0 3 9.9 1.5 4 13.2 2.0 5 16.5 2.5 6 19.8 3.0 7 23.1 3.5 8 26.4 4.0 9 29.7 4.5 From the top : For 20°+ or 200°+, read as printed; for 110°+ or 290°+, read co-function. From the bottom : For 69°+ or 249°+, read as printed ; for 159°+ or 339°+, read co-function. LCos LGtn d L Tan LSin Prop. Pts. fi9° -TiOo*a,ritliTns nf Trie^ononnpifrin Fiinr»tioTis Ill] 21° — Logarithms of Trigonometric Functions 67 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 L Sin 9.55 433 9.55 46() 9.55 499 9.55 532 9.55 564 9.55 597 9.55 630 9.55 663 9.55 695 9.55 728 9.55 761 9.55 793 9.55 826 9.55 858 9.55 891 9.55 923 9.55 956 9.55 988 9.56 021 9.56 053 9.56 085 9.56 118 9.56 150 9.56 182 9.56 215 9.56 247 9.56 279 9.56 311 9.56 343 9.56 375 9.56 408 9.56 440 9.56 472 9.56 504 9.56 536 9.56 568 9.56 599 9.56 631 9.56 663 9.56 695 9.56 727 9.56 759 9.56 790 9.56 822 9.56 854 9.56 886 9.56 917 9.56 949 9.56 980 9.57 012 9.57 044 9.57 075 9.57 107 9.57 138 9.57 169 9.57 201 9.57 232 9.57 264 9.57 295 9.57 326 9.57 358 L Tan c d L Ctn 9.58 418 9.58 455 9.58 493 9.58 531 9.58 569 9.58 606 9.58 644 9.58 681 9.58 719 9.58 757 9.58 794 9.58 832 9.58 869 9.58 907 9.58 944 9.58 981 9.59019 9.59056 9 59 094 9.59 131 9.59 168 9.59 205 9.59 243 9.59 280 9.59 317 9.59 354 9.59 391 9.59 429 9.59 466 9.59 503 9.59 540 9.59 577 9.59 614 9.59 651 9.59 688 9.59 725 9.59 762 9.59 799 9.59 835 9.59 872 9.59 909 9.59 946 9.59 983 9.60 019 9.60 056 9.60 093 9.60 130 9.60 166 9.60 203 9.60 240 9.60 276 9.60 313 9.60 349 9.60 386 9.60 422 9.60 459 9.60 495 9.60 532 9.60 568 9.60 605 9.60 641 0.41 582 0.41 545 0.41 507 0.41 469 0.41 431 0.41 394 0.41 356 0.41 319 0.41 281 0.41 243 0.41 206 0.41 168 0.41 131 0.41 093 0.41 056 0.41019 0.40 981 0.40 944 0.40 906 0.40 869 0.40 832 0.40 795 0.40 757 0.40 720 0.40 683 0.40 646 0.40 609 0.40 571 0.40 534 0.40 497 0.40 460 0.40 423 0.40 386 0.40 349 0.40 312 0.40 275 0.40 238 0.40 201 0.40 165 0.40 128 0.40091 0.40 054 0.40 017 0.39 981 0.39 944 0.39 907 0.39 870 0.39 834 0.39 797 0.39 760 0.39 724 0.39 687 0.39 651 0.39 614 0.39 578 0.39 541 0.39 505 0.39 468 0.39 432 0.39 395 0.39 359 L Cos 9.97 015 9.97 010 9.97 005 9.97 001 9.96 996 9.96 991 9.96 986 9.96 981 9.96 976 9.96 971 9.96 966 9.96 962 9.96 957 9.96 952 9.96 947 9.96 942 9.96 937 9.96 932 9.96 927 9.96 922 9.96 917 9.96 912 9.96 907 9.96 903 9.96 898 9.96 893 9.96 888 9.f)6 883 9.96 878 9.96 873 9.96 868 9.96 863 9.96 858 9.96 853 9.96 848 9.96 843 9.96 838 9.96 833 9.96 828 9.96 823 9.96 818 9.96 813 9.96 808 9.96 803 9.96 798 9.96 793 9.96 788 9.96 783 9.96 778 9.96 772 9.96 767 9.96 762 9.96 757 9.96 752 9.96 747 9.96 742 9.96 737 9.96 732 9.96 727 9.96 722 9.96 717 Prop. Pts. 38 37 2 7.6 7.4 3 11.4 11.1 4 15.2 14.8 5 19.0 18.5 6 22.8 22.2 7 26.6 25.9 8 30.4 29.6 9 34.2 33.3 33 32 2 6.6 6.4 3 9.9 9.6 4 13.2 12.8 5 16.5 16.0 6 19.8 19.2 7 23.1 22.4 8 26.4 25.6 9 29.7 28.8 6 5 2 1.2 1.0 3 1.8 1.5 4 2.4 2.0 5 3.0 2.5 6 3.6 3.0 7 4.2 3.5 8 4.8 4.0 9 5.4 4.5 36 7.2 10.8 14.4 18.0 21.6 25.2 28.8 32.4 31 6.2 9.3 12.4 15.5 18.6 21.7 24.8 27.9 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 From the top: For 21°+ or 201°+, read as printed ; for ^11°+ or 291°+, read co-function. From the bottom: For 68°+ or 248°+, read as printed ; for 158°+ or 338°+, read co-function. LGos L Ctn c d L Tan L Sin Prop. Pts. 68°— Logarithms of Trigonometric Functions 68 22° — Logarithms of Trigonometric Functions [HI L Sin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.57 358 9.57 389 9.57 420 9.57 451 9.57 482 9.57 514 9 57 545 9.57 576 9.57 607 9.57 638 9.57 669 9.57 700 9.57 731 9.57 762 9.57 793 9.57 824 9.57 855 9.57 885 9.57 916 9.57 947 9.57 978 9.58 008 9.58 039 9.58 070 9.58 101 9.58 131 9.58 162 9.58 192 9.58 223 9.58 253 9.58 284 9.58 314 9.58 345 9.58 375 9.58 406 9.58 436 9.58 467 9.58 497 9.58 527 9.58 557 9.58 588 9.58 618 9.58 648 9.58 678 9.58 709 9.58739 9.58 769 9.58 799 9.58 829 9.58 859 9.58 889 9.58 919 9 58 949 9.58 979 9.59 009 9 59 039 9.59069 9.59 098 9.59128 9.59 158 9.59 188 9.60'641 9.60 677 9.60 714 9.60 750 9.60 786 9.60 823 9.60 859 9.60 895 9.60 931 9.60 967 9.61 004 9.61 040 9.61 076 9.61 112 9.61 148 9.61 184 9.61 220 9.61 256 9.61 292 9.61 328 9.61 364 9.61 400 9.61 436 9.61 472 9.61 508 9.61 544 9.61 579 9.61 615 9.61 651 9.61 687 9.61 722 9.61 758 9.61 794 9.61 830 9.61 865 9.61 901 9.61 936 9.61 972 9.62 008 9.62 043 9.62 079 9.62 114 9.62 150 9.62 185 9.62 221 9.62 256 9.62 292 9.62 327 9.62 362 9.62 398 9.62 433 9.62 468 9.62 504 9.62 539 9.62 574 9.62 609 9.62 645 9.62 680 9.62 715 9.62 750 9.62 785 0.39 359 0.39 323 0.39 286 0.39 250 0.39 214 0.39 177 0.39 141 0.39 105 0.39 069 0.39 0.J3 0.38 996 0.38 960 0.38 924 0.38 888 0.38 852 0..38 816 0.38 780 0.38 744 0.38 708 0.38 672 0.38 636 0.38 600 0.38 564 0.38 528 0.38 492 0.38 456 0.38 421 0.38 385 0.38 349 0.38 313 0.38 278 0.38 242 0.38 206 0.38 170 0.38 135 0.38 099 0.38 064 0.38 028 0.37 992 0.37 957 0.37 921 0.37 886 0.37 850 0.37 815 0.37 779 0.37 744 0.37 708 37 673 0.37 638 0.37 602 0.37 567 0.37 532 0.37 496 0.37 461 0.37 426 0.37 391 0.37 355 0.37 320 0.37 285 0.37 250 0.37 215 9.96 717 9.96 711 9.96 706 9.96 701 9.96 696 9.96 691 9.96 686 9.96 681 9.96 676 9.96 670 9.96 665 9.96 660 9.96 655 9.96 650 9.96 645 9.96 640 9.96 634 9.96 629 9.96 624 9.96 619 9.96 614 9.96 608 9.96 603 9.96 598 9.96 593 9.96 588 9.96 582 9.96 577 9.96 572 9.96 567 9.96 562 9.96 556 9.96 551 9.96 546 9.96 541 9.96 535 9.96 530 9.96 525 9.96 520 9.96 514 9.96 509 9.96 504 9.96 498 9.96 493 9.96 488 9.96 483 9.96 477 9.96 472 9.96 467 9.96 461 9.96 456 9.96 451 9.96 445 9.96 440 9.96435 9.96 429 9.96 424 9.96 419 9.96 413 9.96 408 9.96403 5 33 5 32 5 31 f] 30 t^ 29 5 28 5 27 6 26 5 25 18 .5 17 5 16 37 86 2 7.4 7.2 3 11.1 10.8 4 14.8 14.4 5 18.5 18.0 6 22.2 21.6 7 25.9 25.2 8 29.6 28.8 9 33.3 32.4 32 31 2 6.4 6.2 3 9.6 9.3 4 12.8 12.4 5 16.0 15.5 6 19.2 18.6 7 22.4 21.7 8 25.6 24.8 9 28.8 27.9 35 7.0 10.5 14.0 17.5 21.0 24.5 28.0 31.5 30 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 From the top : For 22°+ or 202°+, read as printed ; for 112°+ or 292°+, read co-function. From the bottom : For 67°+ or 247°+, read as printed; for 157°+ or 337°+, read co-function. 29 6 2 5.8 1.2 3 8.7 1.8 4 11.6 2.4 6 14.5 3.0 6 17.4 3.6 7 20.3 4.2 8 23.2 4.8 9 26.1 5.4 L Cos d L Ctn c d L Tan L Sin d ' Prop. Pts. 67° — Losrarithnis of Trigonometric Functions Ill] 23° — Logarithms of Trigonometric Functions 69 L Sin L Tan c d L Gtn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.59188 9.59 218 9.59 247 9.59 277 9.59 307 9.59 336 9.59 366 9.59 396 9.59 425 9.59455 9.59 484 9.59 514 9.59 543 9.59 573 9.59 602 9.59 632 9.59 661 9.59 690 9.59 720 9.59 749 9.59 778 9.59 808 9.59 837 9.59 866 9.59 895 9.59 924 9.59 954 9.59 983 9.60 012 9.60 041 9.60 070 9.60 099 9.60 128 9.60 157 9.60 186 9.60 215 9.60 244 9.60 273 9.60 302 9.60 331 9.60 359 9.60 388 9.60 417 9.60 446 9.60 474 9.60 503 9.60 532 9.60 561 9.60 589 9.60 618 9.60 646 9.60 675 9.60 704 9.60 732 9.60 761 9.60 789 9.60 818 9.60 84(5 9.60 875 9.60 903 9.60 931 9.62 785 9.62 820 9.62 855 9.62 890 9.62 926 9.62 961 9.62 996 9.63031 9.63 066 9.63 101 9.63 135 9.63170 9.63 205 9.63 240 9.63 275 9.63 310 9.63 345 9.63 379 9.63 414 9.63 449 9.63 484 9.63 519 9.63 553 9.63 588 9.63 623 9.63 657 9.63 692 9.63 726 9.63 761 9.63 796 9.63 830 9.63 865 9.63 899 9.63 934 9.63 968 9.64 003 9.64 037 9.64 072 9.64 106 9.64 140 9.64 175 9.()4 209 9.64 243 9.64 278 9.64 312 9.64 346 9.64 381 9.64 415 9.64 449 9.64 483 9.64 517 9.64 552 9.64 586 9.64 620 9.64 654 9.64 688 9.64 722 9.64 756 9.64 790 9.61824 9.64 858 0.37 215 0.37 180 0.37 145 0.37 110 0.37 074 0.37 039 0.37 004 0.36 969 0.36 934 0.36 899 0.36 865 0.36 830 0.36 795 0.36 760 0.36 725 0.36 im 0.36 655 0.36 621 0.36 586 0.36 551 0.36 516 0.36 481 0.36 447 0.36 412 0.36 377 0.36 343 0.36 308 0.36 274 0.36 239 0.36 204 0.36 170 0.36 135 0.36 101 0.36 066 0.36 032 0.35 997 0.35 963 0.35 928 0.35 894 0.35 860 0.35 825 0.35 791 0.35 757 0..35 722 0.35 688 0.35 654 0.35 619 0.35 585 0.35 551 0.35 517 0.35 483 0.35 448 0.35 414 0.35 380 0.35 346 0.35 312 0.35 278 0.35 244 0.35 210 0.35 176 0.35 142 9.96 403 9.96 397 9.96 392 9.96 387 9.96 381 9.96 376 9.96 370 9.96 365 9.96 360 9.96 354 9.96 349 9.96 343 9.96 338 9.96 333 9.96 327 9.96 322 9.96 316 9.96 311 9.96 305 9.96 300 9.96 294 9.96 289 9.96 284 9.96 278 9.96 273 9.96 267 9.96 262 9.96 256 9.96 251 9.96 245 9.96 240 9.96 234 9.96 229 9.96 223 9.96 218 9.96 212 9.96 207 9.96 201 9.96 196 9.96 190 9.96185 9.96 179 9.96 174 9.96 168 9.96 162 9.96 157 9.96 151 9.96 146 9.96 140 9.96 135 9.96 129 9.96 123 9.96 118 9.96112 9.96 107 9.96101 9.96 095 9.96 090 9.96 084 9.96 079 9.96 073 36 35 2 7.2 7.0 3 10.8 10.5 4 14.4 14.0 5 18.0 17.5 6 21.6 21.0 7 25.2 24.5 8 28.8 28.0 9 32.4 31.5 30 29 2 6.0 5.8 3 9.0 8.7 4 12.0 11.6 5 15.0 14.5 6 18.0 17 4 7 21.0 20.3 8 24.0 23.2 9 27.0 26.1 34 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 5.6 8.4 11.2 140 16.8 19.6 22.4 25.2 2 6 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 6.4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 From the top : For 23°+ or 203°+, read as printed; for 113°+ or 293°+, read co-function. From the bottom : For 66°+ or 246°+, read as printed ; for 156°+ or 336°+, read co-function. L Cos LGtn c d L Tan L Sin Prop. Pts. 66° — Logarithms of Trigonometric Functions 70 34° — Logarithms of TrigonomeMc Functions [in LSin L Tan c d L Gtn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 63 54 55 56 57 58 59 60 9.60 931 9.60 960 9.60 988 9.61016 9.61045 9.61073 9.61 101 9.61 129 9.61 158 9.61 186 9.61214 9.61 242 9.61 270 9.61 298 9.61 326 9.61 354 9.61 382 9.61 411 9.61 438 9.61466 9.61 494 9.61 522 9.61 550 9.61 578 9.61 606 9.61 634 9.61 662 9.61 689 9.61 717 9.61 745 9.61 773 9.61 800 9.61 828 9.61 856 9.61 883 9.61 911 9.61 939 9.61 966 9.61 994 9.62 021 9.62.049 9.62 076 9.62 104 9.62 131 9.62 159 9.62 186 9.62 214 9.62 241 9.62 268 9.62 296 9.62 323 9.62 350 9.62 377 9.62 405 9.62 432 9.62 459 9.62 486 9.62 513 9.62 541 9.62 568 9.62 595 9.64 858 9.64 892 9.64 926 9.64 960 9.64 994 9.65 028 9.65 062 9.65 096 9.65 130 9.65 164 9.65 197 9.65 231 9.65 265 9.65 299 9.65 333 9.65 366 9.65 400 9.65 434 9.65 467 9.65 501 9.65 535 9.65 568 9.65 602 9.65 636 9.65 669 9.65 703 9.65 736 9.65 770 9.65 803 9.65 837 9.65 870 9.65 904 9.65 937 9.65 971 9.66 004 9.66 038 9.66 071 9.66 104 9.66 138 9.66 171 9.66204 9.66 238 9.66 271 9.66 304 9.66 337 9.66 371 9.66 404 9.66 437 9.66 470 9.66503 9.66 537 9.66 570 9.66 603 9.66 636 9.66 669 9.66 702 9.66 735 9.66 768 9.66 801 9.66 834 9.66 867 0.35 142 0.35 108 0.35074 0.35 040 0.35 006 0.34 972 0.34 938 0.34 904 0.34 870 0.34 836 0.34 803 0.34 769 0.34 735 0.34 701 0.34 667 0.34 634 0.34 600 0.34 566 0.34 533 0.34 499 0.34 465 0.34 432 0.34 398 0.34 364 0.34 331 0.34 297 0.34 264 0.34 230 0.34 197 0.34 163 0.34130 0.34 096 0.34 063 0.34 029 0.33 996 0.33 962 0.33 929 0.33 896 0.33 862 0.33 829 0.33 796 0.33 762 0.33 729 0.33 696 0.33 663 0.33629 0.33 596 0.33 563 0.33 530 0.33497 0.33463 0.33430 0.33 397 0.33 364 0.33 331 0.33 298 0.33 265 0.33 232 0.33 199 0.33 166 0.33 133 9.96 073 9.96067 9.96 062 9.96 056 9.96050 9.96 045 9.96 039 9.96034 9.96 028 9.96 022 9.96 017 9.96 Oil 9.96 005 9.96 000 9.95 994 9.95 988 9.95 982 9.95 977 9.95 971 9.95 965 9.95 960 9.95 954 9.95 948 9.95 942 9.95 937 9.95 931 9.95 925 9.95 920 9.95 914 9.95 908 9.95 902 9.95 897 9.95 891 9.95 885 9.95 879 9.95 873 9.95 868 9.95 862 9.95 856 9.95 850 9.95 844 9.95 839 9.95 833 9.95 827 9.95 821 9.95 815 9.95 810 9.95 804 9.95 798 9.95 792 9.95 786 9.95 780 9.95 775 9.95 769 9.95 763 9.95 757 9.95 751 9.95 745 9.95 739 9.95 733 9.95 728 34 33 2 6.8 6.6 3 10.2 9.9 4 13.6 13.2 5 17.0 16.5 6 20.4 19.8 7 23.8 23.1 8 27.2 26.4 9 30.6 29.7 29 5.8 8.7 11.6 14.5 17.4 20.3 23.2 26.1 28 27 2 5.6 5.4 3 8.4 8.1 4 11.2 10.8 5 14.0 13.5 6 16.8 16.2 7 19.6 18.9 8 22.4 21.6 9 25.2 24.3 6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 From the top : For 24°+ or 204°+, read as printed; for 114°+ or 294°+, read co-function. From the bottom : For 65°+ or 245°+, read as printed; for 155°+ or 335°+, read co-function. L Cos LCtn cd LTan L Sin d / Prop. Pts. 6/i° — TiOerarifliiris of Tri iron om ft trie Functions nq 25° — Logarithms of Trigonometric Functions 71 LSin L Tan led L Gtn L Cos Prop. Pts. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.62 595 9.62 622 9.62 649 9.62 676 9.62 703 9.62 730 9.62 757 9.62 784 9.62 811 9.62 838 9.62 865 9.62 892 9.62 918 9.62 945 9.62 972 9.62 999 9.63 026 9.63 052 9.63 079 9.63 106 9.63 133 9.63 159 9.63 186 9.63 213 9.63 239 9.63 266 9.63 292 9.63 319 9.63 345 9.63 372 9.63 398 9.63 425 9.63451 9.63 478 9.63 504 9.63 531 9.63 557 9.63 583 9.63 610 9.63636 9.63 662 9.63689 9.63 715 9.63 741 9.63 767 9.63 794 9.63 820 9.63 846 9.63 872 9.63 898 9.63 924 9.63 950 9.63 976 9.64002 9.64028 9.64 054 9.64080 9.64 106 9.64 132 9.64 158 9.64 184 LGos 9.66 867 9.66 900 9.66 933 9.66 966 9.66 999 9.67 032 9.67 065 9.67 098 9.67 131 9.67 163 9.67 196 9.67 229 9.67 262 9.67 295 9.67 327 9.67 360 9.67 393 9.67 426 9.67 458 9.67 491 9.67 524 9.67 556 9.67 589 9.67 622 9.67 654 9.67 687 9.67 719 9.67 752 9.67 785 9.67 817 9.67 850 9.67 882 9.67 915 9.67 947 9.67 980 9.68 012 9.68 044 9.68 077 9.68 109 9.68 142 9.68 174 9.68 206 9.68 239 9.68 271 9.68 303 9.68 336 9.68 368 9.68 400 9.68 432 9.68 465 9.68 497 9.68 529 9.68 561 9.68 593 9.68 626 9.68 658 9.68 690 9.68 722 9.68 754 9.68 786 9.68 818 L Ctn c d L Tan 0.33 133 0.33 100 0.33067 0.33 034 0.33 001 0.32 968 0.32 935 0.32 902 0.32 869 0.32 837 0.32 804 0.32 771 0.32 738 0.32 705 0.32 673 0.32 640 0.32 607 0.32 574 0.32 542 0.32 509 0.32 476 0.32 444 0.32 411 0.32 378 0.32 346 0.32 313 0.32 281 0.32 248 0.32 215 0.32 183 0.32 150 0.32 118 0.32 085 0.32 053 0.32 020 0.31 988 0.31 956 0.31 923 0.31 891 0.31 858 0.31 826 0.31 794 0.31 761 0.31 729 0.31 697 0.31 664 0.31 632 0.31 600 0.31 568 0.31 535 0.31 503 0.31 471 0.31 439 0.31 407 0.31 374 0.31 342 0.31 310 0.31 278 0.31246 0.31 214 0.31 182 9.95 728 9.95 722 9.95 716 9.95 710 9.95 704 9.95 698 9.95 692 9.95 686 9.95 680 9.95 674 9.95 668 9.95 663 9.95 657 9.95 651 9.95 645 9.95 639 9.95 633 9.95 627 9.95 621 9.95 615 9.95 609 9.95 603 9.95 597 9.95 591 9.95 585 9.95 579 9.95 573 9.95 567 9.95 561 9.95 555 9.95 549 9.95 543 9.95 537 9.95 531 9.95 525 9.95 519 9.95 513 9.95 507 9.95 500 9.95 494 9.95 488 9.95 482 9.95 476 9.95 470 9.95 464 9.95 458 9.95 452 9.95 446 9.95 440 9.95 434 9.95 427 9.95 421 9.95 415 9.95 409 9.95 403 9.95 397 9.95 391 9.95 384 9.95 378 9.95 372 9.95 366 L Sin 33 32 2 6.6 6.4 3 9.9 9.6 4 13.2 12.8 5 16.5 16.0 6 19.8 19.2 7 23.1 22.4 8 26.4 25.6 9 29.7 28.8 27 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 d ' 26 2 5.2 3 7.8 4 10.4 5 13.0 6 15.6 7 18.2 8 20.8 9 23.4 6 2 1.2 3 1.8 4 2.4 5 3.0 6 3.6 7 4.2 8 4.8 9 5.4 7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 From the top : For 25°+ or 205^+, read as printed; for 115°+ or 295°+, read co-functiou. From the bottom: For 64°+ or 244°+, read as printed ; for 154°+ or 334°+, read co-function. Prop. Pts. 64°— Logarithms of Trigonometric Functions 72 26°— Logarithms of Trigonometric Functions [ii, L Sin LTan c d L Gtn LCos Prop. Pts. 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.64 184 9.64 210 9.64 236 9.64 262 9.64 288 9.64 313 9.64 339 9.64 3(i5 9.64 391 9 64 417 9.64 442 9.61468 964 494 9.64 519 9.64 545 9.64 571 9.64 596 9.64 622 9.64 647 9.64 673 9.64 698 9.64 724 9.64 749 9.64 775 9.64 800 9.64 826 9.64 851 9.64 877 9.64 902 9.64 927 9.64 953 9.64 978 9.65 003 9.65 029 9.65 054 9.65 079 9.65 104 9.65 130 9.65 155 9.65 180 9.65 205 9.65 230 9.65 255 9.65 281 9.65 306 9.65 331 9.65 356 9.65 381 9.65 406 9.65 431 9.65 456 9.65 481 9.65 506 9.65 531 9.65 556 9.65 580 9.65 605 9.65 630 9.65 655 9.65 680 9.65 705 26 26 26 26 25 26 26 26 26 25 26 26 25 26 26 25 26 25 26 25 26 25 26 25 26 25 26 25 25 26 25 25 26 25 25 25 26 25 25 25 25 25 26 25 25 25 25 25 25 25 25 25 25 25 24 25 25 25 25 25 9.68 818 9.()8 850 9.68 882 9.68 914 9.68 94() 9.68 978 9.69010 9.69 042 9.69 074 9.69 106 9.69 138 9.69 170 9.69 202 9.69 234 9.69 266 9.69298 9.69 329 9.69 361 9.69 393 9.69425 9.69457 9.69488 9.69 520 9.69 552 9.69 584 9.69 615 9.69 647 9.69 679 9.69 710 9.69 742 9.69 774 9.69 805 9.69 837 9.69 868 9.69 900 9.69 932 9.69 963 9.69 995 9.70 026 9.70 058 9.70 089 9.70 121 9.70 152 9.70 184 9.70 215 9.70 247 9.70 278 9.70 309 9.70 341 9.70 372 9.70 404 9.70 435 9.70 466 9.70 498 9.70 529 9J0 560 9.70 592 9 JO 623 9.70 654 9.70 685 9.70 717 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 31 32 32 32 32 31 32 32 32 31 32 32 31 32 32 31 32 31 32 32 31 32 31 32 31 32 31 32 31 32 31 31 32 31 32 31 31 32 31 31 32 31 31 31 32 0.31 182 0.31 150 0.31 118 0.31 086 0.31 054 0.31 022 0.30 990 0.30 958 0.30 926 0.30 894 0.30 862 0.30 830 0.30 798 0.30 766 0.30 734 0.30 702 0.30 671 0.30 639 0.30 607 0.30 575 0.30 543 0.30 512 0.30 480 0.30 448 0.30 416 0.30 385 0.30 353 0.30 321 0.30 290 0.30 258 0.30 226 0.30 195 0.30 163 0.30 132 0.30 100 0.30068 0.30 037 0.30 005 0.29 974 0.29 942 0.29 911 0.29 879 0.29 848 0.29 816 0.29 785 0.29 753 0.29 722 0.29 691 0.29 659 0.29 628 0.29 596 0.29 565 0.29 534 0.29 502 0.29 471 0.29 440 0,29 408 0.29 377 0.29 346 0.29 315 0.29 283 9.95 366 9.95 360 9.95 354 9.9.) 348 9.95 311 9.95 335 9.95 329 9.95 323 9.95 317 9.95 310 9.95 304 9.95 298 9.95 292 9.95 286 9.95 279 9.95 273 9.95 267 9.95 261 9.95 254 9.95 248 9.95 242 9.95 236 9.95 229 9.95 223 9.95 217 9.95 211 9.95 204 9.95 198 9.95 192 9.95 185 9.95 179 9.95 173 9.95 167 9.95 160 9.95 154 9.95 148 9.95 141 9.95 135 9.95 129 9.95 122 9.95116 9.95 110 9.95 103 9.95 097 9.95 090 9.95 084 9.95 078 9.95 071 9.95 065 9.95 059 9.95 052 9.95 046 9.95 039 9.95 033 9.95027 9.95 020 9.95 014 9.95 007 9.95 001 9.94 995 9.94 988 60 59 58 57 56 55 54 53 62 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 32 31 2 64 6.2 3 9.6 9.3 4 12.8 12.4 5 16.0 15.5 6 19.2 18.6 7 22.4 21,7 8 25.6 24.8 9 28.8 27.9 26 5.2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 25 2 5.0 3 7.5 4 10.0 5 12.5 6 15.0 7 17.5 8 20.0 9 22.5 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 24 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.6 6 1.2 1.8 ' 2.4 3.0 3.6 4.2 4.8 5.4 From the top : For 26°+ or 206°+, read, as printed ; for 116°+ or 296°+, read co-function. From the bottom : For 63°+ or 243°+, read as printed; for 153°+ or 333°+, read co-function. L Cos L Ctn c d L Tan L Sin Prop. Pts. 63° — Logarithms of Trigonometric Functions Ill] 27° — Logarithms of Trigonometric Functions 73 LSin L Tan cd LCtn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 .57 58 59 60 9.()5 705 9.65 729 9.65 754 9.65 779 9.65 804 9.65 828 9.65 853 9.65 878 9.65 902 9.65 927 9.65 952 9.65 976 9.66001 9.66 025 9.66 050 9.66 075 9.66 099 9.r>6 124 9.66 148 9.66 173 9.66 197 9.66 221 9.66 246 9.66 270 9.66 295 9.66 319 9.66 343 9.66 368 9.66 392 9.66 416 9.66 441 9.66 465 9.66 489 9.66 513 9.66 537 9.66 562 9.66 586 9.66 610 9.66 634 9.66 658 9.66 682 9.66 706 9.66 731 9.66 755 9.66 779 9.66 803 9.66 827 9.66 851 9.66 875 9.66 899 m 922 66 946 66 970 66 994 .67 018 67 042 67 066 67 090 .67 113 67 137 67 161 9.70 717 9.70 748 9.70 779 9.70 810 9.70 841 9.70 873 9.70 904 9.70 935 9.70 966 9.70 997 9.71 028 9.71 059 9.71 090 9.71 121 9.71 153 9.71 184 9.71 215 9.71 246 9.71 277 9.71 308 9.71 339 9.71 370 9.71 401 9.71 431 9.71 462 9.71 493 9.71 524 9.71 555 9.71 586 9.71 617 9.71 648 9.71679 9.71709 9.71 740 9.71 771 9.71 802 9.71 833 9.71 863 9.71 894 9.71 925 9.71 955 9.71 986 9.72 017 9.72 048 9.72 078 9.72 109 9.72 140 9.72 170 9.72 201 9.72 231 9.72 262 9.72 293 9.72 323 9.72 354 9.72 384 9.72 415 9.72 445 9.72 476 9.72 506 9.72 537 9.72 567 0.29 283 0.29 252 0.29 221 0.29 190 0.29 159 0.29 127 0.29 096 0.29065 0.29034 0.29003 0.28 972 0.28 941 0.28 910 0.28 879 0.28 847 0.28 816 0.28 785 0.28 754 0.28 723 0.28 692 0.28 661 0.28 630 0.28 599 0.28 569 0.28 538 0.28 507 0.28 476 0.28 445 0.28 414 0.28 383 0.28 352 0.28 321 0.28 291 0.28 260 0.28 229 0.28 198 0.28 167 0.28 137 0.28 106 0.28 075 0.28 045 0.28 014 0.27 983 0.27 952 0.27 922 0.27 891 0.27 860 0.27 830 0.27 799 0.27 769 0.27 738 0.27 707 0.27 677 0.27 646 0.27 616 0.27 585 0.27 555 0.27 524 0.27 494 0.27 463 0.27 433 9.94 988 9.94 982 9.94 975 9.94 969 9.94 962 9.94 956 9.94 949 9.94 943 9.94 936 9.94 930 9.94 923 9.94 917 9.94 911 9.94 904 9.94 898 9.94 891 9.94 885 9.94 878 9.94 871 9.94 865 9.94 858 9.91 852 9.94 845 9.94 839 9.94 832 9.94 826 9.94 819 9.94 813 9.94 806 9.94 799 9.94 793 9.94 786 9.94 780 9.94 773 9.94 767 9.94 760 9.94 753 9.94 747 9.94 740 9.94 734 9.94 727 9.94 720 9.94 714 9.94 707 9.94 700 9.94 694 9.94 687 9.94 680 9.94 674 9.94 667 9.94 660 9.94 654 9.94 647 9.94 640 9.94 634 9.^)4 627 9.94 620 9.94 614 9.94 607 9.94600 9.94 593 32 31 2 6.4 6.2 3 9.6 9.3 4 12.8 12.4 5 16.0 15.5 6 19.2 18.6 7 22.4 21.7 8 25.6 24.8 9 28.8 27.9 25 24 2 5.0 4.8 3 7.5 7.2 4 10.0 9.6 5 12.5 12.0 6 15.0 14.4 7 17.5 16.8 8 20.0 19.2 9 22.5 21.6 30 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 23 4.6 6.9 9.2 11.5 13.8 16.1 18.4 20.7 7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 1.2 1.8 2.4 3.0 3.6 4.2 4.8 6.4 From the top : For 27°+ or 207°+, read as printed; for 117°+ or 297°+, read co-function. From the bottom : For 62°+ or 242°+, read as printed; for 152°+ or 332°+, read co-function. L Cos LCtn cd L Tan L Sin Prop. Pts. 63°— Logarithms of Trigonometric Functions 74 38 — Logarithms of Trigonometric Functions pii ' LSin L Tan c d L Ctn LGos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.67161 9.67 185 9.67 208 9.67 232 9.67 256 9.67 280 9.67 303 9.67 327 9.67 350 9.67 374 9.67 398 9.67 421 9.67 445 9.67 468 9.67 492 9.67 515 9.67 539 9.67 562 9.67 586 9.67 609 9.67 633 9.67 656 9.67 680 9.67 703 9.67 726 9.67 750 9.67 773 9.67 796 9.67 820 9.67 843 9.67 866 9.67 890 9.67 913 9.67 936 9.67 959 9.67 982 9.68 006 9.68 029 9.68052 9.68 075 9.68098 9.68 121 9.68 144 9.68 167 9.68 190 9.68 213 9.68 237 9.68 260 9.68 283 9.68 305 9.68 328 9.68 351 9.68 374 9.68 397 9.68 420 9.68443 9.68466 9.68 489 9.68 512 9.68 534 9.68 557 9.72 567 9.72 598 9.72 628 9.72659 9.72689 9.72 720 9.72 750 9.72 780 9.72 811 9.72 841 9.72 872 9.72 902 9.72 932 9.72 963 9.72 993 9.73023 9.73 054 9.73084 9.73 114 9.73 144 9.73 175 9.73 205 9.73 235 9.73 265 9.73 295 9.73 326 9.73 356 9.73 386 9.73 416 9.73 446 9.73 476 9.73 507 9.73 537 9.73 567 9.73 597 9.73 627 9.73 657 9.73 687 9.73 717 9.73 747 9.73 777 9.73 807 9.73 837 9.73 867 9.73 897 9.73 927 9.73 957 9.73 987 9.74 017 9.74 047 9.74 077 9.74 107 9.74 137 9.74 166 9.74 196 9.74 226 9.74 256 9.74 286 9.74 316 9.74 345 9.74 375 0.27 433 0.27 402 0.27 372 0.27 341 0.27 311 0.27 280 0.27 250 0.27 220 0.27 189 0.27 159 0.27 128 0.27 098 0.27 068 0.27 037 0.27 007 0.26 977 0.26 946 0.26 916 0.26 886 0.26 856 0.26 825 0.26 795 0.26 765 0.26 735 0.26 705 0.26674 0.26 644 0.26 614 0.26 584 0.26 554 0.26 524 0.26 493 0.26 463 0.26433 0.26 403 0.26 373 0.26 343 0.26 313 0.26 283 0.26 253 0.26 223 0.26 193 0.26 163 0.26 133 0.26 103 0.26073 0.26 043 0.26 013 0.25 983 0.25 953 0.25 923 0.25 893 0.25 863 0.25 834 0.25 804 0.25 774 0.25 744 0.25 714 0.25 684 0.25 655 0.25 625 9.94 593 9.94 587 9.94 580 9.94 573 9.94 567 9.94 560 9.94 553 9.94 546 9.94 540 9.94 533 9.94 526 9.94 519 9.94 513 9.94 506 9.94499 9.94 492 9.94 485 9.94 479 9.94 472 9.94465 9.94 458 9.94 451 9.94 445 9.94438 9.94 431 9.94 424 9.94 417 9.94410 9.94 404 9.94 397 9.94 390 9.94 383 9.94 376 9.94 369 9.94 362 9.94 355 9.94 349 9.94 342 9.94 335 9.94328 9.94 321 9.94 314 9.94 307 9.94 300 9.94 293 9.94 286 9.94 279 9.94 273 9.94 266 9.94 259 9.94 252 9.94245 9.94 238 9.94 231 9.94 224 9.94 217 9.94210 9.94 203 9.94 196 9.94189 9.94 182 31 30 2 6.2 6.0 3 9.3 9.0 4 12.4 12.0 5 15.5 15.0 6 18.6 18.0 7 21.7 21.0 8 24.8 24.0 9 27.9 27.0 24 23 4.8 4.6 7.2 6.9 9.6 9.2 12.0 11.5 14.4 13.8 16.8 16.1 19.2 18.4 21.6 20.7 29 5.8 8.7 11.6 14.5 17.4 20.3 23.2 26.1 22 4.4 6.6 8.8 11.0 13.2 15.4 17.6 19.8 7 2 1.4 3 2.1 4 2.8 5 3.5 6 4.2 7 4.9 8 5.6 9 6.3 6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 From the top : For 28°+ or 208°+, read as printed; for 118°+ or 298°+, read co-function. From the bottom : For 61°+ or 241°+, read as printed; for 151°+ or 331°+, read co-function. LGos LCtn cd LTan L Sin Prop. Pts. 61° — Logarithms of Trigonometric Functions Ill] 29° — Logarithms of Trigonometric Functions 75 L Sin LTan c d L Ctn L Cos Prop. Pts. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3(] 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.68 557 9.68 580 9.68 603 9.68 625 9.68 648 9.68 671 9.68 694 9.68 716 9.68 739 9.68 762 9.68 784 9.68 807 9.68 829 9.68 852 9.68 875 9.68 897 9.68 920 9.68 942 9.68 965 9.68 987 9.69010 9.69 032 9.69 055 9.69 077 9.69 100 9.69 122 9.69 144 9.69 167 9.69 189 9.69 212 9.69 234 9.69 256 9.69 279 9.69 301 9.69 323 9.69 345 9.69 368 9.69 390 9.69 412 9.69 434 9.69456 9.69 479 9.69 501 9.69 523 9.69 545 9.69 567 9.69 589 9.69 611 9.69 633 9.69 655 9.69 677 9.69 699 9.69 721 9.69 743 9.69 765 9.69 787 9.69 809 9.69 831 9.69 853 9.69 875 9.69 897 9.74 375 9.74 405 9 74 435 9.74 465 9.74 494 9.74 524 9.74 554 9.74 583 9.74 613 9.74 643 9.74 673 9.74 702 9.74 732 9.74 762 9.74 791 9.74 821 9.74 851 9.74 880 9.74 910 9.74 939 9.74 969 9.74 998 9.75 028 9.75 058 9.75 087 9.75 117 9.75 14^ 9.75 176 9.75 205 9.75 235 9.75 264 9.75 294 9.75 323 9.75 353 9.75 382 9.75 411 9.75 441 9.75 470 9.75 500 9.75 529 9.75 558 9.75 588 9.75 617 9.75 647 9.75 676 9.75 705 9.75 735 9.75 764 9.75 793 9.75 822 9.75 852 9.75 881 9.75 910 9.75 939 9.75 969 9.75 998 9.76 027 9.76 056 9.76 086 9.76 115 9.76144 0.25 625 0.25 595 0.25 565 0.25 535 0.25 506 0.25 476 0.25 446 0.25 417 0.25 387 0.25 357 0.25 327 0.25 298 0.25 268 0.25 2;58 0.25 209 0.25 179 0.25 149 0.25 120 0.25 090 0.25 061 0.25 031 0.25 002 0.24 972 0.24 912 0.24 913 0.24 883 0.24 854 24 824 0.24 795 0.24 765 0.24 736 0.24 706 0.24 677 0.24 647 0.24 618 0.24 589 0.24 559 0.24 530 0.24 500 0.24 471 0.24 442 0.24 412 0.24 383 0.24 353 0.24 324 0.24 295 0.24 265 0.24 236 0.24 207 0.24 178 0.24 148 0.24 119 0.24 090 0.24(61 0.24 031 0.24 002 0.23 973 0.23 944 0.23 914 0.23 885 0.23 856 9.94 182 9.94 175 9.94 168 9.94 161 9.94 154 9.94 147 9.94 140 9.94133 9.94 126 9.94 119 9.94 112 9.94 105 9.94 098 9.94 090 9.94 083 9.94 076 9.94 069 9.94 062 9.94 055 9.94 048 9.94041 9.94 034 9.94 027 9.94 020 9.94 012 9.94 005 9.93 998 9.93 991 9.93 984 9.93 977 9.93 970 9.93 963 9.93 955 9.93 948 9.93 941 9.93 934 9.93 927 9.93 920 9.93 912 9.93 905 9.93 898 9.93 891 9.93 884 9.93 876 9.93 869 9.93 862 9.93 855 9.93 847 9.93 840 9.93 833 9.93 826 9.93 819 9.93 811 9.93 804 9.93 797 9 93 789 9.93 782 9.93 775 9.93 768 9.93 760 9.93 753 30 29 2 6.0 5.8 3 9.0 8.7 4 12.0 11.6 5 15.0 14.5 6 18 17.4 7 21.0 20.3 8 24.0 23.2 9 27.0 26.1 22 8 2 4.4 1.6 3 6.6 2.4 4 8.8 3.2 5 11.0 4.0 6 13.2 4.8 7 15.4 5.6 8 17.6 6.4 9 19.8 7.2 4.6 6.9 9.2 11.5 13.8 16.1 18.4 20.7 7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 From the top : For 29°+ or 209°+, read as printed; for 119°+ or 299°+, read co-function. From the bottom: For 60°+ or 240°+, read as printed ; for 150°+or 330°+, read co-function. L Cos L Ctn I c d LTan L Sin Prop. Pts. 60°— Logarithms of Trigonometric Functions 76 30° — Logarithms of Trigonometric Functions [in L Sin L Tan c d L Gtn L Cos Prop. PtB. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.69 897 9.69 919 9.69941 9.69 963 9.69 984 9.70 006 9.70028 9.70050 9.70 072 9.70 093 9.70 115 9.70 137 9.70 159 9.70 180 9.70 202 9.70 224 9.70 245 9.70 267 9.70 288 9.70 310 9.70 332 9.70 353 9.70 375 9.70 396 9.70 418 9.70439 9.70 461 9.70 482 9.70 504 9.70 525 9.70 547 9.70 568 9.70 590 9.70 611 9.70 633 9.70 654 9.70 675 9.70 697 9.70 718 9.70 739 9.70 761 9.70 782 9.70 803 9.70 824 9.70 846 9.70 867 9.70 888 9.70 909 9.70 931 9.70 952 9.70 973 9.70 994 9.71 015 9.71 036 9.71 058 9.71079 9.71 100 9.71 121 9.71 142 9.71 163 9.71 184 9.76 144 9.76 173 9.76 202 9.76 231 9.76 261 9.76 290 9.76 319 9.76 348 9.76 377 9.76 406 9.76 435 9.76 464 9.76 493 9.76 522 9.76 551 9.76 580 9.76 609 9.76 639 9.76 668 9.76 697 9.76 725 9.76 754 9.76 783 9.76 812 9.76 841 9.76 870 9.76 89^) 9.76 928 9.76 957 9.76 986 9.77 015 9.77 044 9.77 073 9.77 101 9.77 130 9.77 159 9.77 188 9.77 217 9.77 246 9.77 274 9.77 303 9.77 3.32 9.77 361 9.77 390 9.77 418 9.77 447 9.77 476 9.77 505 9.77 533 9.77 562 9.77 591 9.77 619 9.77 648 9.77 677 9.77 706 9.77 734 9.77 763 9.77 791 9.77 820 9.77 849 9.77 877 0.23 856 0.23 827 0.23 798 0.23 769 0.23 739 0.23 710 0.23 681 0.23 652 0.23 623 0.23 594 0.23 565 9.23 536 0.23 507 0.23 478 0.23 449 0.23420 0.23 391 0.23 361 0.23 332 0.23 303 0.23 275 0.23 246 0.23 217 0.23 188 0.23 159 0.23 130 0.23 101 0.23 072 0.23 043 0.23 014 0.22 985 0.22 956 0.22 927 0.22 899 0.22 870 0.22 841 0.22 812 0.22 783 0.22 754 0.22 726 0.22 697 0.22 668 0.22 639 0.22 610 0.22 582 0.22 553 0.22 524 0.22 495 0.22 467 0.22 438 0.22 409 0.22 381 0.22 352 0.22 323 0.22 294 0.22 266 0.22 237 0.22 209 0.22 180 0.22 151 0.22 123 9.93 753 9.93 746 9.93 738 9.93 731 9.93 724 9.93 717 9.93 709 9.93 702 9.93 695 9.93 687 9.93 680 9.93 673 9.93 665 9.93 658 9.93 650 9.93 643 9.93 636 9.93 628 9.93 621 9.93 614 9.93 606 9.93 599 9.93 591 9.93 584 9.93 577 9.93 569 9.93 562 9.93 554 9.93 547 9.93 539 9.93 532 9.93 525 9.93 517 9.93 510 9.93 502 9.93 495 9.93 487 9.93480 9.93 472 9.93 465 9.93 457 9.93 450 9.93 442 9.93 435 9.93 427 9.93 420 9.93 412 9.93 405 9.93 397 9.93 390 9.93 382 9.93 375 9.93 367 9.93 360 9.93 352 9.93 344 9.93 337 9.93 329 9.93 322 9.93 314 9.93 307 30 29 2 6.0 5.8 3 9.0 8.7 4 12.0 11.6 5 15.0 14.5 6 18.0 17.4 7 21.0 20.3 8 24.0 23.2 9 27.0 26.1 28 6.6 8.4 11.2 14.0 16.8 19.6 22.4 25.2 22 2 4.4 3 6.6 4 8.8 5 11.0 6 13.2 7 15.4 8 17.6 9 19.8 8 2 1.6 3 2.4 4 3.2 5 4.0 6 4.8 7 5.6 8 6.4 9 7.2 21 4.2 6.3 8.4 10.5 12.6 14.7 16.8 18.9 7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 From the top : For 30°+ or 210°+, read as printed ; for 120°+ or 300°+, read co-function. From the bottom : For 59°+ or 239°+, read as printed; for 149°+ or 329°+, read co-function. L Cos L Ctn c d L Tan L Sin d I Prop. Pts. 59° — Losraritlims of Trigonometric Functions Ill] 31° — Logarithms of Trigonometric Functions 77 L Sin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 .57 58 59 60 9.71 184 9.71 205 9.71 226 9.71 247 9.71 268 9.71 289 9.71 310 9.71 331 9.71 352 9.71 373 9.71 393 9.71 414 9.71 435 9.71 456 9.71 477 9.71498 9.71 519 9.71 539 9.71 560 9.71 581 9.71 602 9.71 622 9.71 643 9.71 664 9.71 685 9.71 705 9.71 726 9.71 747 9.71 767 9.71 788 9.71 809 9.71 829 9.71 850 9.71 870 9.71 891 9.71 911 9.71 932 9.71 952 9.71 973 9.71 994 9.72 014 9.72 03i 9.72 055 9.72 075 9.72 096 9.72 116 9.72 137 9.72 157 9.72 177 9.72 198 9.72 218 9.72 238 9.72 259 9.72 279 9.72 299 9.72 320 9.72 340 9.72 360 9.72 381 9.72 401 9.72 421 9.77 877 9.77 906 9.77 935 9.77 9(53 9.77 992 9.78 020 9.78 049 9.78 077 9.78 106 9.78 135 9.78 163 9.78 192 9.78 220 9.78 249 9.78 277 9.78 306 9.78 334 9.78 363 9.78 391 9.78 419 9.78 448 9.78 476 9.78 505 9.78533 9.78 uG2 9.78 590 9.78 618 9.78 647 9.78 675 9.78 704 9.78 732 9.78 760 9.78 789 9.78 817 9.78 845 9.78 874 9.78 902 9.78 930 9.78 959 9.78 987 9.79015 9.79043 9.79 072 9.79 100 9.79 128 9.79 156 9.79 185 9.79 213 9.79 241 9.79 269 9.79 2i)7 9.79 326 9.79 354 9.79 382 9.79 410 9.79 438 9.79 466 9.79 495 9.79 523 9.79551 9.79 579 0.22 123 0.22 094 0.22 065 0.22 037 0.22 008 0.21 980 0.21951 0.21 923 0.21 894 0.21 865 0.21 837 0.21 808 0.21 780 0.21 751 0.21 723 0.21694 0.21 666 0.21 637 0.21 609 0.21 581 0.21 552 0.21 524 0.21 495 0.21 467 0.21 438 0.21 410 0.21 382 0.21 353 0.21 325 0.21 296 0.21 268 0.21 240 0.21 311 0.21 383 0.21 155 0.21 126 0.21 098 0.21 070 0.21 041 0.21 013 0.20 985 0.20 957 20 928 0.20 900 0.20 872 0.20 844 0.20 815 0.20 787 0.20 759 0.20 731 0.20 703 0.20 674 0.20 646 0.20618 0.20590 0.20 562 0.20 534 0.20505 0.20 477 0.20 449 0.20 421 9.93 307 9.93 299 9.93 291 9.93 284 9.93 276 9.93 269 9.93 261 9.93 253 9.93 246 9.93 238 9.93230 9.93 223 9.93 215 9.93 207 9.93 200 9.93 192 9.93 184 9.93 177 9.93 169 9.93 161 9.93154 9.93 146 9.93 138 9.93 131 9.93 123 9.93 115 9.93 108 9.93 100 9.93 092 9.93084 9.93077 9.93 069 9.93 061 9.93 053 9.93 046 9.93 038 9.93030 9.93 022 9.93 014 9.93 007 9.92 999 9.92 991 9.92 983 9.92 976 9.92 968 9.92 960 Q.92 952 9,92 944 9.92 936 9.92 929 9.92 921 9.92 913 9.92 905 9.92 897 9.92 889 9.92 881 9.92 874 9.92 866 9.92 858 9.92 850 9.92 842 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 29 28 2 5.8 5.6 3 8.7 8.4 4 11.6 11.2 5 14.5 14.0 6 17.4 16.8 7 20.3 19.6 8 23.2 22.4 9 26.1 25.2 20 8 4.0 1.6 6.0 2.4 8.0 3.2 10.0 4.0 12.0 4.8 14.0 5.6 16.0 6.4 18.0 7.2 21 4.2 6.3 8.4 10.5 1^.6 14.7 16.8 18.9 7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 From the top : For 31°+ or 211°+, read as printed; for 121°+ or 301°+, read co-f unctioD . From the bottom : For 58°+ or 238°+, read as printed; for 148°+ or 328°+, read co-function. LGos L Ctn c d L Tan L Sin d ' Prop. Pts. 58°— Logarithms of Trigonometric Functions 78 32° — Logarithms of Trigonometric Functions [in L Sin L Tan c d L Ctn LCos Prop. Pts. 10 11 12 13 14 16 16 17 IH 19 20 21 22 23 24 25 2() 27 28 29 30 31 32 33 34 35 3i) 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.72 421 9.72 441 9.72 4()1 9.72 482 9.72 502 9.72 522 9.72 542 9.72 562 9.72 582 9.72 602 9.72 622 9.72 ()43 9.72 663 9.72 683 9.72 703 9.72 723 9.72 743 9.72 763 9.72 783 9.72 803 9.72 823 9.72 843 9.72 863 9 72 883 9.72 902 9.72 922 9 72 942 9.72 962 9.72 982 9.73 002 9 73 022 9.73 041 9.73 061 9.73 081 9.73 101 9.73 121 9.73 140 9.73 160 9.73 180 9.73 200 9.73 219 9.73 239 9.73 259 9.73 278 9.73 298 9.73 318 9.73 337 9.73 357 9.73 377 9.73 396 9.73 416 9.73435 9.73455 9.73 474 9.73 494 9.73513 9.73 533 9.73 552 9.73 572 9.73 591 9.73 611 9.79 9.79 9.79 9 79 9.79 9.79 9.79 9.79 9.79 9.79 9.79 9.79 9.79 9.79 9.79 9.80 9.80 9.80 9.8") 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.80 9.81 9.81 9.81 9.81 9.81 9.81 9.81 9.81 9.81 9.81 579 607 (>35 663 691 719 747 776 804 832 860 888 916 944 972 000 028 056 084 112 140 168 195 223 251 279 307 335 3(53 391 419 447 474 502 530 558 586 614 642 669 697 725 753 781 808 836 864 892 919 947 975 003 030 058 086 113 141 169 196 224 252 0.20 421 0.20 393 0.20 365 0.20 337 0.20 309 0.20 281 0.20 253 0.20 224 0.20 196 0.20 168 0.20 140 0.20112 0.20084 0.20 056 0.20 028 0.20 000 0.19 972 0.19 944 0.19 916 0.19 888 0.19 860 0.19 832 0.19 805 0.19 777 0.19 749 0.19 721 0.19 693 0.19 665 0.19 637 0.19 609 0.19 581 0.19 553 0.19 52() 0.19 498 0.19 470 0.19 442 0.19 414 0.19 386 0.19 358 0.19 331 0.19 303 0.19 275 0.19 247 0.19 219 0.19 192 0.19164 0.19136 0.19 108 0.19081 0.19 053 0.19 025 0.18 997 0.18 970 0.18 942 0.18 914 0.18 887 0.18 859 0.18 831 0.18 804 0.18 776 0.18 748 9.92 842 9.92 834 9.92 826 9.92 818 9.92 810 9.92 803 9.92 795 9.92 787 9.92 779 9.92 771 9.92 763 9.92 755 9.92 747 9.92 739 9.92 731 9.92 723 9.92 715 9.92 707 9.92 699 9.92 691 9.92 683 9.92 675 9.92 667 9.92 659 9.92 651 9.92 643 9.92 635 9.92 627 9.92 619 9.92 611 9.92 603 9.92 595 9.92 587 9.92 579 9.92 571 9.92 563 9.92 555 9.92 546 9.92 538 9.92 530 9.92 522 9.92 514 9.92 506 9.92 498 9.92 490 9.92 482 9.92 473 9.92 465 9.92 457 9.92 449 9.92 441 9.92 433 9.92 425 9.92 416 9.92 408 9.92 400 9.92 392 9.92 384 9.92 376 9.92 367 9.92 359 29 28 2 5.8 5.6 3 8.7 8.4 4 11.6 11.2 5 14.5 14.0 6 17.4 16.8 7 20.3 19.6 8 23.2 22.4 9 26.1 25.2 21 20 2 4.2 4.0 3 6.3 6.0 4 8.4 8.0 5 10.5 10.0 6 12.6 12.0 7 14.7 14.0 8 16.8 16.0 9 18.9 18.0 27 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 19 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 Frotn the top : For 32°+ or 212°+, read as printed; for 122°+ or 302°+, read co-function. Froyn the "bottom : For 57°+ or 237°+, read as printed; for 147°+ or 327°+, read co-function. 9 8 2 1.8 1.6 3 2.7 2.4 4 3.6 3.2 5 4.5 4.0 6 5.4 4.8 7 6.3 5.6 8 7.2 6.4 9 8.1 7.2 LCos L Ctn c d L Tan L Sin Id Prop. Pts. 57° — Logarithms of Trigonometric Functions 33° — Logarithms of Trigonometric Functions 79 L Sin L Tan , c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1() 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3(3 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 '57 58 59 60 9.73 611 9.73630 9.73650 9.73 669 9.73 689 9.73 708 9.73 727 9.73 747 9.73 766 9.73 785 9.73 805 9.73 824 9.73 843 9.73 863 9.73 882 9.73 901 9.73 921 9.73 940 9.73 959 9.73 978 9.73 997 9.74 017 9.74036 9.74 055 9.74074 9.74093 9.74 113 9.74 132 9.74 151 9.74 170 9.74 189 9.74 208 9.74 227 9.74246 9.74265 9.74 284 9.74 303 9.74 322 9.74 341 9.74 360 9.74 379 9.74 398 9.74 417 9.74 436 9.74455 9.74 474 9.74 493 9.74 512 9.74 531 9.74 549 9.74568 9.74 587 9.74 606 9.74 625 9.74 644 9.74 662 9.74 681 9.74 700 9.74 719 9.74 737 9.74 756 9.81 252 9.81 279 9.81 307 9.81 335 9.81 362 9.81 390 9.81 418 9.81 445 9.81 473 9.81 500 9.81528 9.81 556 9.81 583 9.81611 9.81 638 9.81 666 9.81 693 9.81 721 9.81 748 9.81776 9.81 803 9.81 831 9.81 858 9.81 886 9.81 913 9.81 941 9.81968 9.81 996 9.82 023 9.82 051 9.82 078 9.82 106 9.82 133 9.82 161 9.82 188 9.82 215 9.82 243 9.82 270 9.82 298 9.82 325 9.82 352 9.82 380 9.82 407 9.82 435 9.82 462 9.82 489 9.82 517 9.82 544 9.82 571 9.82 599 9.82 626 9.82 653 9.82 681 9.82 708 9.82735 9.82 762 9.82 790 9.82 817 9.82 844 9.82 871 9.82 899 0.18 748 0.18 721 0.18 693 0.18 665 0.18 638 0.18 610 0.18 582 0.18555 0.18 527 0.18 500 0.18 472 0.18 444 0.18417 0.18 389 0.18 362 0.18 334 0.18 307 0.18 279 0.18 252 0.18 224 0.18 197 0.18 169 0.18 142 0.18 114 0.18 087 0.18 059 0.18 032 0.18 004 0.17 977 0.17 949 0.17 922 0.17 894 0.17 867 0.17 839 0.17 812 0.17 785 0.17 757 0.17 730 0.17 702 0.17 675 0.17648 0.17 620 0.17 593 0.17 565 0.17538 0.17511 0.17483 0.17 456 0.17 429 0.17 401 0.17 374 0.17 347 0.17 319 0.17 292 0.17 265 0.17 238 0.17 210 0.17 183 0.17 156 0.17 129 0.17 101 9.92 359 9.92 351 9.92 343 9.92 335 9.92 326 9.92 318 9.92 310 9.92 302 9.92 293 9.92 285 9.92 277 9.92 269 9.92 260 9.92 252 9.92 244 9.92 235 9.92 227 9.92 219 9.92 211 9.92 202 9.92 194 9.92 186 9.92 177 9.i;2 169 9.92 161 9.92 152 9.92 144 9.92 136 9.92 127 9.92 119 9.92 111 9.92 102 9.92 094 9.92 086 9.92 077 9.92 069 9.92 060 9.92 052 9.92 044 9.92 035 9.92 027 9.92 018 9.92 010 9.92002 9.91 993 9.91 985 9.91 976 9.91 968 9.91 959 9.91 951 9.91942 9.91 934 9.91 925 9.91 917 9.91908 9.91900 9.91 891 9.91 883 9.91 874 9.91 866 9.91 857 28 27 2 5.6 5.4 3 8.4 8.1 4 11.2 10.8 5 14.0 13.5 6 16 8 16.2 7 19.6 18.9 8 22.4 21.6 9 25.2 24.3 20 40 6.0 8.0 10.0 12.0 14.0 16.0 18.0 19 2 3.8 3 5.7 4 7.6 5 9.5 6 11.4 7 13.3 8 15.2 9 17.1 9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 18 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 From the top : For 33°+ or 213°+, read as printed; for 123°+ or 303°+, read co-function. From the bottom: For 66°+ or 236°+, read as printed; for 146°+ or 326°+, read co-function. L Cos LCtn c d L Tan L Sin | d ' Prop. Pta. 56°— Logarithms of Trigonometric Functions 80 34° — Logarithms of Trigonometric Functions [in ' LSin LTan c d L Gtn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2$ 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.74 756 9.74 775 9.74 794 9.74 812 9.74 831 9.74 850 9.74 868 9.74 887 9.74 906 9.74 924 9.74 943 9.74 961 9.74 980 9.74 999 9.75 017 9.75 036 9.75 054 9.75 073 9.75 091 9.75 110 9.75 128 9.75 147 9.75 165 9.75 184 9.75 202 9.75 221 9.75 239 9.75 258 9.75 276 9.75 294 9.75 313 9.75 331 9.75 350 9.75 368 9.75 386 9.75 405 9.75 423 9.75 441 9.75 459 9.75 478 9.75 496 9.75 514 9.75 533 9.75 551 9.75 569 9.75 587 9.75 605 9.75 624 9.75 642 9.75 660 9.75 678 9.75 696 9.75 714 9.75 733 9.75 751 9.75 769 9.75 787 9.75 805 9.75 823 9.75 841 9.75 859 9.82 899 9.82 926 9.82 953 9.82 980 9.83008 9.83 035 9.83 062 9.83 089 9.83 117 9.83 144 9.83 171 9.83 198 9.83 225 9.83 252 9.83 280 9.83 307 9.83 334 9.83 361 9.83 388 9.83 415 9.83442 9.83 470 9.83 497 9.83 524 9.83 551 9.83 578 9.83 605 9.83 632 9.83 659 9.83 686 9.83 713 9.83 740 9.83 768 9.83 795 9.83 822 9.83 849 9.83 876 9.83 903 9.83 930 9.83 957 9.83 984 9.84 011 9.84 038 9.84 065 9.84092 9.84 119 9.84 146 9.84 173 9.84 200 9.84 227 9.84 254 9.84 280 9.84 307 9.84 334 9.84 361 9.84 388 9.84415 9.84 442 9.84 469 9.84 496 9.84 523 0.17 101 0.17 074 0.17 047 0.17 020 0.16 992 0.16 <)65 0.16 938 0.16 911 0.16 883 0.16 856 0.16 829 0.16 802 0.16 775 0.16 748 0.16 720 0.16 693 0.16 666 0.16 639 0.16 612 0.16 585 0.16 558 0.16 5:30 0.16 503 0.16 476 0.16 449 0.16 422 0.16 395 0.16 368 0.16 341 0.16 314 0.16 287 0.16 260 0.16 232 0.16 205 0.16 178 0.16 151 0.16 124 0.16 097 0.16 070 0.16 043 0.16 016 0.15 989 0.15 962 0.15 935 0.15 908 0.15 881 0.15 854 0.15 827 0.15 800 0.15 773 0.15 746 0.15 720 0.15 693 0.15 666 0.15 639 0.15 612 0.15 585 0.15 558 0.15 531 0.15 504 0.15 477 9.91 857 9.91 849 9.91 840 9.91 832 9.91 823 9.91 815 9.91 806 9.91 798 9.91 789 9.91 781 9.91 772 9.91 763 9.91 755 9.91 746 9.91 738 9.91 729 9.91 720 9.91 712 9.91 703 9.91 695 9.91 686 9.91 677 9.91 669 9.91 660 9.91 651 9.91 643 9.91 634 9.91 625 9.91 617 9.91 608 9.91 599 9.91 591 9.91 582 9.91 573 9.91 565 9.91 556 9.91 547 9.91 538 9.91 530 9.91 521 9.91 512 9.91 504 9.91 495 9.91 486 9.91 477 9.91 469 9.91 460 9.91 451 9.91 442 9.91 433 9.91 425 9.91 416 9.91 407 9.91 398 9.91 389 9.91 381 9.91 372 9.91 363 9.91 354 9.91 345 9.91 336 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 28 27: 2 5.6 5.4 3 8.4 8.1 4 11.2 10.8 5 14.0 13.5 6 16.8 16.2 7 19.6 18.9 8 22.4 21.6 9 25.2 24.3 26 5.2 7.8 10.4 13.0 15.6 18.2 20.8 23.4 19 18 2 3.8 3.6 3 5.7 5.4 4 7.6 7.2 5 9.5 9.0 6 11.4 10.8 7 13.3 12.6 8 15.2 14.4 9 17.1 16.2 9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 From the top : For 34°+ or 214°+, read as printed ; for 124°+ or 304°+, read co-function. From the bottom : For 55°+ or 235°+, read as printed; for 145°+ or 325°+, read co-function. L Cos L Gtn c d L Tan L Sin d ' Prop. Pts. 65° — Lofirarithms of Trisronomfttric Functions 35° — logarithms of Trigonometric Functions 81 L Sin L Tan c d L Gtn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 P7 58 59 60 9.75 859 9.75 877 9.75 895 9.75 913 9.75 931 9.75 949 9.75 967 9.75 985 9.76 003 9.76 021 9.76 039 9.76 057 9.76 075 9.76 093 9.76 111 9.76 129 9.76 146 9.76 164 9.76 182 9.76 200 9.76 218 9.76 236 9.76 253 9.76 271 9.76 289 9.76 307 9.76 324 9.76 342 9.76 360 9.76 378 9.76 395 9.76 413 9.76 431 9.76 448 9.76 466 9.76 484 9.76 501 9.76 519 9.76 537 9.76 554 9.76 572 9.76 590 9.76 607 9.76 625 9.76 642 9.76 660 9.76 677 9.76 695 9.76 712 9.76 730 9.76 747 9.76 765 9.76 782 9.76 800 9.76 817 9.76 835 9.76 852 9.76 870 9.76 887 9.76 904 9.76 922 9.84 523 9.84 550 9.84 576 9.84 603 9.84 630 9.84657 9.84 684 9.84 711 9.84 738 9.84 764 9.84 791 9.84 818 9.84 845 9.84 872 9.84 899 9.84 925 9.84 952 9.84 979 9.85 006 9.85 033 9.85 059 9.85 086 9.85 113 9.85 140 9.85 166 9.85 193 9.85 220 9.85 247 9.85 273 9.85 300 9.85 327 9.85 354 9.85 380 9.85 407 9.85 434 9.85 460 9.85 487 9.85 514 9.85 540 9.85 567 9.85 594 9.85 620 9.85 647 9.85 674 9.85 700 9.85 727 9.85 754 9.85 780 9.85 807 9.85 834 9.85 860 9.85 887 9.85 913 9.85 940 9.85 967 9.85 993 9.86 020 9.86 046 9.86 073 9.86 100 9.86 136 0.15 477 0.15 450 0.15 424 0.15 397 0.15 370 0.15 343 0.15 316 0.15 289 0.15 262 0.15 236 0.15 209 0.15 182 0.15 155 0.15 128 0.15 101 0.15 075 0.15 048 0.15 021 0.14 994 0.14 967 0.14 941 0.14 914 0.14 887 0.14 8(K) 0.14 834 0.14807 0.14 780 0.14 753 0.14 727 0.14 700 0.14 673 0.14 646 0.14 620 0.14 593 0.14 566 0.14 540 0.14 513 0.14 486 0.14 460 0.14 433 0.14 406 0.14 380 0.14 353 0.14 326 0.14 300 0.14 273 0.14 246 0.14 220 0.14 193 0.14 166 0.14 140 0.14 113 0.14 087 0.14 060 0.14 033 0.14 007 0.13 980 0.13 954 0.13 927 0.13 900 0.13 874 9.91 336 9.91 328 9.91 319 9.91 310 9.91 301 9.91 292 9.91 283 9.91 274 9.91 266 9.91 257 9.91 248 9.91 239 9.91 230 9.91 221 9.91 212 9.91 203 9.91 194 9.91 185 9.91 176 9.91 167 9.91 158 9.91 149 9.91 141 9.91 132 9.91 123 9.91 114 9.91 105 9.91 096 9.91 087 9.91 078 9.91 069 9.91 060 9.91 051 9.91 042 9.91 033 9.91 023 9.91 014 9.91 005 9.90 996 9.90 987 9.90 978 9.^)0 969 9.f)0 960 9.90 951 9.90 942 9.90 933 9.90 924 9.90 915 9.90 906 9.90 896 9.90 887 9.90 878 9.90 869 9.90 860 9.90 851 9.90 842 9.90 832 9.()0 823 9.90 814 9.90 805 9.90 796 27 26 2 5.4 5.2 3 8.1 7.8 4 10.8 10.4 5 13.5 13.0 6 16.2 15.6 7 18.9 18.2 8 21.6 20.8 9 24.3 23.4 18 3.6 6.4 7.2 9.0 10.8 12.6 14.4 16.2 17 2 3.4 3 5.1 4 6.8 5 8.5 6 10.2 7 11.9 8 13.6 9 15.3 9 2 1.8 3 2.7 4 3.6 5 4.5 6 5.4 7 6.3 8 7.2 9 8.1 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 From the top : For 35°+ or 215°+, read as printed ; for 125°+ or 305°+, read co-function. From the bottom : For 54°+ or 234°+, read as printed ; for 144°+ or 324°+, read co-function. L Goa LCtn c d L Tan L Sin Prop. Pts. 54°— Logarithms of Trigonometric Functions 82 36° — Logarithms of Trigonometric Functions [in L Sin d L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.76 922 9.76 939 9.76 9r)7 9.76 974 9.76 991 9.77 009 9.77 026 9.77 043 9.77 061 9.77 078 9.77 095 9.77 112 9.77 130 9.77 147 9.77 164 9.77 181 9.77 199 9.77 216 9.77 233 9.77 250 9.77 268 9.77 285 9.77 302 9.77 319 9.77 336 9.77 353 9.77 370 9.77 387 9.77 405 9.77 422 9.77 439 9.77 456 9.77 473 9.77 490 9.77 507 9.77 524 9.77 541 9.77 558 9.77 575 9.77 592 9.77 609 9.77 626 9.77 643 9.77 660 9.77 677 9.77 694 9.77 711 9.77 728 9.77 744 9.77 761 9.77 778 9.77 795 9.77 812 9.77 829 9.77 846 9.77 862 9.77 879 9.77 896 9.77 913 9.77 930 9.77 946 9.86 126 9.86 153 9.86 179 9.86 206 9.86 232 9.86 259 9.86 285 9.86 312 9.86 338 9.86 365 9.86 392 9.86 418 9.86 445 9.86 471 9.86 498 9.86 524 9.86 551 9.86 577 9.86 603 9.86 630 9.86 656 9.86 683 9.86 709 9.86 736 9.86 762 9.86 789 9.86 815 9.86 842 9.86 868 9.86 894 9.86 921 9.86 947 9.86 974 9.87 000 9.87 027 9.87 053 9.87 079 9.87 106 9.87 132 9.87 158 9.87 185 9.87 211 9.87 238 9.87 264 9.87 290 9.87 317 9.87 343 9.87 369 9.87 396 9.87 422 9.87 448 9.87 475 9.87 501 9.87 527 9.87 554 9.87 580 9.87 606 9.87 633 9.87 659 9.87 685 9.87 711 27 26 27 26 27 26 27 26 27 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 27 26 27 26 26 27 26 27 26 27 26 26 27 26 26 27 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 27 26 26 26 0.13 874 0.13 847 0.13 821 0.13 794 0.13 768 0.13 741 0.13 715 0.13 688 0.13 662 0.13 635 0.13 608 0.13 582 0.13 555 0.13 529 0.13 502 0.13 476 0.13 449 0.13 423 0.13 397 0.13 370 0.13 344 0.13 317 0.13 291 0.13 264 0.13 238 0.13 211 0.13 185 0.13 158 0.13 132 0.13106 0.13 079 0.13 053 0.13 026 0.13 000 0.12 973 0.12 947 0.12 921 0.12 894 0.12 868 0.12 842 0.12 815 0.12 789 0.12 762 0.12 736 0.12 710 0.12 683 0.12 657 0.12 631 0.12 604 0.12 578 0.12 552 0.12 525 0.12 499 0.12 473 0.12 446 0.12 420 0.12 394 0.12 367 0.12 341 0.12 315 0.12 289 9.90 796 9.90 787 9.90 777 9.90 768 9.90 759 9.90 750 9.90 741 9.90 731 9.90 722 9.90 713 9.90 704 9.90 694 9.90 685 9.90 676 9.90 667 9.90 657 9.90 648 9.90 639 9.90 630 9.90 620 9.90 611 9.<)0 602 9.90 592 9.90 583 9.90 574 9.90 565 9.90 555 9.90 546 9.90 537 9.90 527 9.90 518 9.90 509 9.90 499 9.90 490 9.90 480 9.90 471 9.90 462 9.90 452 9.90 443 9.90434 9.90 424 9.90 415 9.90 405 9.90 396 9.90 386 9.90 377 9.90 368 9.90 358 9.90349 9.90 339 9.90 330 9.90320 9.90 311 9.90 301 9.90 292 9.90 282 9.90 273 9.90 263 9.90 254 9.90 244 9.90 235 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 5.4 5.2 8.1 7.8 10.8 10.4 13.5 13.0 16.2 15.6 18.9 18.2 21.6 20.8 24.3 23.4 18 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 17 2 3.4 3 5.1 4 6.8 5 8.5 6 10.2 7 11.9 8 13.6 9 15.3 10 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8.0 9 9.0 16 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 From the top : For 36°+ or 216°+, read as printed ; for 126°+ or 306°+, read co-function. From the bottom : For 53°+ or 233°+, read as printed; for 143°+ or 323°+, read co-function. LGos L Ctn I c d L Tan L Sin d ' Prop. Pts. 63°— Logarithms of Trigonometric Functions Ill] 37° — Logarithms of Trigonometric Functions 83 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 m 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 ■57 58 59 60 L Sin 9.77 946 9.77 963 9.77 980 9.77 997 9.78 013 9.78 030 9.78 047 9.78 063 9.78 080 9.78 097 9.78 113 9.78 130 9.78 147 9.78 163 9.78 180 9.78 197 9.78213 9.78 230 9.78 246 9.78 263 9.78 280 9.78 296 9.78 313 9.78 329 9.78 346 9.78 362 9.78 379 9.78 395 9.78 412 9.78 428 9.78 445 9.78 461 9.78 478 9.78 494 9.78 510 9.78 527 9.78 543 9.78 560 9.78 576 9.78 592 9.78 609 9.78 625 9.78 642 9.78 658 9.78 674 9.78 691 9.78 707 9.78 723 9.78 739 9.78 756 78 772 78 788 78 805 78 821 78 837 78 853 78 869 78 886 78 902 78 918 78 934 L Tan c d L Ctn 9.87 711 9.87 738 9.87 764 9.87 790 9.87 817 9.87 843 9.87 869 9.87 895 9.87 922 9.87 948 9.87 974 9.88 000 9.88 027 9.88 053 9.88 079 9.88 105 9.88 131 9.88 158 9.88 184 9.88 210 9.88 236 9.88 262 9.88 289 9.88 315 9.88 341 9.88 367 9.88 393 9.88 420 9.88 446 9.88 472 9.88 498 9.88 524 9.88 550 9.88 577 9.88 603 9.88 629 9.88 655 9.88 681 9.88 707 9.88 733 9.88 759 9.88 786 9.88 812 9.88 838 9.88 864 9.88 890 9.88 916 9.88 942 9.88 968 9.88 994 9.89 020 9.89 046 9.89 073 9.89099 9.89 125 9.89 151 9.89 177 9.89 203 9.89 229 9.89 255 9.89 281 0.12 289 0.12 262 0.12 236 0.12 210 0.12 183 0.12 157 0.12 131 0.12105 0.12 078 0.12 052 0.12 026 0.12 000 0.11 973 0.11947 0.11 921 0.11 895 0.11 869 0.11 842 0.11 816 0.11 790 0.11 764 0.11 738 0.11711 0.11685 0.11659 0.11 633 0.11 607 0.11 580 0.11 554 0.11 528 0.11 502 0.11 476 0.11 450 0.11 423 0.11 397 0.11 371 0.11 345 0.11 319 0.11 293 0.11 267 0.11241 0.11214 0.11 188 0.11 162 0.11 136 0.11 110 0.11 084 0.11 058 0.11 032 0.11 006 0.10 980 0.10 954 0.10 927 0.10 901 0.10 875 0.10 849 0.10 823 0.10 797 0.10 771 0.10 745 0.10 719 L Cos 9.90 235 9.90 225 9.90 216 9.W 206 9.90 197 9.90 187 9.90 178 9.90 168 9.90 159 9.90 149 9.90 139 9.90130 9.90 120 9.^)0 111 9.90101 9.90 091 9.90 082 9.90 072 9.90 063 9.90 053 9.90 043 9.90 034 9i)0 024 9.90 014 9.90 005 9.89 995 9.89 985 9.89 976 9.89 966 9.89 956 9.89 947 9.89 937 9.89 927 9.89 918 9.89 908 9.89 898 9.89 888 9.89 879 9.89 869 9.89 859 9.89 849 9.89 840 9.89 830 9.89 820 9.89 810 9.89 801 9.89 791 9.89 781 9.89 771 9.89 761 9.89 752 9.89 742 9.89 732 9.89 722 9.89 712 9.89 702 9.89 693 9.89 683 9.89 673 9.89 663 9.89653 I Prop. Pts. 27 26 2 5.4 5.2 3 8.1 7.8 4 10.8 10.4 5 13.5 13.0 6 16.2 15.6 7 18.9 18.2 8 21.6 20.8 9 24.3 23.4 17 3.4 6.1 6.8 8.5 10.2 11.9 13.6 15.3 9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 From the top : For37°+or217°+, read as printed; for 127°+ or 307°+, read co-function. From the bottom : For 52°+ or 232°+, read as printed ; for 142°+ or 322°+, read co-function. 16 10 2 3.2 2.0 3 4.8 3.0 4 6.4 4.0 6 8.0 5.0 6 9.6 6.0 7 11.2 7.0 8 12.8 8.0 9 14.4 9.0 LGos L Ctn c d LTan L Sin Prop. Pts. 52°— Logarithms of Trigonometric Functions 84 38° — Logarithms of Trigonometric Functions [in L Sin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.78 934 9.78 950 9.78 967 9.78 983 9.78 999 9.79 015 9.79 031 9.79 047 9.79063 9.79 079 9.79095 9.79 111 9.79 128 9.79 144 9.79 160 9.79 176 9.79192 9.79 208 9.79 224 9.79 240 9.79 256 9.79272 9.79 288 9.79 304 9.79 319 9.79 335 9.79 351 9.79 367 9.79 383 9.79 399 9.79415 9.79431 9.79 447 9.79 463 9.79478 9.79494 9.79 510 9.79526 9.79 542 9.79 558 9.79 573 9.79589 9.79605 9.79 621 9.79 636 9.79652 9.79668 9.79 684 9.79 699 9.79 715 9.79 731 9.79 746 9.79 762 9.79 778 9.79 793 9.79 809 9.79 825 9.79 840 9.79 856 9.79 872 9.79 887 9.89 281 9.89 307 9.89 333 9.89 359 9.89 385 9.89 411 9.89 437 9.89463 9.89 489 9.89 515 9.89 541 9.89 567 9.89 593 9.89 619 9.89645 9.89 671 9.89 697 9.89 723 9.89 749 9.89 775 9.89 801 9.89 827 9.89 853 9.89 879 9.89 905 9.89 931 9.89 957 9.89 983 9.90009 9.90 035 9.90 061 9.90 086 9.90112 9.90 138 9.90 164 9.90190 9.90 216 9.90 242 9.90 268 9.90 294 9.90 320 9.90 346 9.90 371 9.90 397 9.90 423 9.90 449 9.90 475 9.90 501 9.90 527 9.90 553 9.90 578 9.90 604 9.90 630 9.90 656 9.90682 9.90 708 9m 734 9.90 759 9.90 785 9.90 811 9.90 837 0.10 719 0.10 693 0.10 667 0.10 641 0.10 615 0.10 589 0.10 563 0.10 537 0.10 511 0.10 485 0.10459 0.10 433 0.10 407 0.10 381 0.10 355 0.10 329 0.10 303 0.10 277 0.10 251 0.10 225 0.10199 0.10 173 0.10147 0.10121 0.10 095 0.10069 0.10 043 0.10 017 0.09 991 0.09 965 0.09 939 0.09 914 0.09 888 0.09 862 0.09 836 0.09 810 0.09 784 0.09 758 0.09 732 0.09 706 0.09 680 0.09 654 0.09 629 0.09 603 0.09 577 0.09 551 0.09 525 0.09499 0.09 473 0.09 447 0.09 422 0.09 396 0.09 370 0.09 344 0.09 318 0.09 292 0.09 266 0.09 241 0.09 215 0.09 189 0.09 163 9.89 653 9.89 643 9.89 633 9.89 624 9.89 614 9.89 604 9.89594 9.89 584 9.89574 9.89 564 9.89 554 9.89 544 9.89534 9.89 524 9.89514 9.89504 9.89495 9.89 485 9.89 475 9.89465 9.89 455 9.89 445 9.89 435 9.89425 9.89 415 9.89405 9.89 395 9.89 385 9.89 375 9.89 364 9.89 354 9.89 344 9.89 334 9.89 324 9.89 314 9.89 304 9.89 294 9.89 284 9.89 274 9.89 264 9.89 254 9.89 244 9.89 233 9.89 223 9.89 213 9.89 203 9.89 193 9.89 183 9.89 173 9.89 162 9,89152 9.89 142 9.89132 9.89 122 9.89112 9.89 101 9.89091 9.89081 9.89071 9.89 060 9.89 050 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 26 25 2 5.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 20.8 20.0 9 23.4 22.5 16 15 2 3.2 3.0 3 4.8 4.5 4 6.4 6.0 5 8.0 7.5 6 9.6 9.0 7 11.2 10.5 8 12.8 12.0 9 14.4 13.5 17 3.4 5.1 6.8 8.5 10.2 11.9 13.6 15.3 11 2.2 'So 4.4 5.5 6.() 7.7 8.8 9.9 10 2 2.0 3 3.0 4 4.0 5 5.0 6 6.0 7 7.0 8 8.0 9 9.0 9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 From the top : For38°+or218°+, read as printed ; for 128°+ or 308°+, read co-function. From the bottom : For 51°+ or 231°+, read as printed; for 141°+ or 321°+, read co-function. LOos L Ctn c d L Tan L Sin Prop. Pts. 51° — Logarithms of Trigonometric Functions rii] 39° — Logarithms of Trigonometric Functions S5 L Sin L Tan c d L Ctn L Cos Prop. Pts. 9.79 887 9.79 903 9.79 918 9.79 934 9.79 950 9.79 %5 9.79 981 9.79 996 9.80 012 9.80 027 9.80 043 9.80 058 9.80 074 9.80 089 9.80 105 9.80 120 9.80 136 9.80 151 9.80 166 9.80 182 9.80 197 9.80 213 9.80 228 9.80 244 9.80 259 9.80 274 9.80 290 9.80 305 9.80 320 9.80 33() 9.80 351 9.80 366 9.80 382 9.80 397 9.80 412 9.80 428 9.80 443 9.80 458 9.80 473 9.80 489 9.80 504 9.80 519 9.80 534 9.80 550 9.80 565 9.80 580 9.80 595 9.80 610 9.80 625 9.80 641 9.80 656 9.80 671 9.80 686 9.80 701 9.80 716 9.80 731 9.80 746 9.80 762 9.80 777 9.80 792 9.80 807 9.90 837 9.90 863 9.90 889 9.90 914 9.90 940 9.90 966 9.90 992 9.91 018 9.91 043 9.91 069 9.91 095 9.91 121 9.91 147 9.91 172 9.91 198 9.91 224 9.91250 9.91 276 9.91 301 9.91 327 9.91 353 9.91 379 9.91 404 9.91 430 9.91 456 9.91 482 9.91 507 9.91 533 9.91 559 9.91 585 9.91 610 9.91 636 9.91 662 9.91 688 9.91 713 9.91 739 9.91 765 9.91 791 9.91 816 9.91 842 9.91 868 9.91 893 9.91 919 9.91 945 9.91 971 9.91 996 9.92 022 9.92 048 9.92 073 9.92 099 9.92 125 9.92 150 9.92 176 9.92 202 9.92 227 9.92 253 9.92 279 9.92 304 9.92 330 9.92 356 9.92 381 0.09 163 0.09 137 0.09 111 0.09 086 0.09 060 0.09034 0.09 0C8 0.08 982 0.08 957 0.08 931 0.08 905 0.08 879 0.08 853 0.08 828 0.08 802 0.08 776 0.08 750 0.08 724 0.08 699 0.08 673 0.08 647 0.08 621 0.08 596 0.08 570 0.08 544 0.08 518 0.08 493 0.08 467 0.08 441 0.08 415 0.08 390 0.08 364 0.08 338 0.08 312 0.08 287 0.08 261 0.08 235 0.08 209 0.08 184 0.08 158 0.08 132 0.08 107 0.08 081 0.08 055 0.08 029 0.08 004 0.07 978 0.07 952 0.07 927 0.07 901 0.07 875 0.07 850 0.07 824 0.07 798 0.07 773 0.07 747 0.07 721 0.07 696 0.07 670 0.07 644 0.07 619 9.89 050 9.89 040 9.89 030 9.89 020 9.89 009 9.88 999 9.88 989 9.88 978 9.88 968 9.88 958 9.88 948 9.88 937 9.88 927 9.88 917 9.88 906 9.88 896 9.88 886 9.88 875 9.88 865 9.88 855 9.88 844 9.88 834 9.88 824 9.88 813 9.88 803 9.88 793 9.88 782 9.88 772 9.88 761 9.88 751 9.88 741 9.88 730 9.88 720 9.88 709 9.88 699 9.88 688 9.88 678 9.88 668 9.88 657 9.88 647 9.88 636 9.88 626 9.88 615 9.88 605 9.88 594 9.88 584 9.88 573 9.88 563 9.88 552 9.88 542 9.88 531 9.88 521 9.88 510 9.88 499 9.88 489 9.88 478 9.88 468 9.88 457 9.88 447 9.88 436 9.88 425 26 25 5.2 5.0 7.8 7.5 10.4 10.0 13.0 12.5 15.6 15.0 18.2 17.5 20.8 20.0 23.4 22.5 15 11 2 3.0 2.2 3 4.5 3.3 4 6.0 4.4 5 7.5 5.5 6 9.0 6.6 7 10.5 7.7 8 12.0 8.8 9 13.5 9.9 16 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 From the top : For 39°+ or 219°+, read as printed ; for 129°+ or 309°+, read co-function. From the bottom : For 60°+ or 230°+, read as printed ; for 140°+ or 320°+, read co-function. LGos L Ctn c d LTan L Sin Prop. Pts. 50° — Logarithms of Trigonometric Functions ^6 40*" — Logarithms of Trigonometric Functions [in LSin L Tan c d L Ctn L Cos Prop. Pts. 1 2 3 4 5 6 7 8 9 LO 11 12 13 14 15 m 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3(3 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.80 807 9.80 822 9.80 837 9.80 852 9.80 867 9.80 882 9.80 897 9.80 912 9.80 927 9.80 942 9.80 957 9.80 972 9.80 987 9.81 002 9.81017 9.81 032 9.81 047 9.81 061 9.81 076 9.81091 9.81 106 9.81 121 9.81 136 9.81 151 9.81 166 9.81 180 9.81 195 9.81 210 9.81 225 9.81 240 9.81 254 9.81 269 9.81 284 9.81 299 9.81 314 9.81 328 9.81 343 9.81 358 9.81 372 9.81 387 9.81 402 9.81 417 9.81 431 9.81 446 9.81 461 9.81 475 9.81 490 9.81 505 9.81 519 9.81 534 9.81 549 9.81 563 9.81 578 9.81 592 9.81 607 9.81 622 9.81 636 9.81 651 9.81 665 9.81 680 9.81 694 9.92 381 9.92 407 9.92 433 9.92 458 9.92 484 9.92 510 9.92 535 9.92 561 9.92 587 9.92 612 9.92 638 9.92 663 9.92 689 9.92 715 9.92 740 9.92 766 9.92 792 9.92 817 9.92 843 9.92 868 9.92 894 9.92 920 9.92 945 9.92 971 9.92 996 9.93022 9.93 048 9.93 073 9.93099 9.93 124 9.93 150 9.93 175 9.93 20r 9.93 227 9.93 252 9.93 278 9.93 303 9.93 329 9.93 354 9.93 380 9.93 406 9.93 431 9.93 457 9.93482 9.93508 9.93 533 9.93 559 9.93 584 9.93 610 9.93 636 9.93 661 9.93 687 9.93 712 9.93 738 9.93 763 9.93 789 9.93 814 9.93 840 9.93 865 9.93 891 9.93 916 0.07 619 0.07593 0.07 567 0.07 542 0.07 516 0.07 490 0.07 465 0.07 439 0.07 413 0.07 388 0.07 362 0.07 337 0.07 311 0.07 285 0.07 260 0.07 234 0.07 208 0.07 183 0.07 157 0.07 132 0.07 106 0.07 080 0.07 055 0.07 029 0.07 004 0.06 978 0.06 952 0.06 927 0.06 901 0.06 876 0.06 850 0.06 825 0.06 799 0.06 773 0.06 748 0.06 722 0.06 697 0.06 671 0.06 646 0.06620 0.06 594 0.06 569 0.06 543 0.06 518 0.06 492 0.06 467 0.06 441 0.06 416 0.06 390 0.06 364 0.06 339 0.06 313 0.06 288 0.06 262 0.06 237 0.06 211 0.06 186 0.06 160 0.06 135 0.06 109 0.06 084 9.88 425 9.88 415 9.88 404 9.88 394 9.88 383 9.88 372 9.88 362 9.88 351 9.88 340 9.88 330 9.88 319 9.88 308 9.88 298 9.88 287 9.88 276 9.88 266 9.88 255 9.88 244 9.88 234 9.88 223 9.88 212 9.88 201 9.88 191 9.88 180 9.88 169 9.88 158 9.88 148 9.88 137 9.88 126 9.88 115 9.88 105 9.88 094 9.88 083 9.88 072 9.88 061 9.88051 9.88 040 9.88 029 9.88 018 9.88007 9.87 996 9.87 985 9.87 975 9.87 964 9.87 953 9.87 942 9.87 931 9.87 920 9.87 909 9.87 898 9.87 887 9.87 877 9.87 866 9.87 855 9.87 844 9.87833 9.87 822 9.87 811 9.87 800 9.87 789 9.87 778 26 25 2 6.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 20.8 20.0 9 23.4 22.5 14 11 2.8 2.2 4.2 3.3 5.6 4.4 7.0 5.5 8.4 6.6 9.8 7.7 11.2 8.8 12.6 9.9 15 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 From the top : For40<^+or220°+, read as printed ; for 130°+ or 310°+, read co-function. ^ From the bottom : For 49°+ or 229°+, read as printed; for 139°+ or 319°+, read co-function. LGos LCtn c d L Tan LSin Prop. Pts. 49° — Logarithms of Trigonometric Functions ill] 41° — Logarithms of Trigonometric Functions LSin L Tan c d L Ctn L Cos Prop. Pts. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 -57 58 59 60 9.81 694 9.81 709 9.81 723 9.81 738 9.81 752 9.81 767 9.81 781 9,81 796 9.81 810 9.81 825 9.81 839 9.81 854 9.81 868 9.81 882 9.81 897 9.81 911 9.81 926 9.81 940 9.81 955 9.81 969 9.81 983 9.81 998 9.82 012 9.82 026 9.82 041 9.82 055 9.82 069 9.82 084 9.82 01)8 9.82 112 9.82 126 9.82 141 9.82 155 9.82 169 9.82 184 9.82 198 9.82 212 9.82 226 9.82 240 9.82 255 9.82 269 9.82 283 9.82 297 9.82 311 9.82 326 9.82 340 9.82 354 9.82 368 9.82 382 9.82 396 9.82 410 9.82 424 9.82 439 9.82 453 9.82 467 9.82 481 9.82 495 9.82 509 9.82 523 9.82 537 9.82 551 L Cos 9.93 916 9.93 942 9.93 967 9.93 993 9.94018 9.94 044 9.94 069 9.94 095 9.94 120 9.94 146 9.94171 9.94 197 9.94 222 9.94 248 9.94 273 9.94 299 9.94 324 9.94 350 9.94 375 9.94401 9.94426 9.94452 9.94 477 9.94 503 9.94528 9.94 554 9.94 579 9.94 604 9.94 630 9.94 655 9.94 681 9.94 706 9.94 732 9.94 757 9.94 783 9.94 808 9.94 834 9.94 859 9.94 884 9.94 910 9.94 935 9.94 961 9.94 986 9.95 012 9.95 037 9.95 062 9.95 088 9.95 113 9.95 139 9.95 164 9.95 190 9.95 215 9.95 240 9.95 266 9.95 291 9.95 317 9.95 342 9.95 368 9.95 393 9.95 418 9.95 444 L Ctn c d 0.06 084 0.06 058 0.06033 0.06 007 0.05 982 0.05 956 0.05 931 0.05 905 0.05 880 0.05 854 0.05 829 0.05 803 0.05 778 0.05 752 0.05 727 0.05 701 0.05 676 0.05 650 0.05 625 0.05 599 0.05 574 0.05 548 0.05 523 0.05 497 0.05472 0.05 446 0.05 421 0.05 396 0.05 370 0.05 345 0.05 319 0.05 294 0.05 268 0.05 243 0.05 217 0.05 192 0.05 166 0.05 141 0.05 116 0.05 090 0.05 065 0.05 039 0.05 014 0.04 988 0.04 963 0.04 938 0.04 912 0.04 887 0.04 861 0.04 836 0.04 810 0.04 785 0.04 760 0.04 734 0.04 709 0.04 683 0.04 658 0.04 632 0.04 607 0.04 582 0.04 556 9.87 778 9.87 767 9.87 756 9.87 745 9.87 734 9.87 723 9.87 712 9.87 701 9.87 690 9.87 679 9.87 668 9.87 657 9.87 646 9.87 635 9.87 624 9.87 613 9.87 601 9.87 590 9.87 579 9.87 668 9.87 557 9.87 546 9.87 635 9.87 524 9.87 613 9.87 601 9.87 490 9.87 479 9.87 468 9.87 457 9.87 446 9.87 434 9.87 423 9.87 412 9.87401 9.87 3C0 9.87 378 9.87 367 9.87 356 9.87 345 9.87 334 9.87 322 9.87 311 9.87 300 9.87 288 9.87 277 9.87 266 9.87 255 9.87 243 9.87 232 9.87 221 9.87 209 9.87 198 9.87 187 9.87 175 9.87 164 9.87 153 9.87 141 9.87 130 9.87 119 9.87 107 26 25 2 6.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 20.8 20.0 9 23.4 22.5 14 12 2 2.8 2.4 3 4.2 3.6 4 6.6 4.8 6 7.0 6.0 6 8.4 7.2 7 9.8 8.4 8 11.2 9.6 9 12.6 10.8 15 3.0 4.6 6.0 7.5 9.0 10.5 12.0 13.5 11 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 From the top : For41°+or221^+, read as printed; for 131°+ or 311°+, read co-function. From the bottom : For 48°+ or 228°+, read as printed ; for 138°+ or 318°+, read co-function. L Tan L Sin Prop. Pts. 48° — Logarithms of Trigonometric Functions 88 42° — Logarithms of Trigonometric Functions [iii L Sin L Tan c d L Ctn L Cos Prop. Pts. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 9.82 551 9.82 565 9.82 579 9.82 593 9.82 607 9.82 621 9.82 635 9.82 649 9.82 663 9.82 677 9.82 691 9.82 705 9.82 719 9.82 733 9.82 747 9.82 761 9.82 775 9.82 788 9.82 802 9.82 816 9.82 830 9.82 844 9.82 858 9.82 872 9.82 885 9.82 899 9.82 913 9.82 927 9.82 941 9.82 955 9.82 968 9.82 982 9.82 996 9.83010 9.83023 9.83037 9.83051 9.83 065 9.83 078 9.83092 9.83106 9.83 120 9.83 133 9.83 147 9.83 161 9.83 174 9.83 188 9.83 202 9.83 215 9.83 229 9, 9, 9, 83 242 83 256 83 270 83 283 83 297 83 310 83 324 83 338 83 351 .83 365 .83 378 9.95 444 9.95 469 9.95 495 9.95 520 9.95 545 9.95 571 9.95 596 9.95 622 9.95 647 9.95 672 9.95 698 9.95 723 9.95 748 9.95 774 9.95 799 9.95 825 9.95 850 9.95 875 9.95 901 9.95 926 9.95 952 9.95 977 9.96002 9.96 028 9.96053 9.96 078 9.96104 9.96 129 9.96 155 9.96180 9.96 205 9.96 231 9.96 256 9.96 281 9.96 307 9.96 332 9.96 357 9.96 383 9.96 408 9.96 433 9.96 459 9.96 484 9.96 510 9.96 535 9.96 560 9.96 586 9.96 611 9.96 636 9.96 662 9.96 687 9.9f)712 9.96 738 9.96 763 9.96 788 9.96 814 9.9i] 839 9.t)6 8<>4 9.96 890 9.96 915 9.96 940 9.96 966 0.04 556 0.04 531 0.04 505 0.04 480 0.04 455 0.04 429 0.04404 0.04 378 0.04 353 0.04 328 0.04 302 0.04 277 0.04 252 0.04 226 0.04 201 0.04 175 0.04 150 0.04 125 0.04 099 0.04 074 0.04 048 0.04 023 0.03 998 0.03 972 0.03 947 0.03 922 0.03 896 0.03 871 0.03 845 0.03 820 0.03 795 0.03 769 0.03 744 0.03 719 0.03 693 0.03 668 0.03 643 0.03 617 0.03 592 0.03 567 0.03 541 0.03 516 0.03 490 0.03 465 0.03440 0.03 414 0.03 389 0.03 364 0.03 338 0.03 313 0.03 288 0.03 262 0.03 237 0.03 212 0.03 186 0.03 161 0.03 136 0.03 110 0.03085 0.03 060 0.03 034 9.87 107 9.87 096 9.87 085 9.87 073 9.87 062 9.87 050 9.87 039 9.87 028 9.87 016 9.87 005 9.86 993 9.86 982 9.86 970 9.86 959 9.86 947 9.86 936 9.86 924 9.86 913 9.86 902 9.86890 9.86 879 9.86 867 9.86 855 9.86 844 9.86 832 9.86 821 9.86 809 9.86 798 9.86 786 9.86 775 9.86 763 9.86 752 9.86 740 9.86 728 9.86 717 9.86 705 9.86 694 9.86 682 9.86 670 9.86659 9.86 647 9.86 635 9.86 624 9.86 612 9.86 600 9.86 589 9.86 577 9.86 565 9.86 554 9.86 542 9.86 530 9.86 518 9.86 507 9.86 495 9.86 483 9.86 472 9.86 460 9.86 448 9.86 436 9.86425 9.86 413 26 25 2 5.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 20.8 20.0 9 23.4 22.5 13 12 2 2.6 2.4 3 3.9 3.6 4 5.2 4.8 5 6.5 6.0 6 7.8 7.2 7 9.1 8.4 8 10.4 9.6 9 11.7 10.8 14 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 11 2.2 8.3 4.4 5.5 6.6 7.7 8.8 9.9 From the top: For42°+or222'^+, read as printed ; for 132°+ or 312°+ co-function. read From the bottom : ror47°+or227°+, read as printed; for 137°+ or 317°+, read co-function. L Cos L Ctn c d L Tan L Sin Prop. Pts. 47° — Logarithms of Trigonometric Functions Ill] 43^ — Logarithms of Trigonometric Functions 89 L Sin L Tan c d L Gtn L Cos d Prop. Pts. 3 4 5 (3 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 '57 58 59 60 9.83 378 9.83 392 9.83 405 9.83 419 9.83 432 9.83 446 9.83 459 9.83 473 9.83 486 9.83 500 9.83513 9.83 527 9.83 540 9.83 554 9.83 567 9.83 581 9.83 594 9.83 608 9.83 621 9 83 634 9.83 648 9.83 661 9.83 674 9.83 688 9.83 701 9.83 715 9.83 728 9.83 741 9.83 755 9.83 768 9.83 781 9.83 795 9.83 808 9.83 821 9.83 834 9.83 848 9.83 861 9.83 874 9.83 887 9.83 901 9 9, 9 9, 9 9, 9, 9 9, 9 83 914 .83 927 83 940 .83 954 .83 967 83 980 83 993 84 006 84 020 .84 033 9.84 046 9.84 059 9.84 072 9.84 085 9.84 098 9.84112 9.84 125 9.84 138 9.84 151 9.84 164 9.84 177 9.96 966 9.96 991 9.97 016 9.97 042 9.97 067 9.97 092 9.97 118 9.97 143 9.97 168 9.97 193 9.97 219 9.97 244 9.97 269 9.97 295 9.97 320 9.97 345 9.97 371 9.97 396 9.97 421 9.97 447 9.97 472 9.97 497 9.97 523 9.97 548 9.97 573 9.97 598 9.97 624 9.97 649 9.97 674 9.97 700 9.97 725 9.97 750 9.97 776 9.97 801 9.97 826 9.97 851 9.97 877 9.97 902 9.97 927 9.97 953 9.97 978 9.98 003 9.98 029 9.98 054 9.98 079 9.98 104 9.98 130 9.98 155 9.98 180 9.98 206 9.98 231 9.98 256 9.98 281 9.98 307 9.98 332 9.98 357 9.98 383 9.98 408 9.98 433 9.98 458 9.98 484 0.03 034 0.03009 0.02 984 0.02 958 0.02 933 0.02 908 0.02 882 0.02 857 0.02 832 0.02 807 0.02 781 0.02 756 0.02 731 0.02 705 0.02 680 0.02 655 0.02 629 0.02 604 0.02 579 0.02 553 0.02 528 0.02 503 0.02 477 0.02 452 0.02 427 02 402 0.02 376 G.02 351 0.02 326 0.02 300 0.02 275 0.02 250 0.02 224 0.02 199 0.02 174 0.02 149 0.02 123 0.02 098 0.02 073 0.02 047 0.02 022 0.01 997 0.01 971 0.01 946 0.01 921 0.01 896 0.01 870 0.01 845 0.01 820 0.01 794 0.01 769 0.01 744 0.01 719 0.01 693 0.01 ms 0.01 643 0.01 617 0.01 592 0.01 567 0.01 542 0.01 516 9.86 413 9.86 401 9.86 389 9.86 377 9.86 366 9.8() 354 9.86 342 9.86 330 9.86 318 9.86 306 9.86 295 9.86 283 9.86 271 9.86 259 9.86 247 9.86 235 9.8() 223 9.86 211 9.86 200 9.86 188 9.8(3 176 9.86 164 9.86 152 9.86 140 9.86 128 9.86 116 9.86 104 9.86 092 9.86 080 9.86 068 9.86 056 9.86 044 9.86 032 9.86 020 9.86 008 9.85 996 9.85 984 9.85 972 9.85 960 9.85 948 9.85 93(3 9.85 924 9.85912 9.85^)00 9.85 888 9.85 876 9.85 864 9.85 851 9.85 839 8.85 827 9.85 815 9.85 803 9.85 791 9.85 779 9.85 766 9.85 754 9.85 742 9.85 730 9.85 718 9.85 706 9.85 693 26 25 2 5.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 .20.8 20.0 9 23.4 22.5 13 12 2 2.6 2.4 3 3.9 3.6 4 5.2 4.8 5 6.5 6.0 H 7.8 7.2 7 9.1 8.4 8 10.4 9.6 9 11.7 10.8 1 14 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 11 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 From the top : For 43°+ or 223°+, read as printed ; for 133°+ or 313°+, read co-function. Frojii the bottom : For 46°+ or 226°+, read as printed ; for 136°+ or 316°+, read co-function. LCos LCtn c d L Tan LSin Id Prop. Pts. 46°— Losrarithms of Trigonometric Functions 90 44° — Logarithms of Trigonometric Functions [in 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3f) 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 LSin 9.84177 9.84 190 9.84 203 9.84 216 9.84 229 9.84 242 9.84 255 9.84 269 9.84 282 9.84 295 9.84 308 9.84 321 9.84 334 9.84 347 9.84 360 9.84 373 9.84 385 9.84 398 9.84 411 9.84 424 9.84 437 9.84 450 9.84 463 9.84 476 9.84 489 9.84 502 9.84 515 9.84 528 9.84 540 9.84 553 9.84 566 9.84 579 9.84 592 9.84 605 9.84 618 9.84 630 9.84 643 9.84 656 9.84 669 9.84 682 9.84 694 9.84 707 9.84 720 9.84 733 9.84 745 9.84 758 9.84 771 9.84 784 9.84 796 9.84 809 9.84 822 9.84 835 9.84 847 9.84 860 9.84 873 9.84 885 9.84 898 9.84 911 9.84 923 9.84 936 9.84 949 LGos L Tan 9.98 484 9.98 509 9.98 534 9.98 560 9.98 585 9.98 610 9.98 635 9.98 661 9.98 686 9.98 711 9.98 737 9.98 762 9.98 787 9.98 812 9.98 838 9.98 863 9.98 888 9.98 913 9.98 939 9.98 964 9.98 989 9.99 015 9.99 040 9.99065 9.99 090 9.99 116 9.99 141 9.99 166 9.99 191 9.99 217 9.99 242 9.99 267 9.99 293 9.99 318 9.99 343 9.99 368 9.99 394 9.99 419 9.99444 9.99 469 9.99 495 9.99 520 9.99 545 9.99 570 9.99 596 9.99 621 9.99 ()46 9.99 672 9.99 697 9.99 722 9.99 747 9.99 773 9.99 798 9.99 823 9.99 848 9.99 874 9.99 899 9.99 924 9.99 949 9.99 975 0.00 000 LCtn c d L Ctn 0.01 516 0.01 491 0.01 466 0.01 440 0.01 415 0.01 390 0.01 365 0.01 339 0.01 314 0.01 289 0.01 263 0.01 2.38 0.01 213 0.01 188 0.01 162 0.01 137 0.01 112 0.01 087 0.01 061 0.01036 0.01011 0.00 985 0.00 960 0.00 935 0.00 910 0.00 884 0.00 859 0.00 834 0.00 809 0.00 783 0.00 758 0.00 733 0.00 707 0.00 682 0.00 657 0.00 632 0.00 606 0.00 581 0.00 556 0.00 531 0.00 505 0.00 480 0.00 455 0.00430 0.00404 0.00 379 0.00 354 0.00 328 0.00 303 0.00 278 0.00 253 0.00 227 0.00 202 0.00 177 0.00 152 0.00 126 0.00 101 0.00 076 0.00 051 0.00 025 0.00 000 c d L Tan LGos 9.85 693 9.85 681 9.85 669 9.85 657 9.85 645 9.85 632 9.85 620 9.85 608 9.85 596 9.85 583 9.85 571 9.85 559 9.85 547 9.85 534 9.85 522 9.85 510 9.85 497 9.85 485 9.85 473 9.85 460 9.85 448 9.85 436 9.85 423 9.85 411 9.85 399 9.85 386 9.85 374 9.85 361 9.85 349 9.85 337 9.85 324 9.85 312 9.85 299 9.85 287 9.85 274 9.85 262 9.85 250 9.85 237 9.85 225 9.85 212 9.85 200 9.85 187 9.85 175 9.85 162 9.85 150 9.85 1.37 9.85 125 9.85 112 9.85 100 9.85 087 9.85 074 9.85 062 9.85 049 9.85 037 9.85 024 9.85 012 9.84 999 9.84 986 9.84 974 9.84 961 9 84949 L Sin Prop. Pts. 26 25 2 5.2 5.0 3 7.8 7.5 4 10.4 10.0 5 13.0 12.5 6 15.6 15.0 7 18.2 17.5 8 20.8 20.0 9 23.4 22.5 14 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 13 2- 2.6 3 3.9 4 5.2 5 6.5 6 7.8 7 9.1 8 10.4 9 11.7 12 2.4 3.6 4.8 6.0 7.2 8.4 9.6 10.8 From the top : For 44°+ or 224°+, read as printed; for 134°+ or 314°+, read co-function. Fro7n the bottom : For 45°+ or 225°+, read as printed ; for 135°+ or 315°+, read co-function. Prop. Pts. 45° — Logarithms of Triaronometric Functions IV] Table IV — Degrees, Minutes, and Seconds to Radians 91 Degrees Minutes Seconds 0° 0.0000000 60° 1.04719 76 120° 2.09439 51 0' 0.00000 00 0" 0.00000 00 1 0.01745 33 61 1.0(;465 08 121 2.11184 84 1 0.00029 09 1 0.00000 48 2 0.03190 66 62 1.08210 41 122 2.12930 17 2 0.00058 18 2 0.00000 97 3 0.05235 99 63 1.09955 74 123 2.14675 50 3 0.00087 27 3 0.00001 45 4 0.06981 32 64 1.11701 07 124 2.16420 83 4 0.00116 36 4 0.00001 94 5 0.08726 65 65 1.13446 40 125 2.18166 16 5 0.00145 44 5 0.00002 42 6 0.1047198 66 1.1519173 126 2.19911 49 6 0.00174 53 6 0.00002 91 7 0.12217 30 67 1.16937 06 127 2.21656 82 7 0.00203 62 7 0.00003 39 8 0.13962 63 68 1.18682 39 128 2.23402 14 8 0.00232 71 8 0.00003 88 9 0.15707 96 69 1.20427 72 129 2.25147 47 9 0.00261 80 9 0.00004 36 10 0.17453 29 70 1.22173 05 130 2.26892 80 10 0.00290 89 10 0.00004 85 11 0.19198 62 71 1.23918 38 131 2.28638 13 11 0.00319 98 11 0.00005 33 12 0.20943 95 72 1.25663 71 132 2.3038346 12 0.00349 07 12 0.00005 82 13 0.22689 28 73 1.27409 04 133 2.32128 79 13 0.00378 15 13 0.00006 30 14 0.24434 61 74 1.29154 36 134 2.33874 12 14 0.00407 24 14 0.00006 79 15 0.26179 94 75 1.30899 69 135 2.35619 45 15 0.00436 33 15 0.00007 27 16 0.27925 27 76 1.32645 02 136 2.37364 78 16 0.00465 42 16 0.00007 76 17 0.29670 60 77 1.31390 35 137 2.39110 11 17 0.00494 51 17 0-00008 24 18 0.31415 93 78 1.36135 68 138 2.40855 44 18 0.00523 60 18 0.00008 73 19 0.33161 26 79 1.3788101 139 2.42600 77 19 0.00552 69 19 0.00009 21 20 0.34906 59 80 1.39626 34 140 2.44346 10 20 0.00581 78 20 0.00009 70 21 0.36651 91 81 1.4137167 141 2.46091 42 21 0.00610 87 21 0.00010 18 22 0.38397 24 82 1.4.3117 00 142 2.47836 75 22 0.00639 95 22 0.00010 67 23 0.40142 57 83 1.44862 33 143 2.49582 08 23 0.00669 04 23 0.00011 15 24 0.41887 90 84 1.46607 66 144 2.51327 41 24 0.00698 13 24 0.0001164 25 0.43633 23 85 1.48352 99 145 2.53072 74 25 0.00727 22 25 0.00012 12 26 0.45378 56 86 1.50098 32 146 2.54818 07 26 0.00756 31 26 0.00012 61 27 0.47123 89 87 1.51843 64 147 2.56563 40 27 0.00785 40 27 0.00013 09 28 0.48869 22 88 1.53588 97 148 2.58308 73 28 0.a)814 49 28 0.00013 57 29 0.50614 55 89 1.55334 30 149 2.60054 06 29 0.00843 58 29 0.00014 06 30 0.52359 88 90 1.57079 63 150 2.61799 39 30 0.00872 66 30 0.00014 54 31 0.54105 21 91 1.58824 9(5 151 2.63544 72 31 0.00901 75 31 0.00015 03 32 0.55850 54 92 1.60570 29 152 2.65290 05 32 0.00930 84 32 0.00015 51 33 0.57595 87 93 1.62315 62 153 2.67035 38 33 0.00959 93 33 0.00016 00 34 0.59341 19 94 1.64060 95 154 2.68780 70 34 0.00989 02 34 0.00016 48 35 0.61086 52 95 1.65806 28 155 2.70526 03 35 0.01018 11 35 0.00016 97 36 0.62831 85 96 1.67551 61 156 2.72271 36 36 0.01047 20 36 0.00017 45 37 0.64577 18 97 1.69296 94 157 2.74016 69 37 0.01076 29 37 0.00017 94 38 0.66322 51 98 1.71042 27 158 2.75762 02 38 0.01105 38 38 0.00018 42 39 0.68067 84 99 1.72787 60 159 2.77507 35 39 0.01134 46 39 0.00018 91 40 0.69813 17 100 1.74532 93 160 2.79252 68 40 0.01163 55 40 0.00019 39 41 0.71558 50 101 1.76278 25 161 2.80998 01 41 0.01192 64 41 0.00019 88 42 0.73303 83 102 1.78023 58 162 2.82743 34 42 0.01221 73 42 0.00020 36 43 0.75049 16 103 1.79768 91 163 2.84488 67 43 0.01250 82 43 0.00020 85 44 0.76794 49 104 1.81514 24 164 2.86234 00 44 0.01279 91 44 0.00021 33 45 0.78530 82 105 1.83259 57 165 2.87979 33 45 0.01309 00 45 0.00021 82 46 0.8028515 106 1.8.5004 90 166 2.89724 66 46 0.01338 09 46 0.00022 30 47 0.82030 47 107 1.86750 23 167 2.91469 99 47 0.01367 17 47 0.00022 79 48 0.83775 80 108 1.88495 56 168 2.93215 31 48 0.01.S96 26 48 0.00023 27 49 0.85521 13 109 1.90240 89 169 2.94960 64 49 0.01425 35 49 0.00023 76 50 0.87266 46 110 1.91986 22 170 2.96705 97 50 0.01454 44 50 0.00024 24 51 0.89011 79 111 1.9373155 171 2.98451 30 51 0.01483 53 51 0.00024 73 52 0.90757 12 112 1.95476 88 172 3.00196 63 52 0.01512 62 52 0.00025 21 53 0.92502 45 113 1.97222 21 173 3.01941 96 53 0.01541 71 53 0.00025 70 54 0.94247 78 114 1.98967 53 174 3.03687 29 54 0.01570 80 54 0.00026 18 55 0.95993 11 115 2.00712 86 175 3.05432 62 55 0.01599 89 55 0.00026 66 56 0.97738 44 116 2.0245819 176 3.07177 95 56 0.01628 97 56 0.00027 15 57 0.99483 77 117 2.04203 52 177 3.08923 28 57 0.01658 06 57 0.00027 63 58 1.0122910 118 2.05948 85 178 3.10668 61 58 0.01687 15 58 0.00028 12 59 1.02974 43 119 2.07694 18 179 3.12413 94 59 0.01716 24 59 0.00028 60 60 1.04719 76 120 2.09439 51 180 3.14159 27 60 0.01745 33 60 0.00029 09 92 V— Radian Measure — Trigonometric Functions [t i 8 Sin x Cos a? Tana; I. .00 .00000 1.0000 .00000 0°00'.0 .01 .02 .03 .04 .05 .06 .07 .08 .09 .01000 .02000 .03000 .03999 .04998 .05996 .06994 .07991 .08988 .999^5 .99980 .99955 .99920 .99875 .99820 .99755 .99680 .99595 .01000 .02000 .03001 .04002 .05004 .06007 .07011 .08017 .09024 0°34'.4 1°08'.8 l°43'.l 2° 17'. 5 2°51'.9 3° 26'. 3 4° 00'. 6 4°.35'.0 5° 09'. 4 .10 .09983 .99500 .10033 5° 43'. 8 .11 .12 .13 .14 .15 .16 .17 .18 .19 .10978 .11971 .12963 .13954 .14944 .15932 .16918 .17903 .18886 .99396 .99281 .99156 .99022 .98877 .98723 .98558 .98384 .98200 .11045 .12058 .13074 .14092 .15114 .16138 .17166 .18197 .19232 6°18'.2 6° 52'. 5 7°26'.9 8° 01'. 3 8°35'.7 9°10'.0 9° 44'. 4 10° 18'. 8 10° 53'. 2 .20 .19867 .98007 .20271 11° 27' .5 .21 .22 .23 .24 .25 .26 .27 .28 .29 .20846 .21823 .22798 .23770 .24740 .25708 .26673 .27636 .28595 .97803 .97590 .97367 .97134 .96891 .96639 .96377 .96106 .95824 .21314 .22362 .23414 .24472 .25534 .26602 .27676' .28755 .29841 12°01'.9 12° 36'. 3 13°10'.7 13°45'.l 14° 19' .4 14° 53'. 8 15°28'.2 16° 02'. 6 16°36'.9 .30 .29552 .95534 .30934 17°11'.3 .31 .32 .33 .34 .35 .36 .37 .38 .39 .30506 .31457 .32404 .33349 .34290 .35227 .36162 .37092 .38019 .95233 .94924 .94604 .94275 .93937 .93590 .93233 .92866 .92491 .32033 .33139 .34252 .35374 .36503 .37640 .38786 .39941 .41106 17°45'.7 18°20'.l 18° 54' .5 19° 28'. 8 20° 03'. 2 20°37'.6 21°12'.0 21° 46'. 3 22° 20'. 7 .40 .38942 .92106 .42279 22°55'.l .41 .42 .43 .44 .45 .46 .47 .48 .49 .39861 .40776 .41687 .42594 .43497 .44395 .45289 .46178 .47063 .91712 .91309 .90897 .90475 .90045 .89605 .89157 .88699 .88233 .43463 .44657 .45862 .47078 .48305 .49545 .50795 .52061 .53339 23° 29' .5 24° 03'. 9 24° 38'. 2 25°12'.6 25°47'.0 26° 21'. 4 26°55'.7 27°30'.l 28° 04' .5 .60 .47943 .87758 .54630 28° 38'. 9 CO 1 8 Sin a; Cos a; Tana; 1 .50 .47943 .87758 .54630 28° 38'. 9 .51 .52 .53 .54 .55 .56 .57 .58 .59 .48818 .49688 .50553 .51414 .52269 .53119 .53963 .54802 .55636 .87274 .86782 .86281 .85771 .85252 .84726 .84190 .83646 83094 .55936 .57256 .58592 .59943 .61311 .62695 .64097 .65517 .66956 29° 13'. 3 29° 47'. 6 30°22'.0 30° 56'. 4 31° 30'. 8 32°05'.l 32° 39'. 5 33° 13'. 9 33° 48'. 3 .60 .56464 .82534 .68414 34° 22'. 6 .61 .62 .63 .64 .65 .66 .67 .68 .69 .57287 .58104 .58914 .59720 .60519 .61312 .62099 .62879 .63654 .81965 .81388 .80803 .80210 .79608 .78999 .78382 .77757 .77125 .69892 .71391 .72911 .74454 .76020 .77610 .79225 .80866 .82533 34°57'.0 35° 31 '.4 36° 05'.8 36° 40'. 2 37° 14'. 5 37° 48'. 9 38° 23'. 3 38°57'.7 39°32'.0 .70 .64422 .76484 .84229 40° 06'. 4 .71 .72 .73 .74 .75 .76 .77 .78 .79 .65183 .65938 .66687 .67429 .68164 .68892 .69614 .70328 .71035 .75836 .75181 .74517 .73847 .73169 .72484 .71791 .71091 .70385 .85953 .87707 .89492 .91309 .93160 .95055 .96967 .98926 1.0092 40° 40'. 8 41°15'.2 41° 49'. 6 42° 23'. 9 42°58'.3 43° 32'. 7 44°07'.l 44°41'.4 45° 15'. 8 .80 .71736 .69671 1.0296 45°50'.2 .81 .82 .83 .84 .85 .86 .87 .88 .89 .72429 .73115 .73793 .74464 .75128 .75784 .76433 .77074 .77707 .68950 .68222 .67488 .66746 .65998 .65244 .64483 .63715 .62941 1.0505 1.0717 1.0934 1.1156 1.1383 1.1616 1.1853 1.2097 1.2346 46° 24' .6 46°59'.0 47° 33'. 3 48° 07'. 7 48°42'.l 49° 16'. 5 49°50'.8 50° 25'. 2 50° 59'. 6 .90 .78333 .62161 1.2602 51°34'.0 .91 .92 .93 .94 .95 .96 .97 .98 .99 .78950 .79560 .80162 .80756 .81342 .81919 .82489 .83050 .83603 .61375 .60582 .59783 .58979 .58168 .57352 .56530 .55702 .54869 1.2864 1.3133 1.3409 1.3692 1.3984 1.4284 1.4592 1.4910 1.5237 52°08'.3 52° 42'. 7 53°17'.l 53°51'.5 54° 25'. 9 55° 00'. 2 55° 34'. 6 56°09'.0 56° 43'. 4 1.00 .84147 .54030 1.5574 57°17'.7 V — Radian Measure — Trigonometric Functions 93 8 Sin a; Cos a? Tana; 1 1.00 .84147 .54030 1.5574 57°17'.7 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 .84683 .85211 .85730 .86240 .86742 .87236 .87720 .88196 .88663 .53186 .52337 .51482 .50622 .49757 .48887 .48012 .47133 .46249 1.5922 1.6281 1.6652 1.7036 1.7433 1.7844 1.8270 1.8712 1.9171 57°52'.l 58° 26'. 5 59° 00'. 9 59° 35'. 3 60° 09'. 6 60° 44'.0 61°18'.4 61°52'.8 62°27'.l 1.10 .89121 .45360 l.<)648 63° 01'. 5 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 .89570 .90010 .90441 .90863 .91276 .91680 .92075 .92461 .92837 .44466 .43568 .42666 .41759 .40849 .39934 .39015 .38092 .37166 2.0143 2.0660 2.1198 2.1759 2.2345 2.2958 2.3600 2.4273 2.4979 63° 35'. 9 64° 10'. 3 64° 44'. 7 65°19'.0 65° 53'. 4 66° 27'.8 67° 02' .2 67° 36'. 5 68° 10'. 9 1.20 .93204 .36236 2.5722 68° 45'. 3 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 .93562 .93910 .94249 .94578 .94898 .95209 .95510 .95802 .96084 .35302 .34365 .33424 .32480 .31532 .30582 .29628 .28672 .27712 2.6503 2.7328 2.8198 2.9119 3.0096 3.1133 3.2236 3.3413 3.4672 69° 19'. 7 69°54'.l 70° 28'. 4 71° 02'. 8 71° 37'. 2 72° 11'. 6 72° 45'. 9 73° 20'. 3 73° 54'. 7 1.30 .96356 .26750 3.6021 74°29'.l 1 8 Sin a; Cos a? Tana; 1 r 1.30 .96356 .26750 3.6021 74°29'.l 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 .96618 .90872 .97115 .97348 .97572 .97786 .97991 .98185 .98370 .25785 .24818 .23848 .22875 .21901 .20924 .19945 .18964 .17981 3.7470 3.9033 4.0723 4.2556 4.4552 4.6734 4.9131 5.1774 5.4707 75° 03'. 4 75° 37 '.8 76° 12'. 2 76°4(r.6 77°21'.0 77° 55'. 3 78°29'.7 79°04'.l 79° 38'. 5 1.40 .98545 .16997 5.7979 80° 12'. 8 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 .98710 .98865 .99010 .99146 .99271 .99387 .99492 .99588 .99674 .16010 .15023 .14033 .13042 .12050 .11057 .10063 .09067 .08071 6.16.^4 6.5811 7.0555 7.6018 8.2381 8.9886 9.8874 10.983 12.350 80°47'.2 81°21'.6 81°56'.0 82° 30'. 4 83° 04'. 7 83°39'.l 84° 13'. 5 84°47'.9 85° 22'. 2 1.50 .99749 .07074 14.101 85° 56'. 6 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 .99815 .99871 .99917 .99953 .99978 .99994 1.0000 .999% .99982 .06076 .05077 .04079 .03079 .02079 .01080 .00080 -.00920 -.01920 16.428 19.670 24.498 32.461 48.078 92.621 1255.8 -108.65 -52.067 86°31'.0 87° 05'. 4 87° 39'. 8 88°14'.l 88° 48'. 5 89° 22'. 9 89° 57'. 3 90° 31 '.6 91°06'.0 1.60 .99957 -.02920 -34.233 91°40'.4 TT radians = 180° ir = 3.14159265 1 radian = 57° 17' 44".806 = 57.° 2957795 3600" = 60' = 1° = .01745329 radian TABLE V a — RADIANS TO DEGREES Radians Tenths Hundredths Thousandths Ten-thousandths 1 57°17'44".8 5°43'46".5 0°34'22".6 0° 3'26".3 0° 0'20".6 2 114°35'29".6 11°27'33".0 1° 8'45".3 0° 6'52".5 0° 0'41".3 3 171°53'14".4 17°11'19".4 1°43'07".9 0°10'18".8 0° 1'01".9 4 229°10'59".2 22°55'05".9 2°17'30".6 0°13'45".l 0° 1'22".5 5 286°28'44".0 28°38'52".4 2°51'53".2 0°17'11".3 0° l'43".l 6 343°46'28".8 34°22'38".9 3°26'15".9 0°20'37".6 0° 2'03".8 7 401° 4' 13" .6 40° 6'25".4 4° 0'38".5 0°24'03".9 0° 2'24".4 8 458°21'58".4 45°50'11".8 4°35'01".2 0°27'30".l 0° 2'45".0 9 515°39'43".3 61°33'58".3 5° 9'23".8 0°30'56".4 0° 3'05".6 94 Table VI- - Powers — Roots — Reciprocals [VI n n2 Vn VlOn n^ ^ ^10 n 1/n ^100 li 1.00 1.0000 1.00000 3.16228 1.00000 1.00000 2.15443 4.64159 1.00000 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.00499 1.00995 1.01489 1.01980 1.02470 1.02956 1.03441 1.03923 1.04403 3.17805 3.19374 3.20936 3.22490 3.24037 3.25576 3.27109 3.28634 3.30151 1.03030 1.06121 1.09273 1.12486 1.15762 1.19102 1.22504 1.25971 1.29503 1.00332 1.00662 1.00990 1.01316 1.01640 1.01961 1.02281 1.02599 1.02914 2.16159 2.16870 2.17577 2.18279 2.18976 2.19669 2.20358 2.21042 2.21722 4.65701 4.67233 4.68755 4.70267 4.71769 4.73262 4.74746 4.76220 4.77686 .9<)0099 .980392 .970874 .961538 .952381 .943396 .934579 .925926 .917431 1.10 1.2100 1.04881 3.31662 1.33100 1.03228 2.22398 4.79142 .90fX)91 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2321 1.2544 1.2769 1.2996 1.3225 1.3456 1.3689 1.3924 1.4161 1.05357 1.05830 1.06:301 1.06771 1.07238 1.07703 1.08167 1.08628 1.09087 3.33167 3.34664 3.36155 3.37639 3.39116 3.40588 3.42053 3.43511 3.44964 1.36763 1.40493 1.44290 1.48154 1.52088 1.56090 1.60161 1.64303 1.68516 1.03540 1.03850 1.04158 1.04464 1.04769 1.05072 1.05373 1.05672 1.05970 2.23070 2.23738 2.24402 2.25062 2.25718 2.26370 2.27019 2.27664 2.28305 4.80590 4.82028 4.83459 4.84881 4.86294 4.87700 4.89097 4.90487 4.91868 .900901 .892857 .884956 .877193 .869565 .862069 .854701 .847458 .840336 1.20 1.4400 1.09545 3.46410 1.72800 1.06266 2.28943 4.93242 .833333 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.4641 1.4884 1.5129 1.5376 1.5625 1.5876 1.6129 1.6384 1.6641 1.10000 1.10454 1.10905 1.11355 1.11803 1.12250 1.12694 1.13137 1.13578 3.47851 3.49285 3.50714 3.52136 3.53553 3.54965 3.56371 3.57771 3.59166 1.77356 1.81585 1.86087 1.90662 1.95312 2.00038 2.04838 2.09715 2.14669 1.06560 1.06853 1.07144 1.07434 1.07722 1.08008 1.08293 1.08577 1.08859 2.29577 2.30208 2.30835 2.31459 2.32079 2.32697 2.33311 2.33921 2.34529 4.94609 4.95968 4.97319 4.98663 5.00000 6.01330 5.02653 5.03968 5.05277 .826446 .819672 .813008 .806452 .800000 .793651 .787402 .781250 .775194 1.30 1.6900 1.14018 3.60555 2.19700 1.09139 2.35133 5.06580 .769231 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.7161 1.7424 1.7689 1.7956 1.8225 1.8496 1.8769 1.9044 1.9321 1.14455 1.14891 1.15326 1.15758 1.16190 1.16619 1.17047 1.17473 1.17898 3.61939 3.63318 3.64692 3.66060 3.67423 3.68782 3.70135 3.71484 3.72827 2.24809 2.29997 2.35264 2.40610 2.46038 2.51546 2.57135 2.62807 2.68562 1.09418 1.096% 1.09972 1.10247 1.10521 1.10793 1.11064 1.11334 1.11602 2.35735 2.36333 2.36928 2.37521 2.38110 2.38697 2.39280 2.39861 2.40439 5.07875 5.09164 6.10447 5.11723 5.12993 5.14256 5.15514 5.16765 5.18010 .763359 .757576 .751880 .746269 .740741 .735294 .729927 .724638 .719424 1.40 1.9600 1.18322 3.74166 2.74400 1.11869 2.41014 5.19249 .714286 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.9881 2.0164 2.0449 2.0736 2.1025 2.1316 2.1609 2.1904 2.2201 1.18743 1.19164 1.19583 1.20000 1.20416 1.20830 1.21244 1.21655 1.22066 3.75500 3.76829 3.78153 3.79473 3.80789 3.82099 3.83406 3.84708 3.86005 2.80322 2.86329 2.92421 2.98598 3.04862 3.11214 3.17652 3.24179 3.30795 1.12135 1.12399 1.12662 1.12924 1.13185 1.13445 1.13703 1.13960 1.14216 2.41587 2.42156 2.42724 2.43288 2.43850 2.44409 2.44966 2.45520 2.46072 5.20483 5.21710 5.22932 5.24148 5.25359 6.26564 5.27763 5.28957 5.30146 .709220 .704225 .699301 .694444 .689655 .684932 .680272 .675676 .671141 1.50 2.2500 1.22474 3.87298 3.37500 1.14471 2.46621 5.31329 .666667 n n2 Vw^ VlOn n^ ^n ^10 w. 1/n ^100 n VI] Powers — Roots — Reciprocals 95 n n^ \/n VlOtfc n^ ^n ^10 w, 1/n 002 8.86566 8.87130 8.87694 8.88257 476.380 478.212 480.049 481.890 483.737 485.588 487.443 489.304 491.169 1.98404 1.98489 1.98573 1.98658 1.98742 1.98826 1.98911 1.98995 1.99079 4.27448 4.27631 4.27813 4.27995 4.28177 4.28359 4.28540 4.28722 4.28903 9.20910 9.21302 9.21695 9.22087 9.22479 9.22871 9.23262 9.23653 9.24043 .128041 .127877 .127714 .127551 .127389 .127226 .127065 .126904 .126743 7.90 62.4100 2.81069 8.88819 493.039 1.99163 4.29084 9.24434 .126582 7.91 7.92 7.93 7.94 7.95 7.96 7.97 7.98 7.99 62.5681 62.7264 62.8849 63.0436 63.2025 63.3616 63.5209 63.6804 63.8401 2.81247 2.81425 2.81603 2.81780 2.81957 2.82135 2.82.312 2.82489 2.82666 8.89382 8.89944 8.90505 8.91067 8.91628 8.92188 8.92749 8.93308 8.93868 494.914 496.793 498.677 500.566 502.460 504.358 506.262 508.170 510.082 1.99247 1.99331 1.99415 1.99499 1.99582 1.99666 1.99750 1.99833 1.99917 4.29265 4.29446 4.29627 4.29807 4.29987 4.30168 4.30348 4.30528 4.30707 9.24823 9.25213 9.25602 9.25991 9.26380 9.26768 9.27156 9.27544 9.27931 .126422 .126263 .126103 .125945 .125786 .125628 .125471 .125313 .125156 8.00 64.0000 2.82843 8.94427 512.000 2.00000 4.30887 9.28318 .125000 n 1*2 y/n n^ ^n ^\^n \ln VlOn ^100 n 108 Powers — Roots — Reciprocals [VI n n^ Vn VlOn n^ ^ ^10 n l/n ^100 n 8.00 64.0000 2.82843 8.94427 512.000 2.00000 4.30887 9.28318 .125000 8.01 8.02 8.03 8.04 8.05 8.06 8.07 8.08 8.09 64.1601 64.3204 64.4809 64.6416 64.8025 64.9636 65.1249 65.2864 65.4481 2.83019 2.83196 2.83373 2.83549 2.83725 2.83901 2.84077 2.84253 2.84429 8.94986 8.95545 8.96103 8.96660 8.97218 8.97775 8.98.332 8.98888 8.99444 513.922 515.850 517.782 519.718 521.660 523.607 525.558 527.514 529.475 2.00083 2.00167 2.00250 2.00333 2.00416 2.00499 2.00582 2.00664 2.00747 4.31066 4.31246 4.31425 4.31604 4.31783 4.31961 4.32140 4.32318 4.32497 9.28704 9.29091 9.29477 9.29862 9.30248 9.30633 9.31018 9.31402 9.31786 .124844 .124688 .124533 .124378 .124224 .124069 .123916 .123762 .123609 8.10 65.6100 2.84605 9.00000 531.441 2.00830 4.32675 9.32170 .123457 8.11 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 65.7721 65.9344 66.0969 66.2596 66.4225 66.5856 66.7489 66.9124 67.0761 2.84781 2.84956 2.85132 2.85307 2.85482 2.85657 2.85832 2.86007 2.86182 9.00555 9.01110 9.01665 9.02219 9.02774 9.03327 9.03881 9.04434 9.04986 533.412 535.387 537.368 639.353 541.343 543.338 545.339 547.343 549.353 2.00912 2.00995 2.01078 2.01160 2.01242 2.01325 2.01407 2.01489 2.01571 4.32853 4.33031 4.33208 4.33386 4.33563 4.33741 4.33918 4.34095 4.34271 9.32553 9.32936 9.33319 9.33702 9.34084 9.34466 9.34847 9.35229 9.35610 .123305 .123153 .123001 .122850 .122699 .122549 .122399 .122249 .122100 8.20 67.2400 2.86356 9.05539 551.368 2.01653 4.34448 9.35990 .121951 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.28 8.29 67.4041 67.5684 67.7329 67.8976 68.0625 68.2276 68.3929 68.5584 68.7241 2.86531 2.86705 2.86880 2.87054 2.87228 2.87402 2.87576 2.87750 2.87924 9.06091 9.06642 9.07193 9.07744 9.08295 9.08845 9.09.395 9.09945 9.10494 553:388 555.412 557.442 559.476 561.516 563.560 565.609 567.664 569.723 2.01735 2.01817 2.01899 2.01980 2.02062 2.02144 2.02225 2.02307 2.02388 4.34625 4.34801 4.34977 4.35153 4.35329 4.35505 4.35681 4.35856 4.36032 9.36370 9.36751 9.37130 9.37510 9.37889 9.38268 9.38646 9.39024 9.39402 .121803 .121655 .121507 .121359 .121212 .121065 .120919 .120773 .120627 8.30 68.8900 2.88097 9.11043 571.787 2.02469 4.36207 9.39780 .120482 8.31 8.32 8.33 8.34 8.35 8.36 8.37 8.38 8.39 69.0561 69.2224 69.3889 69.5556 69.7225 69.8896 70.0569 70.2244 70.3921 2.88271 2.88444 2.88617 2.88791 2.88964 2.89137 2.89310 2.89482 2.8f)655 9.11592 9.12140 9.12688 9.13236 9.13783 9.14330 9.14877 9.15423 9.15969 573.856 575.930 578.010 580.094 582.183 584.277 586.376 588.480 590.590 2.02551 2.02632 2.02713 2.02794 2.02875 2.02956 2.03037 2.03118 2.03199 4.36382 4.36557 4.36732 4.36907 4.37081 4.37256 4.37430 4.37604 4.37778 9.40157 9.40534 9.40911 9.41287 9.41663 9.42039 9.42414 9.42789 9.43164 .120337 .120192 .120048 .119904 .119760 .119617 .119474 .119332 .119190 8.40 70.5600 2.89828 9.16515 592.704 2.03279 4.37952 9.43539 .119048 8.41 8.42 8.43 8.44 8.45 8.46 8.47 8.48 8.49 70.7281 70.8964 71.0649 71.2336 71.4025 71.5716 71.7409 71.9104 72.0801 2.90000 2.90172 2.90345 2.90517 2.90689 2.90861 2.91033 2.91204 2.91376 9.17061 9.17606 9.18150 9.18695 9.19239 9.19783 9.20326 9.20869 9.21412 594.823 596.948 599.077 601.212 603.351 605.496 607.645 609.800 611.960 2.03360 2.03440 2.03521 2.03601 2.03682 2.03762 2.03842 2.03923 2.04003 4.38126 4.38299 4.38473 4.38646 4.38819 4.38992 4.39165 4.39338 4.39510 9.43913 9.44287 9.44661 9.45034 9.45407 9.45780 9.46152 9.46525 9.46897 .118906 .118765 .118624 .118483 .118343 .118203 .118064 .117925 .117786 8.50 72.2500 2.91548 9.21954 614.125 2.04083 4.39683 9.47268 .117647 n n^ Vn n^ ^ ^10^ l/n VlOn ^100 n %rl] Powers — Roots — Reciprocals 109 n n^ Vn VlOn n^ 9.084 U 9.307 1 9.489 9.108 9.327 9.506 9.132 9.346 9.522 9.156 9.365 9.538 9.179 9.384 9.554 9.201 9.402 9.569 9.223 9.420 9.584 9.245 9.438 9.600 9.266 9.455 9.614 9.287 9.472 9.629 0.7 0.8 0.9 f 9.643 -^ 9.777 H 9.895 9.658 9.789 9.906 9.671 9.802 9.917 9.685 9.814 9.927 9.699 9.826 9.938 9.712 9.837 9.949 9.726 9.849 9.959 9.739 9.861 9.970 9.752 9.872 9.980 9.764 9.883 9.990 1.0 0.00000 0995 1980 2956 3922 4879 5827 6766 7696 8618 1.1 1.2 1.3 9531 0.1 8232 0.2 6236 *0436 9062 7003 *1333 9885 7763 *2222 *0701 8518 *3103 *1511 9267 *3976 *2314 *0010 *4842 *3111 *0748 *5700 *3t)02 *1481 *6551 *4686 *2208 *7395 *5464 *2930 1.4 1.5 1.6 0.3 3647 0.4 0547 7000 4359 1211 7623 5066 1871 8243 5767 2527 8858 6464 3178 9470 7156 3825 *0078 7844 4469 *0682 8526 5108 *1282 9204 5742 *1879 9878 6373 *2473 1.7 1.8 1.9 0.5 3063 8779 0.6 4185 3649 9333 4710 4232 9884 5233 4812 *0432 5752 5389 *0977 6269 5962 *1519 6783 6531 *2058 7294 7098 *2594 7803 7661 *3127 8310 8222 *3658 8813 2.0 9315 9813 *0310 *0804 *1295 *1784 *2271 *2755 *3237 *371G 2.1 2.2 2.3 0.7 4194 8846 0.8 3291 4669 9299 3725 5142 9751 4157 5612 *0200 4587 6081 *0648 5015 6547 *1093 5442 7011 *1536 5866 7473 *1978 6289 7932 *2418 6710 8390 *2855 7129 2.4 2.5 2.6 7547 0.9 1629 5551 7963 2028 5935 8377 2426 6317 8789 2822 6698 9200 3216 7078 9609 3609 7456 *0016 4001 7833 *0422 4391 8208 *0826 4779 8582 *1228 5166 8954 2.7 2.8 2.9 9325 1.0 2962 6471 9695 3318 6815 *0063 3674 7158 *0430 4028 7500 *0796 4380 7841 *1160 4732 8181 *1523 5082 8519 *1885 5431 8856 *2245 5779 9192 *2604 6126 9527 3.0 9861 *0194 *0526 *0856 *1186 *1514 *1841 *2168 *2493 *2817 3.1 3.2 3.3 1.1 3140 6315 9392 3462 6627 9695 3783 6938 9996 4103 7248 *0297 4422 7557 *0597 4740 7865 *0896 5057 8173 *1194 5373 8479 *1491 5688 8784 *1788 6002 9089 *2083 3.4 3.5 3.6 1.2 2378 5276 8093 2671 5562 8371 2964 5846 8647 3256 6130 8923 3547 6413 9198 3837 6695 9473 4127 6976 9746 4415 7257 *0019 4703 7536 *0291 4990 7815 *0563 3.7 3.8 3.9 1.3 0833 3500 6098 1103 3763 6354 1372 4025 6609 1641 4286 6864 1909 4547 7118 2176 4807 7372 2442 5067 7624 2708 5325 7877 2972 5584 8128 3237 5841 8379 4.0 8629 8879 9128 9377 9624 9872 *0118 *0364 *0610 *0854 4.1 4.2 4.3 1.4 1099 3508 5862 1342 3746 6094 1585 3984 6326 1828 4220 6557 2070 4456 6787 2311 4692 7018 2552 4927 7247 2792 5161 7476 3031 5395 7705 3270 5629 7933 4.4 4.5 4.6 8160 1.5 0408 2606 8387 0630 2823 8614 0851 3039 8840 1072 3256 9065 1293 3471 9290 1513 3687 9515 1732 3902 9739 1951 4116 9962 2170 4330 *0185 2388 4543 4.7 4.8 4.9 4756 6862 8924 4969 7070 9127 5181 7277 9331 5393 7485 9534 5604 7691 9737 5814 7898 9939 6025 8104 *0141 6235 8309 *0342 6444 8515 *0543 6653 8719 *0744 5.0 1.6 0944 1144 1343 1542 1741 1939 2137 2334 2531 2728 N 1 2 3 4 5 6 7 8 9 vir\ Napierian or Natural Logarithms 113 N 1 2 3 4 5 6 7 8 9 5.0 1.6 0944 1144 1343 1542 1741 1939 2137 2334 2531 2728 5.1 5.2 5.3 2924 4866 6771 3120 5058 6959 3315 5250 7147 3511 5441 7335 3705 6632 7523 3900 5823 7710 4094 6013 7896 4287 6203 8083 4481 6393 8269 4673 6582 8455 5.4 5.5 5.6 8640 1.7 0475 2277 8825 0;)56 2455 9010 0838 2633 9194 1019 2811 9378 1199 2988 9562 1380 3166 9745 1560 3342 9928 1740 3519 *0111 1919 3695 *0293 2098 3871 5.7 5.8 5.9 4047 5786 7495 4222 5958 7665 4397 6130 7834 4572 6302 8002 4746 6473 8171 4920 6644 8339 6094 6815 8507 6267 6985 8675 6440 7156 8842 6613 7326 9009 6.0 9176 9342 9509 9(575 . 9840 *0006 *0171 *0336 *0500 *0665 6.1 6.2 6.3 1.80829 2455 4055 0993 2616 4214 1156 2777 4372 1319 2938 4530 1482 3098 4688 1645 3258 4845 1808 3418 5003 1970 3578 5160 2132 3737 5317 2294 3896 5473 6.4 6.5 6.6 5630 7180 8707 5786 7334 8858 5942 7487 9010 6097 7641 9160 6253 7794 9311 6408 7947 9462 6563 8099 9612 6718 8251 9762 6872 8403 9912 7026 8556 *0061 6.7 6.8 6.9 7.0 1.90211 1692 3152 0360 1839 3297 0509 1986 3442 0658 2132 3586 0806 2279 3730 0954 2425 3874 1102 2571 4018 1250 2716 4162 1398 2862 4305 1645 3007 4448 4501 4734 4876 5019 5161 5303 5445 5586 5727 6869 7.1 7.2 7.3 6009 7408 8787 6150 7547 8924 6291 7685 9061 6431 7824 9198 6571 7962 9334 6711 8100 9470 6851 8238 9606 6991 8376 9742 7130 8513 9877 7269 8650 *0013 7.4 7.5 7.6 2.00148 1490 2815 0283 1624 2946 0418 1757 3078 0553 1890 3209 0687 2022 3340 0821 2155 3471 0956 2287 3601 1089 2419 3732 1223 2551 3862 1367 2683 3992 7.7 7.8 7.9 4122 5412 6686 4252 5540 6813 4381 5668 6939 4511 5796 7065 4640 5924 7191 4769 6051 7317 4898 6179 7443 5027 6306 7568 5156 6433 7694 6284 6560 7819 8.0 7944 8069 8194 8318 8443 8567 8691 8815 8939 9063 8.1 8.2 8.3 9186 2.10413 1626 9310 0535 1746 9433 0657 1866 9556 0779 1986 9679 0900 2106 9802 1021 2226 9924 1142 2346 *0047 1263 2465 *0169 1384 2585 *0291 1505 2704 8.4 8.5 8.6 2823 4007 5176 2942 4124 5292 3061 4242 5409 3180 4359 5524 3298 4476 5640 3417 4593 5756 3535 4710 5871 3653 4827 5987 3771 4943 6102 3889 5060 6217 8.7 8.8 8.9 6332 7475 8605 6447 7589 8717 6562 7702 8830 6677 7816 8942 6791 7929 9054 6905 8042 9165 7020 8155 9277 7134 8267 9389 7248 8380 9500 7361 8493 9611 9.0 9722 9834 9944 *0055 *0166 *0276 *0387 *0497 *0607 *0717 9.1 9.2 9.3 2.2 0827 1920 3001 0937 2029 3109 1047 2138 3216 1157 2246 3324 1266 2354 3431 1375 2462 3538 1485 2570 3645 1594 2678 3751 1703 2786 3858 1812 2894 3965 9.4 9.5 9.6 4071 6129 6176 4177 5234 6280 4284 5339 6384 4390 5444 6488 4496 5549 6592 4601 5654 6696 4707 5759 6799 4813 5863 6903 4918 5968 7006 6024 6072 7109 9.7 9.8 9.9 7213 8238 9253 7316 8340 9354 7419 8442 9455 7521 8544 9556 7624 8646 9657 7727 8747 9757 7829 8849 9858 7932 8950 9958 8034 9051 *0058 8136 9152 *0158 10.0 2.30259 0358 0458 0558 0658 0757 0857 0956 1055 1154 N 1 2 3 4 5 6 7 8 9 114 Napierian or Natural Logarithms — -10 to 99 [VII 10 2.30259 25 3.21888 40 3.68888 55 4.00733 70 4.24850 85 4.44265 11 12 13 14 2.39790 2.48491 2.56495 2.63906 26 27 28 29 3.25810 3.29584 3.33220 3.36730 41 42 43 44 3.71357 3.73767 3.76120 3.78419 56 57 68 59 4.02535 4.04305 4.06044 4.07764 71 72 73 74 4.26268 4.27667 4.29046 4.30107 86 87 88 89 4.45435 4.46591 4.47734 4.48864 4.49981 15 2.70805 30 3.40120 45 3.80666 60 4.09434 75 4.31749 90 16 17 18 19 2.77259 2.83321 2.89037 2.94444 31 32 33 34 3.43399 3.46574 3.49651 3.5263e 46 47 48 49 3.82864 3.85015 3.87120 3.89182 61 62 63 64 4.11087 4.12713 4.14313 4.15888 76 77 78 79 4.33073 4.34381 4.35671 4.36945 91 92 93 94 4.51086 4.52179 4.53260 4.54329 20 2.99573 35 3.55535 50 3.91202 65 4.17439 80 4.38203 95 4.55388 21 22 23 24 3.04452 3.09104 3.1:3549 3.17805 36 37 38 39 3.58352 3.61092 3.63759 3.66356 51 52 63 54 3.93183 3.95124 3.97029 3.98898 6ij 67 68 69 4.18965 4.20469 4.21951 4.23411 81 82 83 84 4.39445 4.40672 4.41884 4.43082 96 97 98 99 4.56435 4.57471 4.68497 4.69512 NAPIERIAN ( OR NATURAL LOGARITHMS — 100 TO 409 N 1 2 3 4 5 6 7 8 9 10 4.6 0517 1612 2497 3473 4439 6396 6344 7283 8213 9135 11 12 13 4.7 0048 8749 4.8 6753 0953 9579 7520 1850 *0402 8280 2739 *1218 9036 3620 *2028 9784 4493 *2831 *0527 5359 *3628 *1265 6217 *4419 *1998 7068 *5203 *2725 7912 *6981 *3447 14 15 16 4.9 4164 5.0 1064 7517 4876 1728 8140 6683 2388 8760 6284 3044 9376 6981 3695 9987 7673 4343 *0695 8361 4986 *1199 9043 5625 *1799 9721 6260 *2396 *0395 6890 *2990 17 18 19 5.1 3580 9296 5.2 4702 4166 9860 6227 4749 *0401 6760 6329 *0949 6269 6906 *1494 6786 6479 *2036 7300 7048 *2575 7811 7616 *3111 8320 8178 *3644 8827 8739 *4176 9330 20 9832 *0330 *0827 *1321 *1812 *2301 *2788 *3272 *3754 *4233 21 22 23 6.34711 9363 5.4 3808 6186 9816 4242 6659 *0268 4674 6129 *0717 6104 6698 *1165 6532 7064 *1610 6959 7628 *2053 6383 7990 *2495 6806 8450 *2935 7227 8907 *3372 7646 24 25 26 8064 5.5 2146 6068 8480 2545 6452 8894 2943 6834 9306 3339 7215 9717 3733 7595 *0126 4126 7973 *0533 4518 8350 *0939 4908 8726 -1343 6296 9099 *1745 5683 9471 27 28 29 9842 5.6 3479 6988 *0212 3835 7332 *0680 4191 7675 *0947 4545 8017 *1313 4897 8358 *1677 6249 8698 *2040 5599 9036 *2402 5948 9373 *2762 6296 9709 *3121 6643 *0044 30 5.7 0378 0711 1043 1373 1703 2031 2359 2685 3010 3334 31 32 33 3667 6832 9909 3979 7144 *0212 4300 7455 *0513 4620 7765 *0814 4939 8074 *1114 5267 8383 *1413 6574 8690 *1711 68<)0 89f)6 *2008 6206 9301 *2305 6519 9606 *2600 34 35 36 5.8 2895 6793 8610 3188 6079 8888 3481 6363 9164 3773 6647 9440 4064 6930 9715 4354 7212 9990 4644 7493 *0263 4932 7774 *0636 6220 8053 *0808 6607 8332 *1080 37 38 39 5.9 1360 4017 6616 1620 4280 6871 1889 4542 7126 2158 4803 7381 2426 6064 7636 2693 5324 7889 2969 6584 8141 3226 6842 8394 3489 6101 8645 3764 6368 8896 40 9146 9396 9646 9894 *0141 *0389 *0635 *0881 *1127 *1372 N 1 2 3 4 5 6 7 8 9 Above 409, use the formula log* 10 ?i = loge n + loge 10 = loge n + 2.30268509, or the formula log* n = loge 10 • logio ^ = 2.30258509 logio n. Table VIII — Multiples of M and of 1/M 115 N N'M N N'M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 0.00000 000 50 21.71472 410 0.43429 448 0.86858 896 1.30288 345 1.73717 793 2.17147 241 2.60576 689 3.04006 137 3.47435 586 3.90865 034 51 52 53 54 55 50 57 58 59 22.14901 858 22.58331 306 23.01760 754 23.45190 202 23.88619 650 24.32049 099 24.75478 547 25.18907 995 25.62337 443 4.34294 482 60 26.05766 891 4.77723 930 5.21153 378 5.64582 826 6.08012 275 6.51441 723 6.94871 171 7.38300 619 7.81730 067 8.25159 516 61 62 63 64 65 66 67 68 69 26.49196 340 26.92625 788 27.36055 236 27.79484 684 28.22914 132 28.66343 581 29.09773 029 29.53202 477 29.96631 925 8.68588 964 70 30.40061 373 9.12018 412 9.55447 860 9.98877 308 10.42306 757 10.85736 205 11.29165 653 11.72595 101 12.16024 549 12.59453 998 71 72 73 74 75 76 77 78 79 30.83490 822 31.26920 270 31.70349 718 32.13779 166 32.57208 614 33.00638 062 38.44067 511 33.87496 959 34.30926 407 13.02883 446 80 34.74355 855 13.46312 894 13.89742 342 14.33171 790 14.76601 238 15.20030 687 15.63460 135 16.06889 583 16.50319 031 16.93748 479 81 82 83 84 85 86 87 88 89 35.17785 303 35.61214 752 36.04644 200 36.48073 648 36.91503 096 37.31932 644 37.78361 993 38.21791 441 38.65220 889 40 41 42 43 44 45 46 47 48 49 17.37177 928 90 39.08650 337 17.80607 376 18.24036 824 18.67466 272 19.10895 720 19.54325 169 19.97754 617 20.41184 065 20.84613 513 21.28042 961 91 92 93 94 95 96 97 98 99 39.52079 785 39.95509 234 40.38938 682 40.82368 130 41.25797 578 41.69227 026 42.12656 474 42.56085 923 42.99515 371 60 21.71472 410 100 43.42944 819 A^ N-i-M N N-^M 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 0.00000 000 50 115.12925 465 2.30258 509 4.60517 019 6.90775 528 9.21034 037 11.51292 546 13.81551 056 16.11809 565 18.42068 074 20.72326 584 51 52 63 54 65 66 57 58 59 117.43183 974 119.73442 481 122.03700 993 124.33959 502 126.64218 Oil 128.94476 621 131.24735 030 133.54993 639 135.85252 049 23.02585 093 60 138.15510 558 25.32843 602 27.63102 112 29.93360 621 32.23619 130 34.53877 639 36.84136 149 39.14394 658 41.44653 167 43.74911 677 61 62 63 64 65 66 67 68 69 140.45769 067 142.76027 577 145.06286 086 147.36544 595 149.66803 104 151.97061 614 154.27320 123 156,57578 632 158.87837 142 46.05170 186 70 161.18095 651 48.35428 695 50.65687 205 62.95945 714 65.26204 223 67.56462 732 59.86721 242 62.16979 751 64.47238 260 66.77496 770 71 72 73 74 75 76 77 78 79 163.48354 160 165.78612 670 168.08871 179 170.39129 688 172.69388 197 174.99646 707 177.29905 216 179.60163 725 181.90422 235 69.07755 279 80 184.20680 744 71.38013 788 73.68272 298 75.98530 807 78.28789 316 80.59047 825 82.89306 335 85.19564 844 87.49823 353 89.80081 863 81 82 83 84 85 86 87 88 89 186.50939 253 188.81197 763 191.11456 272 193.41714 781 195.71973 290 198.02231 800 200.32490 309 202.62748 818 204.93007 328 40 41 42 43 44 45 46 47 48 49 92.10340 372 90 207.23265 837 91.40598 881 96.70857 391 99.01115 900 101.31374 409 103.61632 918 105.91891 428 108.22149 937 110.52408 446 112.82666 956 91 92 93 94 95 96 97 98 99 209.53524 346 211.83782 856 214.14041 365 216.44299 874 218.74558 383 221.04816 893 223.35075 402 225.65333 911 227.95592 421 50 115.12925 465 100 230.25850 930 M= log^^e = .43429 44819 03261 82765 -^ = log 10 = 2.30258 50929 94045 68402 if ^« ^ log^n = log^^n . log^lO = j^ los,,n. 116 Table IX — Logarithms of Hyperbolic Functions a? Value Logio 6- Value Sinhcc Value Logio Coshac Value Logio Tanhi» Value 0.00 1.0000 .00000 1.0000 0.0000 — 00 1.0000 .00000 .00000 0.01 0.02 0.03 1.0101 1.0202 1.0305 .00434 .00869 .01303 .99005 .98020 .97045 0.0100 0.0200 0.0300 .00001 .30106 .47719 1.0001 1.0002 1.0005 .00002 .00009 .00020 .01000 .02000 .02999 0.04 0.05 0.06 1.0408 1.0513 1.0618 .01737 .02171 .02606 .96079 .95123 .94176 0.0400 0.0500 00600 .60218 .69915 .77841 1.0008 1.0013 1.0018 .00035 .00054 .00078 .03998 .04996 .05993 0.07 0.08 0.09 1.0725 1.0833 1.0942 .03040 .03474 .03909 .93239 .92312 .91393 0.0701 0.0801 0.0901 .84545 .90355 .95483 1.0025 1.0032 1.0041 .00106 .00139 .00176 .06989 .07983 .08976 0.10 1.1052 .04343 .90484 0.1002 .00072 1.0050 .00217 .09967 0.11 0.12 0.13 1.1163 1.1275 1.1388 .04777 .05212 .05646 .89583 .88692 .87809 0.1102 0.1203 0.1304 .04227 .08022 .11517 1.0061 1.0072 1.0085 .00262 .00312 .00366 .10956 .11943 .12927 0.14 0.15 0.16 1.1503 1.1618 1.1735 .06080 .06514 .06949 .86936 .86071 .85214 0.1405 0.1506 0.1607 .14755 .17772 .20597 1.0098 1.0113 1.0128 .00424 .00487 .00554 .13909 .14889 .15865 0.17 0.18 0.19 1.1853 1.1972 1.2092 .07383 .07817 .08252 .84366 .83527 .82696 0.1708 0.1810 0.1911 .23254 .25762 .28136 1.0145 1.0162 1.0181 .00625 .00700 .00779 .16838 .17808 .18775 0.20 1.2214 .08686 .81873 0.2013 .30392 1.0201 ,00863 .19738 0.21 0.22 0.23 1.2337 1.2461 1.2586 .09120 .09554 .09989 .81058 .80252 .79453 0.2115 0.2218 0.2320 .32541 .34592 .36555 1.0221 1.0243 1.0266 .00951 .01043 .01139 .20697 .21652 .22603 0.24 0.25 0.26 1.2712 1.2840 1.2969 .10423 .10857 .11292 .78663 .77880 .77105 0.2423 0.2526 0.2629 .38437 .40245 .41986 1.0289 1.0314 1.0340 .01239 .01343 .01452 .23550 .24492 .25430 0.27 0.28 0.29 1.3100 1.3231 1.3364 .11726 .12160 .12595 .76338 .75578 .74826 0.2733 0.2837 0.2941 .43663 .45282 .46847 1.0367 1.0395 1.0423 .01564 .01681 .01801 .26362 .27291 .28213 0.30 1.3499 .13029 .74082 0.3045 .48362 1.0453 .01926 .29131 0.31 0.32 0.33 1.3634 1.3771 1.3910 .13463 .13897 .14332 .73345 .72015 .71892 0.3150 0.3255 0.3360 .49830 .51254 .52637 1.0484 1.0516 1.0549 .02054 .02107 .02323 .30044 .30951 .31852 0.34 0.35 0.36 1.4049 1.4191 1.4333 .14766 .15200 .15635 .71177 .70469 .69768 0.3466 0.3572 0.3678 .53981 .55290 .56564 1.0584 1.0619 1.0655 .02463 .02607 .02755 .32748 .33638 .34521 0.37 0.38 0-39 1.4477 1.4623 1.4770 .16069 .16503 .16937 .69073 .68386 .67706 .67032 0.3785 0.3892 0.4000 .57807 .59019 .60202 1.0692 1.0731 1.0770 .02907 .03063 .03222 .35399 .36271 .37136 0.40 1.4918 .17372 0.4108 .61358 1.0811 .03385 .37995 0.41 0.42 0.43 1.5068 1.5220 1.5373 .17806 .18240 .18675 .66365 .65705 .65051 0.4216 0.4325 0.4434 .62488 .63594 .64677 1.0852 1.0895 1.0939 .03552 .03723 .03897 .38847 .39693 .40532 0.44 0.45 0.46 1.5527 1.5683 1.5841 .19109 .19543 .19978 .64404 .63763 .63128 0.4543 0.4653 0.4764 .65738 .66777 .67797 1.0984 1.1030 1.1077 .04075 .04256 .04441 .41364 .42190 .43008 0.47 0.48 0.49 1.6000 1.6161 1.6323 .20412 .20846 .21280 .62500 .61878 .61263 0.4875 0.4986 0.5098 .68797 .69779 .70744 1.1125 1.1174 1:1225 .04630 .04822 .05018 .43820 .44624 .45422 050 1.6487 .21715 .60653 0.5211 .71692 1.1276 .05217 .46212 Values and Logarithms of Hyperbolic Functions 117 X Value Log.o Value Sin}] Value I a? Logio Cosl Value Logio Tanha? Value 0.50 1.6487 .21715 .60653 0.5211 .71692 1.1276 .05217 .46212 .46995 .47770 .48538 0.51 0.52 0.53 1.6653 16820 1.6989 .22149 .22583 .23018 .60050 .59452 .58860 0.5324 0.5438 0.5552 .72624 .73540 .74442 1.1329 1.1383 1.1438 .05419 .05625 .05834 0.54 0.55 0.56 1.7160 1.7333 1.7507 .23452 .23886 .24320 .58275 .57695 .57121 0.5666 0.5782 0.5897 ,75330 .76204 .77065 1.1494 1.1551 1.1609 .06046 .06262 .06481 .49299 .50052 .50798 0.57 0.58 C.59 1.7683 1-7860 1.8040 .24755 .25189 .25623 .56553 .55990 .55433 0.6014 0.6131 0.6248 .77914 .78751 .79576 1.1669 1.1730 1.1792 .06703 .06929 .07157 .51536 .52267 .52990 0.60 1.8221 .26058 .54881 0.6367 .80390 1.1855 .07389 .53705 0.61 0.62 0.63 1.8404 1.8589 1.8776 .26492 .26926 .27361 .54335 .53794 .53259 0.6485 0.6605 0.6725 .81194 .81987 .82770 1.1919 1.1984 1.2051 .07624 .07861 .08102 .54413 .55113 .55805 0.64 0.65 0.66 1.8965 1.9155 1.9348 .27795 .28229 .28664 .52729 .52205 .51685 0.6846 0.6967 0.7090 .83543 .84308 .85063 1.2119 1.2188 1.2258 .08346 .08593 .08843 .56490 .57167 .57836 0.67 0.68 0.69 1.9542 1.9739 1.9937 .29098 .29532 .29966 .51171 .50662 .50158 0.7213 0.7336 0.7461 .85809 .86548 .87278 1.2330 1.2402 1.2476 .09095 .09351 .09609 .58498 .59152 .59798 0.70 2.0138 .30401 .49659 0.7586 .88000 1.2552 .09870 .60437 0.71 0.72 0.73 2.0340 2.0544 2.0751 .30835 .31269 .31703 .49164 .48675 .48191 0.7712 0.7838 0.7966 .88715 .89423 .90123 1.2628 1.2706 1.2785 .10134 .10401 .10670 .61068 .61691 .62307 0.74 0.75 0.76 2.0959 2.1170 2.1383 .32138 .32572 .33006 .47711 .47237 .46767 0.8094 0.8223 0.8353 .90817 .91504 .92185 1.2865 1.2947 1.3030 • .10942 .11216 .11493 .62915 .63515 .64108 0.77 0.78 0.79 2.1598 2.1815 2.2034 .33441 ..33875 .34309 .46301 .45841 .45384 0.8484 0.8615 0.8748 .92859 .93527 .94190 1.3114 1.3199 1.3286 .11773 [ .12055 .12340 .64693 .65271 .65841 0.80 2.2255 ..34744 .44933 0.8881 .94846 1.3374 .12627 .66404 0.81 0.82 0.83 2.2479 2.2705 2.2933 ..35178 .35612 .36046 .44486 .44043 .43605 0.9015 0.9150 0.9286 .95498 .96144 .96784 1.3464 1.3555 1.3647 .12917 .13209 .13503 .66959 .67507 .68048 0.84 0.85 0.86 2.3164 2.3396 2.3632 .36481 .36915 .37349 .43171 .42741 .42316 0.9423 0.9561 0.9700 .97420 .98051 .98677 1.3740 1.3835 1.3932 .13800 .14099 .14400 .68581 .69107 .69626 0.87 0.88 0.89 2.3869 2.4109 2.4351 .37784 .38218 .38652 .41895 .41478 .41066 0.9840 0.9981 1.0122 .99299 .99916 .00528 1.4029 1.4128 1.4229 .14704 .15009 .15317 .70137 .70642 .71139 .71630 0.90 2.4596 .39087 .40657 1.0265 .01137 1.4331 .15627 0.91 0.92 0.93 2.4843 2.5093 2.5345 .39521 .39955 .40389 .40252 .39852 .39455 1.0409 1.0554 1.0700 .01741 .02341 .02937 1.4434 1.4539 1.4645 .15939 .16254 .16570 .72113 .72590 .73059 0.94 0.95 0.96 2.5600 2.5857 2.6117 .40824 .41258 .41692 .39063 .38674 .38289 1.0847 1.0995 1.1144 .03530 .01119 .04704 1.4753 1.4862 1.4973 .16888 .17208 .17531 .73522 .73978 .74428 0.97 0.98 0.99 2.6.379 2.6645 2.6912 .42127 .42561 .42995 .37908 .37531 .37158 1.1294 1.1446 1.1598 .05286 .05864 .06439 1.5085 1.5199 1.5314 1.5431 .17855 .18181 .18509 .18839 .74870 .75307 .75736 1.00 2.7183 .43429 .36788 1.1752 .07011 .76159 118 Yalues and Logarithms of Hyperbolic Functions 00 Value Logio Value Value Logio CoshiK Value Logio Tanha; Value 1.00 2.7183 .43429 .36788 1.1752 .07011 1.5431 .18839 .76159 1.01 1.02 1.03 2.7456 2.7732 2.8011 .43864 .44298 .44732 .36422 .36060 .35701 1.1907 1.2063 1.2220 .07580 .08146 .08708 1.5549 1.5669 1.5790 .19171 .19504 .19839 .76576 .76987 .77391 1.04 1.05 1.06 2.8292 2.8577 2.8864 .45167 .45601 .46035 .35345 .34994 .34646 1.2379 1.2539 1.2700 .09268 .09825 .10379 1.5913 1.6038 1.6164 .20176 .20515 .20855 .77789 .78181 .78566 1.07 1.08 1.09 1.10 2.9154 2.9447 2.9743 .46470 .46904 .47338 .34301 .339(i0 .33622 1.2862 1.3025 1.3190 .10930 .11479 .12025 1.6292 1.6421 1.6552 .21197 .21541 .21886 .78946 .79320 .79688 3.0042 .47772 .33287 1.3356 .12569 1.6685 .22233 .80050 1.11 1.12 1.13 3.0344 3.0649 3.0957 .48207 .48641 .49075 .32956 .32628 .32303 1.3524 1.3693 1.3863 .13111 .13649 .14186 1.6820 1.6956 1.7093 .22582 .22931 .23283 .80406 .80757 .81102 1.14 1.15 1.16 3.1268 3.1582 3.1899 .49510 .49944 .50378 .31982 .31664 .31349 1.4035 1.4208 1.4382 .14720 .15253 .15783 1.7233 1.7374 1.7517 .23636 .23990 .24346 .81441 .81775 .82104 1.17 1.18 1.19 3.2220 3.2544 3.2871 .50812 .51247 .51681 .31037 .30728 .30422 1.4558 1.4735 1.4914 .16311 .1683() .17360 1.7662 1.7808 1.7957 .24703 .25062 .25422 .82427 .82745 .83058 1.20 3.3201 .52115 .30119 1.5085 .17882 1.8107 .25784 .83365 1.21 1.22 1.23 3.3535 3.3872 3.4212 .52550 .52984 .58418 .29820 .29523 .29229 1.5276 1.5460 1.5645 .18402 .18920 .19437 1.8258 1.8412 1.8568 .26146 .26510 .26876 .83668 .83965 .84258 1.24 1.25 1.26 3.4556 3.4903 3.5254 .53853 .54287 .54721 .28938 .28650 .28365 1.5831 1.6019 1.6209 .19951 .20464 .20975 1.8725 1.8884 1.9045 .27242 .27610 .27979 .84546 .84828 .85106 1.27 1.28 1.29 3.5609 3.5966 3.6328 .55155 .55590 .56024 .28083 .27804 .27527 1.6400 1.6593 1.6788 .21485 .21993 .22499 1.9208 1.9373 1.9540 .28349 .28721 .29093 .85380 .85648 .85913 1.30 3.6693 .56458 .27253 1.6984 .23004 1.9709 .29467 .86172 1.31 1.32 1.33 3.7062 3.7434 3.7810 .56893 .57327 .57761 .26982 .26714 .26448 1.7182 1.7381 1.7583 .23507 .24009 .24509 1.9880 2.0053 2.0228 .29842 .30217 .30594 .86428 .86678 .86925 1.34 1.35 1.36 3.8190 3.8574 3.8962 .58195 .58630 .59064 .26185 .25924 .25666 1.7786 1.7991 1.8198 .25008 .25505 .26002 2.0404 2.0583 2.0764 .30972 .31352 .31732 .87167 .87405 .87639 1.37 1.38 1.39 3.9354 3.9749 4.0149 .59498 .59933 .60367 .25411 .25158 .24908 1.8406 1.8617 1.8829 .26496 .26990 .27482 2.0947 2.1132 2.1320 .32113 .32495 .32878 .87869 .88095 .88317 1.40 4.0552 .60801 .24660 1.9043 .27974 2.1509 .33262 .88535 1.41 1.42 1.43 4.0960 4.1371 4.1787 .61236 .61670 .62104 .24414 .24171 .23931 1.9259 1.9477 1.9697 .28464 .28952 .29440 2.1700 2.1894 2.2090 .33647 .34033 .34420 .88749 .88960 .89167 1.44 1.45 1.46 4.2207 4.2631 4.3060 .62538 .62973 .63407 .23693 .23457 .23224 1.9919 2.0143 2.0369 .29926 .30412 .30896 2.2288 2.2488 2.2691 .34807 .35196 .35585 .89370 .89569 .89765 1.47 1.48 1.49 4.3492 4.3929 4.4371 .63841 .64276 .64710 .22993 .22764 .22537 2.0597 2.0827 2.1059 .31379 .31862 .32343 2.2896 2.3103 2.3312 .35976 .36367 .36759 .89958 .90147 .90332 1.50 4.4817 .65144 .22313 2,1293 .32823 2.3524 .37151 .90515 Values and Logarithms of Hyperbolic Functions 119 oc e Value Value Sinha? Value Logio Cosho? Value Logio TanhiT Value 1.50 4.4817 .65144 .22313 2.1293 .32823 2.3524 .37151 .90515 1.51 1.52 1.53 4.5267 4.5722 4.6182 .65578 .66013 .66447 .22091 .21871 .21654 2.1529 2.1768 2.2008 .33303 .33781 .34258 2.3738 2.3955 2.4174 .37545 .37939 .38334 .90694 .90870 .91042 1.54 1.55 1.56 4.6646 4.7115 4.7588 .66881 .67316 .67750 .21438 .21225 .21014 2.2251 2.2496 2.2743 .34735 .35211 .35686 2.4395 2.4619 2.4845 .38730 .39126 .39524 .91212 .91379 .91542 1.57 1.58 1.59 4.8066 4.8550 4.9037 .68184 .68619 .69053 .20805 .20598 .20393 2.2993 2.3245 2.3499 .36160 .36633 .37105 2.5073 2.5305 2.5538 .39921 .40320 .40719 .91703 .91860 .92015 1.60 4.9530 .69487 .20190 2.3756 .37577 2.5775 .41119 .92167 1.61 1.62 1.63 5.0028 5.0531 5.1039 .69921 .70356 .70790 .19989 .19790 .19593 2.4015 2.4276 2.4540 .38048 .38518 .38987 2.6013 2.6255 2.6499 .41520 .41921 .42323 .92316 .92462 .92606 1.64 1.65 1.66 5.1552 5.2070 5.2593 .71224 .71659 .72093 .19398 .19205 .19014 2.4806 2.5075 2.5345 .39456 .39923 .40391 2.6746 2.6995 2.7247 .42725 .43129 .43532 .92747 .92886 .93022 1.67 1.68 1.69 5.3122 5.3656 5.4195 .72527 .72961 .73396 .18825 .18637 .18452 2.5620 2.5896 2.6175 .40857 .41323 .41788 2.7502 2.7760 2.8020 .43937 .44341 .44747 .93155 .93286 .93415 1.70 5.4739 .73830 .18268 2.6456 .42253 2.8283 .45153 .93541 1.71 1.72 1.73 5.5290 5.5845 5.6407 .74264 .74699 .75133 .18087 .17907 .17728 2.6740 2.7027 2.7317 .42717 .43180 .43643 2.8549 2.8818 2.9090 .45559 .45966 .46374 .93665 .93786 .93906 1.74 1.75 1.76 5.6973 5.7546 5.8124 .75567 .76002 .76436 .17552 .17377 .17204 2.7609 2.7904 2.8202 .44105 .44567 .45028 2.9364 2.9642 2.9922 .46782 .47191 .47600 .94023 .94138 .94250 1.77 1.78 1.79 5.8709 5.9299 5.9895 .76870 .77304 .77739 .17033 .16864 .16696 2.8503 2.8806 2.9112 .45488 .45948 .46408 3.0206 3.0492 3.0782 .48009 .48419 .48830 .94361 .94470 .94576 1.80 6.0496 .78173 .16530 2.9422 .46867 3.1075 .49241 .94681 1.81 1.82 1.83 6.1104 6.1719 6.2339 .78(507 .79042 .79476 .16365 .16203 .16041 2.9734 3.0049 3.0367 .47325 .47783 .48241 3.1371 3.1669 3.1972 .49652 .50064 .50476 .94783 .94884 .94983 1.84 1.85 1.86 6.2965 6.3598 6.4237 .79910 .80344 .80779 .15882 .15724 .15567 3.0689 3.1013 3.1340 .48698 .49154 .49610 3.2277 3.2585 3.2897 .50889 .51302 .51716 .9.5080 .95175 .95268 1.87 1.88 1.89 6.4883 6.5535 6.6194 .81213 .81647 .82082 .15412 .15259 .15107 3.1671 3.2005 3.2341 .50066 .50521 .50976 3.3212 3.3530 3.3852 .52130 .52544 .52959 .95359 .95449 .95537 1.90 6.6859 .82516 .14957 3.2682 .51430 3.4177 .53374 .95624 1.91 1.92 1.93 6.7531 6.8210 6.8895 .82950 .83385 .83819 .14808 .14661 .14515 3.3025 3.3372 3.3722 .51884 .52338 .52791 3.4506 3.4838 3.5173 .53789 .54205 .54621 .95709 .95792 .95873 1.94 1.95 1.96 6.9588 7.0287 7.0993 .84253 .84687 .85122 .14370 .14227 .14086 3.4075 3.4432 3.4792 .53244 .53696 .54148 3.5512 3.5855 3.6201 .55038 .55455 .55872 .95953 .96032 .96109 1.97 1.98 1.99 7.1707 7.2427 7.3155 .85556 .85990 .86425 .13946 .13807 .13670 3.5156 3.5523 3.58M .54600 .55051 .55502 3.6551 3.6904 3.7261 .56290 .56707 .57126 .96185 .96259 .96331 2.00 7.3891 .86859 .13534 3.6269 .55953 3.7622 .57544 .96403 120 Values and Logarithms of Hyperbolic Functions nc e Value X Value Sinhi» Value Logio Cosh a? Value Logio Tanhiz; Value 200. 7.3891 .86859 .13534 3.6269 .55953 3.7622 .57544 .96403 2.01 2.02 2.03 7.4633 7.5383 7.6141 .87293 .87727 .88162 .13399 .13266 .13134 3.6647 3.7028 3.7414 .56403 .56853 .57303 3.7987 3.8355 3.8727 .57963 .58382 .58802 .96473 .96541 .96609 2.04 2.05 2.06 7.6906 7.7679 7.8460 .88596 .89030 .89465 .13003 .12873 .12745 3.7803 3.8196 3.8593 .57753 .58202 .58650 3.9103 3.9483 3.9867 .59221 .59641 .60061 .96675 .96740 .96803 2.07 2.08 2.09 7.9248 80045 8.0849 .89899 .90333 .90768 .12619 .12493 .12369 3.8993 3.9398 3.9806 .59099 .59547 .59995 4.0255 4.0647 4.1043 .60482 .60903 .61324 .96865 .96926 .96986 2.10 8.1662 .91202 .12246 4.0219 .60443 4.1443 .61745 .97045 2.11 2.12 2.13 8.2482 8.3311 8.4149 .91636 .92070 .92505 .12124 .12003 .11884 4.0635 4.1056 4.1480 .60890 .61337 .61784 4.1847 4.2256 4.2669 .62167 .62589 .63011 .97103 .97159 .97215 2.14 2.15 2.16 8.4994 8.5849 8.6711 .92939 .93373 .93808 .11765 .11648 .11533 4.1909 4.2342 4.2779 .62231 .62677 .63123 4.3085 4.3507 4.3932 .63433 .63856 .64278 .97269 .97323 .97375 2.17 2.18 2.19 8.7583 8.8463 8.9352 .94242 .94676 .95110 .11418 .11304 .11192 4.3221 4.3666 4.4116 .63569 .64015 .64460 4.4362 4.4797 4.5236 .64701 .65125 .65548 .97426 .97477 .97526 2.20 9.0250 .95545 .11080 4.4571 .64905 4.5679 .65972 .97574 2.21 2.22 2.23 9.1157 9.2073 9.2999 .95979 .96413 .96848 .10970 .10861 .10753 4.5030 4.5494 4.5962 .65350 .65795 .66240 4.6127 4.6580 4.7037 .66396 .66820 .67244 .97622 .97668 .97714 2.24 2.25 2.26 9.3933 9.4877 9.5831 .97282 .97716 .98151 .10646 .10540 .10435 4.6434 4.6912 4.7394 .66684 .67128 .67572 4.7499 4.7966 4.8437 .67668 .68093 .68518 .97759 .97803 .97846 2.27 2.28 2.29 9.6794 9.7767 9.8749 .98585 .99019 .99453 .10331 .10228 .10127 4.7880 4.8372 4.8868 .68016 .68459 .68903 4.8914 4.9395 4.9881 .68943 .69368 .69794 .97888 .97929 .97970 2.30 9.9742 .99888 .10026 4.9370 .69346 5.0372 .70219 .98010 2.31 2.32 2.33 10.074 10.176 10.278 .00322 .00756 .01191 .09926 .09827 .09730 4.9876 5.0387 6.0903 .69789 .70232 .70675 5.0868 5.1370 5.1876 .70645 .71071 .71497 .98049 .98087 .98124 2.34 2.35 2.36 10.381 10.486 10.591 .01625 .02059 .02493 .09633 .09537 .09442 5.1425 5.1951 5.2483 .71117 .71559 .72002 5.2388 5.2905 5.3427 .71923 .72349 .72776 .98161 .98197 .98233 2.37 2.38 2.39 10.697 10.805 10.913 .02928 .03362 .03796 .09348 .09255 .09163 5.3020 5.3562 5.4109 .72444 .72885 .73327 5.3954 5.4487 5..W26 .73203 .73630 .74056 .98267 .98301 .98335 2 40 11.023 04231 .09072 5.4662 .73769 5.5569 .74484 .98367 2.41 2.42 2.43 11.134 11.246 11.359 .04665 .05099 .05534 .08982 .08892 .08804 5.5221 5.5785 5.6354 .74210 .74652 .75093 5.6119 5.6674 5.7235 .74911 .75338 .75766 .98400 .98431 .98462 2.44 2.45 2.46 11.473 11.588 11.705 .05968 .06402 .06836 .08716 .08629 .08543 5.6929 5.7510 5.8097 .75534 .75975 .76415 5.7801 5.8373 5.8951 .76194 .76621 .77049 .98492 .98522 .98551 2.47 2.48 2.49 11.822 11.941 12.061 .07271 .07705 .08139 .08458 .08374 .08291 5.8689 5.9288 5.9892 .76856 .77296 .77737 5.9535 6.0125 6.0721 .77477 .77906 .78334 .98579 .98607 .98635 2.50 12.182 .08574 .08208 6.0502 .78177 6.1323 .78762 .98661 Values and Logarithms of Hyperbolic Functions 121 0? Value Logio Value Sinha? Value Logio Cosh a; Value Logio Tanha? Value 2.60 12.182 .08574 .08208 6.0502 .78177 6.1323 .78762 .98661 2.51 2.52 2.53 12.305 12.429 12.554 .09008 .09442 .09877 .08127 .08046 .07966 61118 6.1741 6.2369 .78617 .79057 .79497 6.1931 6.2545 6.3166 .79191 .79619 .80048 .98688 .98714 .98739 2.54 2.55 2.56 12.680 12.807 12.936 .10311 .10745 .11179 .07887 .07808 .07730 6.3004 6.3645 6.4293 .79937 .80377 .80816 6.3793 6.4426 6.5066 .80477 .80906 .81335 .98764 .98788 .98812 2.57 2.58 2.59 13.0(J6 13.197 13.330 .11614 .12048 .12482 .07654 .07577 .07502 6.4946 6.5607 6.6274 6.6947 6.7628 6.8315 6.9008 .81256 .81695 .82134 .82573 .83012 .83451 .83890 6.5712 6.6365 6.7024 .81764 .82194 .82623 .98835 .98858 .98881 2.60 2.61 2.62 2.63 13.464 .12917 .07427 6.7690 6.8363 6.9043 6.9729 .83052 .83482 .83912 .84341 .98903 13.599 13.736 13.874 .13351 .13785 .14219 .07353 .07280 .07208 .98924 .98946 .98966 2.64 2.65 2.66 14.013 14.154 14.296 .14654 .15088 .15522 .07136 .07065 .06995 6.9709 7.0417 7.1132 .84329 .84768 .85206 7.0423 7.1123 7.1831 .84771 .85201 .85631 .98987 .99007 .99026 2.67 2.68 2.69 2.70 2.71 2.72 2*73 14.440 14.585 14.732 .15957 .16391 .16825 .06925 .06856 .06788 7.1854 7.2583 7.3319 .85645 .86083 .86522 7.2546 7.3268 7.3998 .86061 .86492 .86922 .99045 .99064 .99083 14.880 .17260 .06721 7.4063 • 7.4814 7.5572 7.6338 .86960 .87398 .87836 .88274 7.4735 7.5479 7.6231 7.6991 .87352 .87783 .88213 .88644 .99101 .99118 .99136 .99153 15.029 15.180 15.333 .17694 .18128 .18562 .06654 .06587 .06522 2.74 2.75 2.76 15.487 15.643 15.800 .18997 .19431 .19865 .06457 .06393 .06329 7.7112 7.7894 7.8683 .88712 .89150 .89588 7.7758 7.8533 7.9316 .89074 .89505 .89936 .99170 .99186 .99202 2.77 2.78 2.79 2.80 15.959 16.119 16.281 .20300 .20734 .21168 .06266 .06204 .06142 7.9480 8.0285 8.1098 .90026 .90463 .90^)01 8.0106 8.0905 8.1712 8.2527 .90367 .90798 .91229 .91660 .99218 .99233 .99248 16.445 .21602 .06081 8.1919 .91339 .99263 2.81 2.82 2.83 16.610 16.777 16.945 .22037 .22471 .22905 .06020 .05961 .05901 8.2749 8.3586 8.4432 .91776 .92213 .92651 8.3351 8.4182 8.5022 .92091 .92522 .92953 .99278 .99292 .99306 2.84 2.85 2.86 17.116 17.288 17.462 .23340 .23774 .24208 .05843 .05784 .05727 8.5287 8.6150 8.7021 .91^88 .93525 .93963 8.5871 8.6728 8.7594 .93385 .93816 .94247 .99320 .99333 .99346 2.87 2.88 2.89 2.90 17.637 17.814 17.993 .24643 .25077 .25511 .05670 .05613 .05558 8.7902 8.8791 8.9689 9.0596 .94400 .94837 .95274 .95711 8.8469 8.9352 9.0244 .94679 .95110 .95542 .99359 .99372 .99384 18.174 .25945 .05502 9.1146 .95974 .^)9396 2.91 2.92 2.93 18.357 18.541 18.728 .26380 .26814 .27248 .05448 .05393 .05340 9.1512 9.2437 9.3371 .96148 .96584 .97021 9.2056 9.2976 9.3905 .96405 .96837 ,97269 .99408 .99420 .99431 2.94 2 95 2.96 18.916 19.106 19.298 .27683 .28117 .28551 .05287 .05234 .05182 9.4315 9.5268 9.6231 .97458 .97895 .98331 9.4844 9.5791 9.6749 .97701 .98133 .98565 .99443 .99454 . .99464 2.97 2.98 2.99 19.492 19.688 19.886 .28985 .29420 .29854 .05130 .05079 .05029 9.7203 9.8185 9.9177 .98768 .99205 .99641 9.7716 9.8693 9.9680 .98997 .99429 .99861 .99475 .99485 .99496 3.00 20.086 .30288 .04979 10.018 .00078 10.068 .00293 .99505 122 Values and Logarithms of Hyperbolic Functions iC e Value Value SinhiT Value Logio Cosh i» Value Logio Tanh oc A'alue 3.00 20.086 .30288 .04979 10.018 .00078 10.068 .00293 .99505 3.05 3.10 3.15 21.115 22.198 23.336 .32460 .34631 .36803 .04736 .04505 .04285 10.534 11.076 11.646 .02259 .04440 .06619 10.581 11.122 11.690 .02454 .04616 .06780 .99552 .99595 .99631 3.20 3.25 3.30 24.533 25.790 27.113 .38974 .41146 .43317 .04076 .03877 .03688 12.246 12.876 13.538 .08799 .10977 .13155 12.287 12.915 13.575 .08943 .11108 .13273 .99668 .99700 .99728 3.35 3.40 3.45 28.503 29.964 31.500 .45489 .47660 .49832 .03508 .03.337 .03175 14.234 14.965 15.734 .15332 .17509 .19685 14.269 14.999 15.766 .15439 .17605 .19772 .99754 .99777 .99799 3.50 33.115 .52003 .03020 16.543 .21860 16.573 .21940 .99818 3.55 3.60 3.65 34.813 ;i6.598 38.475 .54175 .56346 .58517 .02872 .02732 .02599 17.392 18.286 19.224 .24036 .26211 .28385 17.421 18.313 19.250 .24107 .26275 .28444 ,99833 .99851 .99865 3 70 3.75 3.80 40.447 42.521 44.701 .60689 .62860 .65032 .02472 .02352 .92237 20.211 21.249 22.339 .30559 .32733 .34907 20.236 21.272 22.362 .30612 .32781 .34951 .99878 .99889 .99900 3.85 3.90 3.95 46.993 49.402 51.935 .67203 .69375 .71546 .02128 .02024 .01925 23.486 24.691 25.958 .37081 .39254 .41427 23.507 24.711 25.977 .37120 .39290 .41459 .99909 99918 .99926 4.00 4.10 4.20 4.30 54.598 .73718 .01832 27.290 .43600 27.308 .43()29 .99933 (50.340 6().()86 73.700 .780(>1 .82404 .86747 .01(]57 .01500 .01357 30.162 33.33() 36.843 .47946 .52291 .56636 30.178 S3.. Sol :3(>.857 .47970 .52310 .56652 .99945 .99955 .99963 4.40 4.50 4.60 81.451 90.017 99.484 .910^)0 .95433 .99775 .01227 .01111 .01005 40.719 45.003 49.737 .60980 .65324 .6J)668 40.732 45.014 49.747 .60993 .65335 .69677 .9^)970 .99^)75 .99980 4.70 4.80 4.90 109.95 121.61 134.29 .04118 .08461 .12804 .00910 .00823 .00745 54.9(i9 ()0.751 67.141 .74012 .78355 .82()99 54.978 60.759 67.149 .74019 .78361 .82704 .99^)83 .99986 .99989 5.00 148.41 .17147 .00674 74.203 .87042 74.210 .87046 .99991 5.10 5.20 5.30 164.02 181.27 200.34 .214^)0 .25833 .30176 .00610 .00552 .00199 82.011 90.633 100.17 .91386 .95729 .00074 82.014 90.(i39 100.17 .91389 .95731 .00074 .99993 .99994 .99995 5.40 5.50 5.60 221.41 244.69 270.43 .34519 .38862 .43205 .00452 .00409 .00370 110.70 122.1U 135.21 .04415 .08768 .13101 110.71 122..T) 135.22 .04417 .087(30 .13103 .99996 .99997 .99997 5.70 5.80 5.t)0 298.87 330.:30 3()5.04 .47548 .51891 .56234 .00335 .00303 .00274 149.43 165.15 182.52 .17444 .21787 .26130 149.44 1(J5.15 182.52 .17445 .21788 .26131 .99998 .99998 .99998 6.00 403.43 .60577 .00248 .00193 .00150 .00117 201.71 .30473 201.72 .30474 .9^)999 6.25 6.50 6.75 518.01 665.14 854.06 .71434 .82291 .93149 259.01 332.57 427.03 .41331 .52188 .63046 259.01 332.57 427.03 .41331 .52189 .63046 .99999 1.0000 1.0000 7.00 7.50 8.00 1096.6 1808.0 2981.0 .04006 .25721 .47436 .00091 .00055 .00034 548.32 904.02 1490.5 .73903 .95618 .17333 548.32 904.02 1490.5 .73903 .95618 .17333 1.0000 1.0000 1.0000 8.50 9.00 9.50 4914.8 8103.1 133(30. .69150 .90865 ,12580 .00020 .00012 .00007 2457.4 4051.5 6679.9 .39047 .60762 .82477 2457.4 4051.5 6679.9 .39047 .60762 .82477 1.0000 1.0000 1.0000 10.00 22026. .34294 .00005 11013. .04191 11013. .04191 1.0000 Table X — Values and Logarithms of Haversines 123 [Characteristics of Logai ithnis omitted - - determine by rule from the value] o 10' 20' 30' 40' 50' Value Logio Value Logio Value I-ogio Value Log.o Value Log,o Value Logio ~o" .0000 .0000 4.3254 .0000 4.9275 .0000 5.2796 .0000 5.5295 .0001 5.7223 1 .0001 5.8817 .0001 6.0156 .0001 6.1315 .0002 .2338 .0002 .3254 .0003 .4081 2 .0003 .4837 .0004 .5532 .0004 .6176 .0005 .6775 .0005 .7336 .0006 .7862 3 .0007 .8358 .0008 .8828 .0008 .9273 .0009 .9697 .0010 .0101 .0011 ,0487 4 .0012 .0856 .0013 .1211 .0014 .1551 .0015 .1879 .0017 .2195 .0018 .2499 5 .0019 .2793 .0020 .3078 .0022 .3354 .0023 .3621 .0024 .3880 .0026 .4132 6 .0027 .4376 .0029 .4614 .0031 .4845 .0032 .5071 .0034 .5290 .0036 .5^504 7 .0037 .5713 .0039 .5918 .0041 .6117 .0043 .6312 .0045 .6503 .0047 .6689 8 .0049 .6872 .0051 .7051 .0053 .7226 .0055 .7397 .0057 .7566 .0059 .7731 9 .0062 .7893 .0064 .8052 .0066 .8208 .0069 .83(51 .0071 .8512 .0073 .8660 10 .0076 .8806 .0079 .8949 .0081 .9090 .0084 .9229 .0086 .9365 .0089 .9499 11 .0092 .9631 .0095 .9762 .0097 .98iX) .0100 .0016 .0103 .0141 .010(5 .0264 12 .0109 .0385 .0112 .0504 .0115 .0622 .0119 .0738 .0122 .0853 .0125 .096(5 13 .0128 .1077 .0131 .1187 .0135 .1296 .0138 .1404 .0142 .1510 .0145 .1614 14 .0149 .1718 .0152 .1820 .0156 .1921 .0159 .2021 .0163 .2120 .0167 .2218 15 .0170 .2314 .0174 .2409 .0178 .2504 .0182 .2597 .0186 .2689 .0190 .2781 k; .0194 .2871 .0198 .2961 .0202 .3049 .0206 .3137 .0210 .3223 .0214 .3309 17 .0218 .3394 .0223 .3478 .0227 .35(51 .0231 .3644 .0236 .3726 .0240 .380() 18 .0245 .3887 .0249 .3966 .0254 .4045 .0258 .4123 .0263 .4200 .0268 .4276 19 .0272 .4352 .0277 .4427 .0282 .4502 .0287 .4576 .0292 .4649 .0297 .4721 20 .0:^2 .4793 .0307 .4865 .0312 .4936 .0317 .5006 .0322 .5075 .0327 .5144 21 .0332 .5213 .0337 .5281 .0.343 .5348 .0348 .5415 .0353 .5481 .0359 .5547 22 .0:364 .5612 .0370 .5677 .0375 .5741 .0381 .5805 .0386 .5868 .0392 .5931 23 .0397 .5993 .0403 .6055 .0409 .6116 .0415 .6177 .0421 .(5238 .0426 .6298 24 .0432 .6357 .0438 .6417 .0444 .6476 .0450 .6534 .0456 .6592 .0462 .6650 25 .0468 .6707 .0475 .6764 .0481 .6820 .0487 .()876 .0493 .6932 .0500 .()987 2lj .0506 .7042 .0512 .7096 .0519 .7151 .0525 .7204 .0532 .7258 .0538 .7311 27 .0545 .73(^4 .0552 .7416 .0.558 .7468 .0565 .7520 .0572 .7572 .0578 .762:5 28 .0585 .7673 .0592 .7724 .0599 .7774 .0(306 .7824 .0613 .7874 .0()20 .7923 29 .0627 .7972 .0634 .8020 .0641 .8069 .0()48 .8117 .0655 .8165 .0(363 .8213 30 .0670 .8260 .0()77 .8307 .0684 .8354 .0(592 .8400 .0699 .8446 .0707 .8492 31 .0714 .8538 .0722 .a^83 .0729 .8629 .0737 .8673 .0744 .8718 .0752 .8763 32 .0760 .8807 .0767 .8851 .0775 .8894 .0783 .8938 .0791 .8981 .0799 .9024 33 .0807 .^67 .0815 .9109 .0823 .9152 .0831 .9194 .0839 .9236 .0847 .9277 34 .0855 .9319 .0863 .93(X) .0871 .9401 .0879 .9442 .0888 .9482 .0896 .9523 35 .0i)04 .9563 .0913 .9603 .0921 .9643 .0929 .9682 .0938 .9722 .0946 .9761 36 .0955 .9800 .0963 .9838 .0972 .9877 .0981 .9915 .0989 .9954 .0998 .9992 37 .1007 .0030 .1016 .00<)7 .1024 .0105 .1033 .0142 .1042 .0179 .1051 .0216 38 .1060 .0253 .10(;9 .0289 .1078 .0326 .1087 .03(52 .1096 .0398 .1105 .0434 39 .1114 .0470 .1123 .0505 .1133 .0541 .1142 .0576 .1151 .0611 .1160 .0646 40 .1170 .0681 .1179 .0716 .1189 .0750 .1198 .0784 .1207 .0817 .1217 .0853 41 .1226 .0887 .1236 .0920 .1246 .0954 .1255 .0987 .1265 .1021 .1275 .1054 42 .1284 .1087 .1294 .1119 .1304 .1152 .1314 .1185 .1323 .1217 .1333 .1249 43 .11^3 .1282 .1353 .1314 .1363 .1345 .1373 .1377 .1383 .1409 .1393 .1440 44 .1403 .1472 .1413 .1503 .1424 .1534 .1434 .1565 .1444 .1596 .1454 .1626 45 .1464 .1657 .1475 .1687 .1485 .1718 .1495 .1748 .1506 .1778 .1516 .1808 46 .1527 .1838 .1538 .1867 .1548 .1897 .1558 .1926 .1569 .1956 .1579 .1985 47 .1590 .2014 .1600 .2043 .1611 .2072 .1622 .2101 .1633 .2129 .1644 .2158 48 .1(>54 .2186 .1665 .2215 .1676 .2243 .1687 .2271 .1698 .2299 .1709 .2327 49 .1720 .2355 .1731 .2382 .1742 .2410 .1753 .2437 .1764 .2465 .1775 .2492 50 .1786 .2519 .1797 .2546 .1808 .2573 .1820 .2600 .1831 .2627 .1842 .2653 51 .1853 .2680 .1865 .2706 .1876 .2732 .1887 .2759 .1899 .2785 .1910 .2811 52 .1922 .2837 .1933 .2863 .1945 .2888 .1956 .2914 .1968 .2940 .1979 .2965 53 .1991 .2991 .2003 .3016 .2014 .3041 .2026 .3066 .2038 .3091 .2049 .3116 54 .2061 .3141 .2073 .3166 .2085 .3190 .2096 .3215 .2108 .3239 .2120 .3264 55 .2132 .3288 .2144 .3312 .2156 .3336 .2168 .3361 .2180 .3384 .2192 .3408 56 .2204 .3432 .2216 .3456 .2228 .3480 .2240 .3503 .2252 .3527 .2265 .3550 57 .2277 .3573 .2289 .3596 .2301 .3620 .2314 .3643 .2326 .3666 .2338 .3689 58 .2350 .3711 .2363 .3734 .2375 .3757 .2388 .3779 .2400 .3802 .2412 .3824 59 .2425 .3847 .2437 .3869 .2450 .3891 .2462 .3913 .2475 .3935 .2487 3957 124 Values and Logarithms of Haversines [Characteristics of Logarithms omitted — determine by rule from the value] [X • 0' 10' 20' 30' 40' 50' Value Logio Value Logio Value Logio Value Logio Value Logio Value Logio 60 .2500 .3979 .2513 .4001 .2525 .4023 .2538 .4045 .2551 .4006 .2563 .4088 61 .2576 .4109 .2589 .4131 .2601 .4152 .2614 .4173 .2627 .4195 .2640 .4216 62 .2653 .4237 .2665 .4258 .2678 .4279 .2691 .4300 .2704 .4320 .2717 .4341 63 .2730 .4362 .2743 .4382 .2756 .4403 .2769 .4423 .2782 .4444 .2795 .4464 64 .2808 .4484 .2821 .4504 .2834 .4524 .2847 .4545 .2861 .4565 .2874 .4584 65 .2887 .4604 .2900 .4624 .2913 .4644 .2927 .4664 .2940 .4683 .2953 .4703 66 .2966 .4722 .2980 .4742 .2993 .4761 .3006 .4780 .3020 .4799 .3033 .4819 67 .3046 .4838 .3060 .4857 .3073 .4876 .3087 .4895 .3100 .4914 .3113 .4932 68 .3127 .4951 .3140 .4970 .3154 .4989 .3167 .5007 .3181 .5026 .3195 .6044 69 .3208 .5063 .3222 .5081 .3235 .5099 .3249 .5117 .3263 .5136 .3276 .5154 70 .3290 .5172 .3304 .5190 .3317 .5208 .3331 .5226 .3345 .5244 .3358 .5261 71 .3372 .5279 .3386 .5297 .3400 .5314 .3413 .5332 .3427 .5349 .3441 .5367 72 .3455 .5384 .34()9 .5402 .3483 .5419 .3496 .5436 .3510 .5454 .3524 .5471 73 .3538 .5488 .3552 .5505 .3566 .5522 .3580 .5539 .3594 .5556 .3608 .5572 74 .3622 .5589 .3636 .5606 .3650 .5623 .3664 .5639 .3678 .5656 .3692 .5672 75 .3706 .5689 .3720 .5705 .3734 .5722 .3748 .5738 .3762 .5754 .3776 .5771 76 .3790 .5787 .3805 .5803 .3819 .5819 .3833 .5835 .3847 .5851 .3861 .5867 77 .3875 .5883 .3889 .5899 .3904 .5915 .3918 .5930 .3932 .5946 .3946 .5962 78 .3960 .5977 .3975 .5993 .3989 .6009 .4003 .6024 .4017 .6039 .4032 .6055 79 .4046 .6070 .4060 .6085 .4075 .6101 .4089 .6116 .4103 .6131 .4117 .6146 80 .4132 .6161 .4146 .6176 .4160 .6191 .4175 .6206 .4189 .6221 .4203 .6236 81 .4218 .6251 .4232 .6266 .4247 .6280 .4261 .6295 .4275 .6310 .4290 .6324 82 .4304 .6339 .4319 .(5353 .4333 .6368 .4347 .6382 .4362 .6397 .4376 .6411 83 .4391 .6425 .4405 .6440 .4420 .6454 .4434 .6468 .4448 .6482 .4463 .6496 84 .4477 .6510 .4492 .6524 .4506 .6538 .4521 .6552 .4535 .6566 .4550 .6580 85 .4564 .6594 .4579 .6607 .4593 .6621 .4608 .6635 .4622 .6649 .4637 .6662 86 .4651 .6676 .4666 .6689 .4680 .6703 .4695 .6716 .4709 .6730 .4724 .6743 87 .4738 .6756 .4753 .6770 .4767 .6783 .4782 .6796 .4796 .6809 .4811 .6822 88 .4826 .6835 .4840 .6848 .4855 .6862 .4869 .6875 .4884 .6887 .4898 .6900 89 .4913 .6913 .4937 .6926 .4942 .6939 .4956 .6952 .4971 .6964 .4985 .6977 90 .5000 .6990 .5015 .7002 .5029 .7015 .5044 .7027 .5058 .7040 .5073 .7052 91 .5087 .7065 .5102 .7077 .5116 .7090 .5131 .7102 .5145 .7114 .5160 .7126 92 .5174 .7139 .5189 .7151 .5204 .7163 .5218 .7175 .5233 .7187 .5247 .7199 93 .5262 .7211 .5276 .7223 .5291 .72.35 .5305 .7247 .5320 .7259 .5334 .7271 94 .5349 .7283 .5363 .7294 .5378 .7306 .5392 .7318 .5407 .7329 .5421 .7341 95 .5436 .7353 .5450 .7364 .5465 .7376 .5479 .7387 .5494 .7399 .5508 .7410 96 .5523 .7421 .5537 .7433 .5552 .7444 .5566 .7455 .5580 .7467 .5595 .7478 97 .5609 .7489 .5624 .7500 .5638 .7511 .5653 .7523 .5667 .7534 .5082 .7545 98 .5696 .7556 .5710 .7567 .5725 .7577 .5739 .7588 .5753 .7599 .5768 .7610 99 .5782 .7621 .5797 .7632 .5811 .7642 .5825 .7653 .5840 .7664 .5854 .7674 100 .5868 .7685 .5883 .7696 .5897 .7706 .5911 .7717 .5925 .7727 .5940 .7738 101 .5954 .7748 .5968 .7759 .5983 .7769 .5997 .7779 .6011 .7790 .6025 .7800 102 .6040 .7810 .6054 .7820 .6068 .7830 .6082 .7841 .6096 .7851 .6111 .7861 103 .6125 .7871 .6139 .7881 .6153 .7891 .6167 .imi .6181 .7911 .6195 .7921 104 .6210 .7931 .6224 .7940 .6238 .7950 .6252 .7960 .6266 .7970 .6280 .7980 105 .6294 .7989 .6308 .7999 .6322 .8009 .6336 .8018 .6350 .8028 .6364 .8037 106 .6378 .8047 .6392 .8056 .6406 .8066 .6420 .8075 .6434 .8085 .6448 .8094 107 .6462 .8104 .6476 .8113 .6490 .8122 .6504 .8131 .6517 .8141 .6531 .8150 108 .6545 .8159 .6559 .8168 .6573 .8177 .6587 .8187 .6600 .8196 .6614 .8205 109 .6628 .8214 .6642 .8223 .6655 .8232 .6669 .8241 .6683 .8250 .6696 .8258 110 .6710 .8267 .6724 .8276 .6737 .8285 .6751 .8294 .6765 .8302 .6778 .8311 111 .6792 .8320 .6805 .8329 .6819 .8337 .6833 .8346 .6846 .8354 .6860 .8363 112 .6873 .8371 .6887 .8380 .6900 .8388 .6913 .8397 .6927 .8405 .6940 .8414 113 .6954 .8422 .6967 .8430 .6980 .8439 .6994 .8447 .7007 .8455 .7020 .8464 114 .7034 .8472 .7047 .8480 .7060 .8488 .7073 .8496 .7087 .8504 .7100 .8513 116 .7113 .8521 .7126 .8529 .7139 .8537 .7153 .8545 .7166 .8553 .7179 .8561 116 .7192 .8568 .7205 .8576 .7218 .8584 .7231 .8592 .7244 .8600 .7257 .8608 117 .7270 .8615 .7283 .8623 .7296 .8631 .7309 .8638 .7392 .8646 .7335 .8654 118 .7347 .8661 .7360 .8669 .7373 .8676 .7386 .8684 .7399 .8691 .7411 .8699 119 .7424 .8706 .7437 .8714 .7449 .8721 .7462 .8729 .7475 .8736 .7487 .8743 X] Values and Logarithms of Haversines [Characteristics of Logarithms omitted — determine by rule from the value] 125 o f 10' 20' 30' 40' 50' Value Logio Value Logio Value Logio Value Logio Value Logjo Value Logxo 120 .7500 .8751 .7513 .8758 .7525 .8765 .7538 .8772 .7550 .8780 .7563 .8787 121 .7575 .8794 .7588 .8801 .7600 .8808 .7612 .8815 .7625 .8822 .7637 .8829 122 .7650 .8836 .7662 .8843 .7674 .8850 .7686 .8857 .7699 .8864 .7711 .8871 123 .7723 .8878 .7735 .8885 .7748 .8892 .7760 .8898 .7772 .8905 .7784 .8912 124 .7796 .8919 .7808 .8925 .7820 .8932 .7832 .8939 .7844 .8945 .7856 .8952 125 .7868 .8959 .7880 .8965 .7892 .8972 .7904 .8978 .7915 .8985 .7927 .8991 12() .7939 .8998 .7951 .9004 .7962 .9010 .7974 .9017 .7986 .9023 .7997 .9030 127 .8009 .9036 .8021 .9042 .8032 .9048 .8044 .9055 .8055 .9061 .8067 .9067 128 .8078 .9073 .8090 .9079 .8101 .9085 .8113 .9092 .8124 .9098 .8135 .9104 129 .8147 .9110 .8158 .9116 .8169 .9122 .8180 .9128 .8192 .9134 .8203 .9140 130 .8214 .9146 .8225 .9151 .8236 .9157 .8247 .9163 .8258 .9169 .8269 .9175 i:n .8280 .9180 .8291 .9186 .8302 .9192 .8313 .9198 .8324 .9203 .8335 .9209 132 .8346 .9215 .8356 .9220 .8367 .9226 .8378 .9231 .8389 .9237 .8399 .9242 133 .8410 .9248 .8421 .9253 .8431 .9259 .8442 .9264 .8452 .9270 .8463 .9275 134 .8473 .9281 .8484 .9286 .8494 .9291 ..8501 .9297 .8515 .9302 .8525 .9307 135 .8536 .9312 .8546 .9318 .8556 .9323 .8566 .9328 .8576 .9333 .8587 .9338 136 .8597 .9343 .8607 .9348 .8617 .9353 .8627 .9359 .8637 .9364 .8647 .9369 137 .8657 .9374 .8667 .9379 .8677 .9383 .8686 .9388 .8696 .9393 .8706 .9398 138 .8716 .9403 .8725 .9408 .8735 .9413 .8745 .9417 .8754 .9422 .8764 .9427 139 .8774 .9432 .8783 .9436 .8793 .9441 .8802 .9446 .8811 .9450 .8821 .9455 140 .8830 .9460 .8840 .9464 .8849 .9469 .8858 .9473 .8867 .9478 .8877 .9482 141 .8886 .9487 .8895 .9491 .8904 .9496 .8913 .9500 .8922 .9505 .8931 .9509 142 .8940 .9513 .8949 .9518 .8958 .9522 .8967 .9526 .8976 .9531 .8984 .9535 143 .8993 .9539 .9002 .9543 .9011 .9548 .9019 .9552 .9028 .9556 .9037 .9560 144 .9045 .9564 .9054 .9568 .9062 .9572 .9071 .9576 .9079 .9580 .9087 .9584 145 .9096 .9588 .9104 .9592 .9112 .9596 .9121 .9600 .9129 .9604 .9137 .9608 146 .9145 .9612 .9153 .9616 .9161 .9620 .9169 .9623 .9177 .9627 .9185 .9631 147 .9193 .9635 .9201 .9638 .9209 .9642 .9217 .9646 .9225 .9650 .9233 .9653 148 .9240 .9657 .9248 .9660 .9256 .9664 .9263 .9668 .9271 .9671 .9278 .9675 149 .9286 .9678 .9293 .9682 .9301 .9685 .9308 .9689 .9316 .9692 .9323 .9695 150 .9330 .9699 .9337 .9702 .9345 .9706 .9352 .9709 .9359 .9712 .9366 .9716 151 .9373 .9719 .9380 .9722 .9387 .9725 .9394 .9729 .9401 .9732 .9408 .9735 152 .9415 .9738 .9422 .9741 .9428 .9744 .9435 .9747 .9442 .9751 .9448 .9754 153 .9455 .9757 .9462 .9760 .9468 .9763 .9475 .9766 .9481 .9769 .9488 .9772 154 .9494 .9774 .9500 .9777 .9507 .9780 .9513 .9783 .9519 .9786 .9525 .9789 155 .9532 .9792 .9538 .9794 .9544 .9797 .9550 .9800 .9556 .9803 .9562 .9805 156 .9568 .9808 .9574 .9811 .9579 .9813 .9585 .9816 .9591 .9819 .9597 .9821 157 .9603 .9824 .9608 .9826 .9614 .9829 .9619 .9831 .9625 .9834 .9630 .9836 158 .9636 .9839 .9641 .9841 .9647 .9844 .9652 .9846 .9657 .9849 .9663 .9851 159 .9668 .9853 .9673 .9856 .9678 .9858 .9683 .9860 .9688 .9863 .9693 .9865 160 .9698 .9867 .9703 .9869 .9708 .9871 .9713 .9874 .9718 .9876 .9723 .9878 161 .9728 .9880 .9732 .9882 .9737 .9884 .9742 .9886 .9746 .9888 .9751 .9890 162 .9755 .9892 .9760 .9894 .9764 .9896 .9769 .9898 .9773 .9900 .9777 .9902 163 .9782 .9904 .9786 .9906 .9790 .9908 .9794 .9910 .9798 .9911 .9802 .9913 164 .9806 .9915 .9810 .9917 .9814 .9919 .9818 .9920 .9822 .9922 .9826 .9923 165 .9830 .9925 .9833 .9927 .9837 .9929 .9841 .9930 .9844 .9932 .9848 .9933 166 .9851 .9935 .9855 .9937 .9858 .9938 .9862 .9940 .9865 .9941 .9869 .9943 167 .9872 .9944 .9875 .9945 .9878 .9947 .9881 .9948 .9885 .9950 .9888 .9951 168 .9891 .9952 .9894 .9954 .9897 .9955 .9900 .9956 .9903 .9957 .9905 .9959 169 .9908 .9960 .9911 .9961 .9914 .9962 .9916 .9963 .9919 .9965 .9921 .9966 170 .9924 .9967 .9927 .9968 .9929 .9969 .9931 .9970 .9934 .9971 .9936 .9972 171 .9938 .9973 .9941 .9974 .9943 .9975 .9945 .9976 .9947 .9977 .9949 .9978 172 .9951 .9979 .9953 .9980 .9955 .9981 .9957 .9981 .9959 .9982 .9961 .9983 173 .9963 .9984 .9964 .9984 .9966 .9985 .9968 .9986 .9969 .9987 .9971 .9987 174 .9973 .9988 .9974 .9988 .9976 .9989 .9977 .9990 .9978 .9991 .9980 .9991 175 .9981 .9992 .9982 .9992 .9983 .9993 .9985 .9993 .9986 .9994 .9987 .9994 176 .9988 .9995 .9989 .9995 .9990 .9996 .9991 .9996 .9992 .9996 .9992 .9997 177 .9993 .9997 .9994 .9997 .9995 .9998 .9995 .9998 .9996 .9998 .9996 .9998 178 .9997 .9999 .9997 .9999 .9998 .9999 .9998 .9999 .9999 .9999 .9999 .9999 179 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .0000 1.0000 .0000 126 Table XI — Factor Table — Logarithms of Primes If ^is prime, its logarithm is given. If JVis not prime, its factors are given. jsr 1 S 7 9 10 0043213738 0128372247 0293837777 0374264979 11 3-37 0530784435 32-13 717 12 112 3-41 1038037210 3-43 13 1172712957 7-19 1367205672 1430148003 14 3-47 11 13 3-72 1731862684 16 1789769473 32-17 1958996524 3-53 16 7-23 2121876044 2227164711 132 17 32-19 2380461031 3-59 2528530310 18 2576785749 3-61 11-17 33-7 19 2810333672 2855573090 2944662262 2988530764 20 3-67 7-29 32-23 11-19 21 3242824553 3-71 7-31 3-73 22 13 17 3483048630 3560258572 3598354823 23 3-711 3673559210 3-79 3783979009 24 3820170426 35 13-19 3-83 25 3996737215 11-23 4099331233 7-37 26 32-29 4199557485 3-89 4297522800 27 4329692909 3-7-13 4424797691 32-31 28 4487063199 4517864355 7-41 172 29 3-97 4668676204 33-11 13-23 30 7-43 3-101 4871383755 3-103 31 4927603890 4955443375 5010592622 11-29 32 3 107 17-19 3-109 7-47 33 5198279938 32-37 5276299009 3113 34 11-31 73 5403294748 5428254270 35 33-13 5477747054 3-7-17 5550944486 36 192 3-112 5646660643 32-41 37 7-53 •5717088318 13-29 5786392100 38 3-127 5831987740 32-43 5899496013 39 17-23 3-131 5987905068 3-719 40 6031443726 13-31 11-37 6117233080 41 3-137 7-59 3-139 6222140230 42 6242820958 32-47 7-61 311-13 43 6:344772702 6364878964 19-23 6424645202 44 32-72 6464037262 3-149 6522463410 45 11-41 3-151 6599162001 33-17 46 6637009254 6655809910 6693168806 7-67 47 3-157 11-43 32-53 6803355134 48 13-37 3-7-23 6875289612 3-163 49 6910814921 17-29 7-71 6981005456 50 3 167 7015679851 3-132 7067177823 51 7-73 33-19 11-47 3-173 52 7168377233 7185016889 17-31 232 53 3259 13-41 3-179 72-11 54 7331972651 3-181 7379873263 32-6I 55 19-29 7-79 7458551952 13-43 56 3-11-17 7505083949 34-7 7551122664 57 7566361082 3-191 7611758132 3 193 58 7-83 11-53 7686381012 19-31 59 3-197 7730546934 3-199 7774268224 60 7788744720 32-67 7831886911 3-7-29 61 13-47 7874604745 7902851640 7916906490 62 33-23 7-89 311-19 17-37 63 8000293592 3-211 72-13 32-71 r 64 8068580295 8082109729 8109042807 11-59 65 3-7-31 8149131813 32-73 8188854146 66 8202014595 3-13-17 23-29 3-223 67 11-61 8280150642 8305886687 7-97 68 3-227 8344207037 3-229 13-53 69 8394780474 32-711 17-41 3-233 jsr 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 LogN 301029995664 477121254720 698970004336 845098040014 041392685158 113943352307 230448921378 278753600953 361727836018 462396997899 491361693834 568201724067 612783856720 633468455580 672097857936 724275869601 770852011642 785329835011 826074802701 851258348719 863322860120 897627091290 919078092376 949390006645 986771734266 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1142772966 1149444157 1162755876 1202447955 1209028176 1228709229 1338581252 1357685146 1376705372 1401936786 1458177145 1489109931 1532049001 1544239731 1550322288 1562461904 1580607939 1604685311 1616674124 1622656143 1640552919 1676126727 1705550585 1711411510 1723109685 1728946978 1740598077 1758016328 1792644643 1826999033 1849751907 1883659261 1900514178 1911714557 1928461152 Factor Table — Logarithms of Primes If ^ is a prime, its logarithm is given. If N is not a prime, its factors are given. 12T N 1 3 7 9 JV roj js^ 70 8457180180 19-37 7-101 8506462352 1567 1950689965 71 32-79 23-31 3-239 8567288904 1571 1961761850 72 7 103 3-241 8615344109 36 1579 1983821300 73 17-43 8651039746 11-67 8686444384 1583 1994809149 74 31319 8709888138 32-83 7-107 1597 2033049161 75 8756399370 3-251 8790958795 3-11-23 1601 2043913319 76 8813846568 7-109 13-59 8859263398 1607 2060158768 77 3-257 8881794939 3 7-37 19-41 1609 2065560441 78 11-71 33-29 8959747324 3-263 1613 2076343674 79 7-113 13-61 9014583214 17-47 1619 2092468488 80 32-89 11-73 3-269 9079485216 1621 2097830148 81 9090208542 3-271 19-43 32-7-13 1627 2113875529 82 9143431571 9153998352 9175055096 9185545306 1637 2140486794 83 3-277 72-17 33-31 9237619608 1657 2193225084 84 292 3-281 7-112 3-283 1663 2208922492 85 23-37 9309490312 9329808219 9339931638 1667 2219355998 86 3-7-41 93(^0107957 3-172 11-79 1669 2224563367 87 13-67 32-97 94299959ai 3-293 1693 2286569581 88 9449759084 945f)607036 9479236198 7-127 1697 2296818423 89 34-11 19-47 3-13-23 29-31 1699 2301933789 90 17-53 3-7-43 9576072871 32-101 1709 2327420627 91 9595183770 11-83 7-131 9633155114 1721 2357808703 92 3-307 13-71 32-103 9680157140 1723 2362852774 93 72-19 3-311 9717395909 3-313 1733 2387985627 94 9735896234 23-41 9763499790 13-73 1741 2407987711 95 3-317 9790929006 311-29 7-137 1747 2422929050 96 312 32-107 9854264741 3-1719 1753 2437819161 97 9872102299 7 139 9898945637 11-89 1759 2452658395 98 32-109 9925535178 3-7-47 23-43 1777 2496874278 99 9960736545 3-331 9986951583 33-37 1783 2511513432 100 7-11-13 17-59 19-53 0038911662 1787 2521245525 101 3-337 0056094454 32-113 0081741840 1789 2526103406 102 0090257421 3-11-31 13-79 3-73 1801 2555137128 103 0132586653 0141003215 17-61 0166155476 1811 2579184503 104 3-347 7-149 3-349 0207754882 1823 2607866687 105 0216027160 34-13 7-151 3-353 1831 2626883443 106 0257153839 0265332645 11-97 0289777052 1847 2664668954 107 32-7-17 29-37 3-359 13-83 1861 2697463731 108 23-47 3-192 0362295441 32-112 1867 2711443179 109 0378247506 0386201619 0402066276 7-157 1871 2720737875 110 3-367 0425755124 33-41 0449315461 1873 2725377774 111 11-101 3-7-53 0480531731 3-373 1877 2734642726 112 19-59 0503797563 72-23 0526939419 1879 2739267801 113 313-29 11 103 3-379 17-67 1889 2762319579 114 7 163 32-127 31-37 3-383 um 2789821169 115 0610753236 0618293073 13-89 19-61 1^)07 2803506930 116 33-43 0655797147 3-389 7-167 1913 2817149700 117 0685568951 3-17-23 11-107 32-131 1931 2857822738 118 0722498976 7-132 0744507190 29-41 1933 2862318540 119 3-397 0766404437 32-7-19 11-109 1949 2898118391 120 0795430074 3-401 17-71 3-13-31 1951 2902572694 121 7-173 0838608009 0852905782 23-53 1973 2961270853 122 3-11-37 0874264570 3-409 0895518829 1979 2964457942 123 0902580529 32-137 0923696996 3-7-59 1987 2981978671 124 17-73 11-113 29-43 0965624384 1993 2995072987 125 32-139 7-179 3-419 1000257301 1997 3003780649 126 13-97 3-421 7-181 33-47 1999 3008127941 127 31-41 19-67 1061908973 1068705445 2003 3016809493 128 3-7-61 1082266564 3211-13 1102529174 2011 3034120706 129 1109262423 3-431 1129399761 3-433 2017 3047058982 128 Table XII a — Compound Interest : ( 1 + r)»» Amount of One Dollar Principal at Compound Interest After n. Yeabs n 2^10 2\<^o 3 fo 5J% 4t^o 4i% 5^0 e^o 7^0 1 2 3 4 5 6 7 8 9 1.0200 1.0404 1.0612 1.0824 1.1041 1.1262 1.1487 1.1717 1.1951 1.0250 1.0506 1.0769 1.1038 1.1314 1.1597 1.1887 1.2184 1.2489 1.0300 1.0609 1.0927 1.1255 1.1593 1.1941 1.2299 1.2668 1.3048 1.0350 1.0712 1.1087 1.1475 1.1877 1.2293 1.2723 1.3168 1.3629 1.0400 1.0816 1.1249 1.1699 1.2167 1.2653 1.3159 1.3686 1.4233 1.0450 1.0920 1.1412 1.1925 1.2462 1.3023 1.3609 1.4221 1.4861 1.0500 1.1025 1.1576 1.2155 1.2763 1.3401 1.4071 1.4775 1.5513 1.0600 1.1236 1.1910 1.2625 1.3382 1.4185 1.5036 1.5938 1.6895 1.0700 1.1449 1.2250 1.3108 1.4026 1.5007 1.6058 1.7182 1 .8385 10 1.2190 1.2801 1.3439 1.4106 1.4802 1.5530 1.6289 1.7908 1 9672 11 12 13 14 15 16 17 18 19 1.2434 1.2682 1.2936 1.3195 1.3459 1.3728 1.4002 1.4282 1.4568 1.3121 1.3449 1.3785 1.4130 1.4483 1.4845 1.5216 1.5597 1.5987 1.3842 1.4258 1.4685 1.5126 1.5580 1.6047 1.6528 1.7024 1.7535 1.4600 1.5111 1.5640 1.6187 1.6753 1.7340 1.7947 1.8575 1.9225 1.5395 1.6010 1.6651 1.7317 1.8009 1.8730 1.9479 2.0258 2.1068 1.6229 1.69.59 1.7722 1.8519 1.9353 2.0224 2.1134 2.2085 2.3079 1.7103 1.7959 1.8856 1.9799 2.0789 2.1829 2.2920 2.4066 2.5270 1.8983 2.0122 2.1329 2.2609 2.3966 2.5404 2.6928 2.8543 3.0256 2.1049 2.2522 2.4098 2.5785 2.7590 2.9522 3.1588 3.3799 3.6165 20 1.4859 1.6386 1.8061 1.9898 2.1911 2.4117 2.6533 3.2071 3.8697 4.1406 4.4.304 4.7405 5.0724 5.4274 5.8074 6.2139 6.6488 7.1143 21 22 23 24 25 26 27 28 29 1.5157 1.5460 1.5769 1.6084 1.6406 1.6734 1.7069 1.7410 1.7758 1.6796 1.7216 1.7646 1.8087 1.8539 1.9003 1.9478 1.9965 2.04(;4 1.8603 1.91()1 1.9736 2.0328 2.0938 2.1566 2.2213 2.2879 2.3566 2.0594 2.1315 2.2061 2.2833 2.3632 2.4460 2.5316 2.6202 2.7119 2.2788 2.3699 2.4647 2.5633 2.6058 2.7725 2.8834 2.9987 3.1187 2.5202 2.6337 2.7522 2.8760 3.0054 3.1407 3.2820 3.4297 3.5840 2.7860 2.9253 3.0715 3.2251 3.3864 3.5557 3.7335 3.9201 4.1161 3.3996 3.6035 3.8197 4.0489 4.2919 4.5494 4.8223 5.1117 5.4184 30 1.8114 2.0976 2.4273 2.8068 3.2434 3.7453 4.3219 5.74.35 7.6123 31 32 33 34 35 36 37 38 39 1.8476 1.8845 1.9222 1.9607 1.9999 2.0399 2.0807 2.1223 2.1647 2.1500 2.2038 2.2589 2.3153 2.3732 2.4325 2.4933 2.5557 2.6196 2.5001 2.5751 2.6523 2.7319 2.8139 2.8983 2.9852 3.0748 3.1670 2.9050 3.0067 3.1119 3.2209 3.3336 3.4503 3.5710 3.6960 3.8254 3.3731 3.5081 3.6484 3.7943 3.9461 4.1039 4.2681 4.4388 4.6164 3.9139 4.0900 4.2740 4.4664 4.6673 4.8774 5.0969 5.3262 5.5659 5.8164 4.5380 4.7649 5.0032 5.2533 5.5160 5.7918 6.0814 6.3855 6.7048 6.0881 6.4534 6.8406 7.2510 7.6861 8.1473 8.6361 9.1543 9.7035 10.2857 8.1451 8.7153 9.3253 9.9781 10.6766 11.4239 12.2236 13.0793 13.9948 40 2.2080 2.6851 3.2620 3.9593 4.8010 7.0400 14.9745 41 42 43 44 45 46 47 48 49 50 2.2522 2.2972 2.3432 2.3901 2.4379 2.4866 2.5363 2.5871 2.6388 2.7522 2.8210 2.8915 2.9638 3.0379 3.1139 3.1917 3.2715 3.3533 3.3599 3.4607 3.5645 3.6715 3.7816 3.8950 4.0119 4.1323 4.2562 4.0978 4.2413 4.3897 4.5433 4.7024 4.8669 5.0373 5.2136 5.3961 4.9931 5.1928 5.4005 5.6165 5.8412 6.0748 6.3178 6.5705 6.8333 6.0781 6.3516 6.6374 6.9361 7.2482 7.5744 7.9153 8.2715 8.(i437 9.0326 7.3920 7.7616 8.1497 8.5572 8.9850 9.4343 9.9060 10.4013 10.9213 10.9029 11.5570 12.2505 12.9855 13.7646 14.5905 15.4659 16.3939 17.3775 18.4202 16.0227 17.1443 18.3444 19.6285 21.0025 22.4726 24.0457 25.7289 27.5299 29.4570 2.6916 3.4371 4.3839 5.5849 7.1067 11.4674 Table XII & — Compound Discount : 1/(1 + r)" 129 Present Value of One Dollar Due at the End of n Years n 2^0 2\^o 31o S\^o 4=^0 4:\^0 S^'/o 6% 7% 1 2 3 4 5 6 7 8 9 10 .98039 .96117 .94232 .92385 .90573 .88797 .87056 .85319 .83676 .97561 .95181 .92860 .90595 .88385 .86230 .84127 .82075 .80073 .78120 .97087 .94260 .91514 .88849 .86261 .83748 .81309 .78941 .76642 .96618 .93351 .90194 .87144 .84197 .81350 .78599 .75941 .73373 .96154 .92456 .88900 .85480 .82193 .79031 .75992 .73069 .70259 .95694 .91573 .87630 .83856 .80245 .76790 .73483 .70319 .67290 .95238 .f)0703 .86384 .82270 .78353 .74622 .71068 .67684 .64461 .94340 .89000 .83962 .79209 .74726 .70496 .66506 .62741 .59190 .93458 .87344 .81630 .76290 .71299 .66634 .62275 .58201 .54393 .82035 .74409 .70892 .67556 .64393 .61391 .55839 .50835 11 12 13 14 15 16 17 18 19 20 .80426 .78849 .77303 .75788 .74301 .72845 .71416 .70016 .68643 .76214 .74356 .72542 .70773 .69047 .67362 .65720 .64117 .62553 .72242 !70138 .68095 .66112 .64186 .62317 .60502 .58739 .57029 .68495 .66178 .63940 .61778 .59689 .57671 .55720 .53836 .52016 .64958 .62460 .60057 .57748 .55526 .53391 .51337 .493()3 .47464 .45639 .61620 .58966 .56427 .53997 .51672 .49447 .47318 .45280 .43330 .58468 .55684 .53032 .50507 • .48102 .45811 .43630 .41552 .39573 .52679 .49697 .46884 .44230 .41727 .39365 .37136 .35034 .33051 .47509 .44401 .41496 .38782 .36245 .33873 .31657 .29586 .27651 .67297 .61027 .55368 .50257 .41464 .37689 .31180 .25842 .24151 .22571 .21095 .19715 .18425 .17220 .16093 .15040 .14056 21 22 23 24 25 26 27 28 29 .65978 .64684 .63416 .62172 .60953 .59758 .58586 .57437 .56311 .59539 .58086 .56670 .55288 .53939 .52623 .51340 .50088 .48866 .53755 .52189 .50669 .49193 .47761 .46369 .45019 .43708 .42435 .48557 .46915 .45329 .43796 .42315 .40884 .39501 .38165 .36875 .43883 .42196 .40573 .39012 .37512 .36069 .34682 .33348 .32065 .39679 .37970 .36335 .34770 .33273 .31840 .30469 .29157 .27902 .35894 .34185 .32557 .31007 .29530 .28124 .26785 .25509 .24295 .29416 .27751 .26180 .24698 .23300 .21981 .20737 .19563 .18456 30 .55207 .47674 .41199 .35628 .30832 .26700 .23138 .17411 .13137 31 32 33 34 35 36 37 38 39 .54125 .53063 .52023 .51003 .50003 .49022 .48061 .47119 .46195 .46511 .45377 .44270 .43191 .42137 .41109 .40107 .39128 .38174 .39999 .38834 .37703 .36604 .35538 .34503 .33498 .32523 .31575 .34423 .33259 .32134 .31048 .29998 .28983 .28003 .27056 .26141 .29646 .28506 .27409 .26355 .25:^2 .24367 .23430 .22529 .21662 .25550 .24450 .23397 .22390 .21425 .20503 .19620 .18775 .17967 .22036 .20987 .19987 .19035 .18129 .17266 .16444 .15661 .14915 .16425 .15496 .14619 .13791 .13011 .12274 .11580 .10924 .10306 .12277- .11474 .10723 .10022 .09366 .08754 .08181 .07646 .07146 40 41 42 43 44 45 46 47 48 49 .45289 .37243 .30656 .25257 .20829 .17193 .14205 .09722 .06678 .44401 .43530 .42677 .41840 .41020 .40215 .39427 .38654 .37896 .36335 .35448 .34584 .33740 .32917 .32115 .31331 .30567 .29822 .29094 .29763 .28896 .28054 .27237 .26444 .25674 .24926 .24200 .23495 .22811 .24403 .23578 .22781 .22010 .21266 .20547 .19852 .19181 .18532 .17905 .20028 .19257 .18517 .17805 .17120 .16461 .15828 .15219 .14634 .16453 .15744 .15066 .14417 .13796 .13202 .12634 .12090 .11569 .11071 .13528 .12884 .12270 .11686 .11130 .10600 .10095 .09614 .09156 .09172 .08653 .08163 .07701 .07265 .06854 .06466 .06100 .05755 .06241 .05833 .05451 .05095 .04761 .04450 .04159 .03887 .03632 50 .37153 .14071 .08720 .05429 .03395 130 Table XII c— Amount of an Annuity Amount of AN Annuity op One Dollar PER Year after n Years n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 2'fo 21^0 5% S\^o 4% 4i% 5^0 6^0 7% 1.0700 2.2149 3.4399 4.7507 6.1533 7.6540 9.2598 10.9780 12.8164 1.0200 2.0604 3.1216 4.2040 5.3081 6.4343 7.5830 8.7546 9.9497 1.0250 2.0756 3.1525 4.2563 5..3877 6.5474 7.7361 8.9545 10.2034 1.0300 2.0909 3.1836 4.3091 5.4684 6.6625 7.8923 9.1591 10.4639 1.0350 2.1062 3.2149 4.3625 5.5502 6.7794 8.0517 9.3685 10.7314 12.1420 1.0400 2.1216 3.2465 4.4163 5.6330 6.8983 8.2142 9.5828 11.0061 1.0450 2.1370 3.2782 4.4707 5.7169 7.0192 8.3800 9.8021 11.2882 1.0500 2.1525 3.3101 4.5256 5.8019 7.1420 8.5491 10.0266 11.5779 1.0600 2.1836 3.3746 4.6371 5.9753 7.3938 8.8975 10.4913 12.1808 11.1687 11.4835 11.8078 12.4864 12.8412 13.2068 13.9716 14.7836 16.8885 19.1406 21.5505 24.1290 26.8881 29.8402 32.9990 36.3790 39.9955 12.4121 13.6803 14.9739 16.2934 17.6393 19.0121 20.4123 21.8406 23.2974 12.7956 14.1404 15.5190 16.9319 18.3802 19.8647 21.3863 22.9460 24.5447 13.1920 14.6178 16.0863 17.5989 19.1569 20.7616 22.4144 24.1169 25.8704 13.6020 15.1130 16.6770 18.2957 19.9710 21.7050 23.4997 25.3572 27.2797 14.0258 15.6268 17.2919 19.0236 20.8245 22.6975 24.6454 26.6712 28.7781 14.4640 16.1599 17.9321 19.7841 21.7193 23.7417 25.8551 28.0636 30.3714 14.9171 16.7130 18.5986 20.5786 22.6575 24.8404 27.1324 29.5390 32.0660 15.8699 17.8821 20.0151 22.2760 24.6725 27.2129 29.9057 32.7600 35.7856 24.7833 26.1833 27.6765 29.2695 30.9692 32.7831 34.7193 38.9927 43.8652 26.2990 27.8450 29.4219 31.0303 32.6709 34.3443 36.0512 37.7922 39.5681 27.8629 29.5844 31.3490 33.1578 35.0117 36.9120 38.8598 40.8563 42.9027 29.5368 31.4529 33.4265 35.4593 37.5530 39.7096 41.9309 44.2189 46.5754 31.3289 33.4604 35.6665 37.9499 40.3131 42.7591 45.2906 47.9108 50.6227 33.2480 35.6179 38.0826 40.6459 43.3117 46.0842 48.9676 51.9663 55.0849 35.3034 37.9370 40.6892 43.5652 46.5706 49.7113 52.9933 56.4230 60.0071 37.5052 40.4305 43.5020 46.7271 50.1135 53.6691 57.4026 61.3227 65.4388 42.3923 45.9958 49.8156 53.8645 58.1564 62.7058 67.5281 72.6398 78.0582 48.0057 62.4361 57.1767 62.2490 67.6765 73.4838 79.6977 86.3465 93.4608 30 41.3794 45.0003 49.0027 53.4295 58.3283 63.7524 69.7608 83.8017 101.0730 31 32 33 34 35 36 37 38 39 40 43.2270 45.1116 47.0338 48.9945 50.9944 53.0343 55.1149 57.2372 59.4020 47.1503 49.3540 51.6129 53.9282 56.3014 58.7339 61.2273 63.7830 66.4026 69.0876 71.8398 74.6608 77.5523 80.5161 83.5540 86.6679 89.8596 93.1311 96.4843 51.5028 54.0778 56.7302 59.4621 62.2759 65.1742 68.1594 71.2342 74.4013 56.3345 59.3412 62.4532 65.6740 69.0076 72.4579 76.0289 79.7249 83.5503 61.7015 65.2095 68.8579 72.6522 76.5983 80.7022 84.9703 89.4091 94.0255 67.6662 71.7562 76.0303 80.4966 85.1640 90.0413 95.1382 100.4644 106.0303 111.8467 74.2988 79.0638 84.0670 89.3203 94.8363 100.6281 106.7095 113.0950 119.7998 89.8898 96.3432 103.1838 110.4348 118.1209 126.2681 134.9042 144.0585 153.7620 109.2182 117.9334 127.2588 137.2369 147.9135 159.3374 171.5610 184.6403 198.6351 61.6100 77.6633 87.5095 98.8265 126.8398 164.0477 213.6096 41 42 43 44 45 46 47 48 49 50 63.8622 66.1595 68.5027 70.8927 73.3306 75.8172 78.3535 80.9406 83.5794 81.0232 84.4839 88.0484 91.7199 95.5015 99.3965 103.4084 107.5406 111.7969 91.6074 95.8486 100.2383 104.7817 109.4840 114.3510 119.3883 124.6018 129.9979 103.8196 109.0124 114.4129 120.0294 125.8706 131.9454 138.2632 144.8337 151.6671 117.9248 124.2764 130.9138 137.8500 145.0982 152.6726 160.5879 168.8594 177.5030 134.2318 141.9933 150.1430 158.7002 167.6852 177.1194 187.0254 197.4267 208.3480 174.9505 186.5076 198.7580 211.7435 225.5081 240.0986 255.5645 271.9584 289.3359 229.6322 246.7765 265.1209 284.7493 305.7518 328.2244 352.2701 377.9990 405.5289 86.2710 99.9215 116.1808 135.5828 158.7738 186.5357 219.8154 307.7561 434.9860 Table XII d — Present Value of an Annuity 131 Present Value OF One Dollar per Year for n Yeai R8 n 1 2 3 4 5 6 7 8 9 2^0 2\^o 3^0 Sl^o 4:416 2.8839 3.8077 4.7135 5.6014 6.4720 7.3255 8.1622 .9756 1.9274 2.8560 3.7620 4.6458 5.5081 6.3494 7.1701 7.9709 .9709 1.9135 2.8286 3.7171 4.5797 5.4172 6.2303 7.0197 7.7861 .9662 1.8997 2.8016 3.6731 4.5151 5.3286 6.1145 6.8740 7.6077 8.3166 .9615 1.88(U 2.7751 3.6299 4.4518 5.2421 6.0021 6.7327 7.4353 .9569 1.87-:7 2.7490 3.5875 4.3900 5.1579 5.8927 6.5959 7.2688 .9524 1.8594 2.7232 3.5460 4.3295 5.0757 5.7864 6.4632 7.1078 7.7217 .9434 1.8334 2.6730 3.4651 4.2124 4.9173 5.5824 6.2098 6.8017 .9346 1.8080 2.6243 3.3872 4.1002 4.7665 5.3893 5.9713 6.5152 10 8.9826 8.7521 8.5302 8.1109 7.9127 7.3601 7.0236 11 12 13 14 15 16 17 18 19 9.7868 10.5753 11.3484 12.1062 12.8493 13.5777 14.2919 14.9920 15.6785 9.5142 10.2578 10.9832 11.6909 12.3814 13.0550 13.7122 14.3534 14.9789 9.2526 9.9540 10.6350 11.2961 11.9379 12.5611 13.1661 13.7535 14.3238 9.0016 9.6633 10.3027 10.9205 11.5174 12.0941 12.6513 13.1897 13.7098 8.7605 9.3851 9.9856 10.5631 11.1184 11.6523 12.1657 12.6593 13.1339 8.5289 9.1186 9.6829 10.2228 10.7395 11.2340 11.7072 12.1600 12.5933 8.3064 8.8633 9.3936 9.8986 10.3797 10.8378 11.2741 11.6896 12.0853 7.8869 8.3838 8.8527 9.2950 9.7122 10.1059 10.4773 10.8276 11.1581 7.4987 7.9427 8.3577 8.7455 9.1079 9.4466 9.7632 10.0591 10.3356 20 21 22 23 24 25 26 27 28 29 30 16.3514 15.5892 14.8775 14.2124 13.5903 14.0292 14.4511 14.8568 15.2470 15.6221 15.9828 16.3296 16.6631 16.9837 13.0079 12.4622 11.4699 10.5940 17.0112 17.6580 18.2922 18.9139 19.5235 20.1210 20.7069 21.2813 21.8444 16.1845 16.7654 17.3321 17.8850 18.4244 18.9506 19.4640 19.9649 20.4535 15.4150 15.9369 16.4436 16.9355 17.4131 17.8768 18.3270 18.7641 19.1885 14.6980 15.1671 15.6204 16.0584 16.4815 16.8904 17.2854 17.6670 18.0358 13.4047 13.7844 14.1478 14.4955 14.8282 15.1466 15.4513 15.7429 16.0219 16.2889 16.5444 16.7889 17.0229 17.2468 17.4610 17.6660 17.8622 18.0500 18.2297 12.8212 13.1630 13.4886 13.7986 14.0939 14.3752 14.6430 14.8981 15.1411 11.7641 12.0416 12.3034 12.5504 12.7834 13.0032 13.2105 13.4062 13.5907 10.8355 11.0612 11.2722 11.4693 11.6536 11.8258 11.9867 12.1371 12.2777 22.3965 20.9303 19.6004 18.3920 18.7363 19.0689 19.3902 19.7007 20.0007 20.2905 20.5705 20.8411 21.1025 21.3551 21.5991 21.8349 22.0627 22.2828 22.4955 22.7009 22.8994 23.0912 23.2766 17.2920 15.3725 13,7648 12.4090 31 32 33 34 35 36 37 38 39 40 22.9377 23.4683 23.9886 24.4986 24.9986 25.4888 25.9695 26.4406 26.9026 21.3954 21.8492 22.2919 22.7238 23.1452 23.5563 23.9573 24.3486 24.7303 20.0004 20.3888 20.7658 21.1318 21.4872 21.8323 22.1672 22.4925 22.8082 17.5885 17.8736 18.1476 18.4112 18.6646 18.9083 19.1426 19.3679 19.5845 15.5928 15.8027 16.0025 16.1929 16.3742 16.5469 16.7113 16.8679 17.0170 13.9291 14.0840 14.2302 14.3681 14.4982 14.6210 14.7368 14.8460 14.9491 12.5318 12.6466 12.7538 12.8540 12.9477 13.0352 13.1170 13.1935 13.2649 27.3555 25.1028 23.1148 19.7928 18.4016 17.1591 15.0463 15.1380 15.2245 15.3062 15.3832 15.4558 15.5244 15.5890 15.6500 15.7076 13.3317 41 42 43 44 45 46 47 48 49 50 27.7995 28.2348 28.6616 29.0800 29.4902 29.8923 30.2866 30.6731 31.0521 25.4661 25.8206 26.1664 26.5038 26.8330 27.1542 27.4675 27.7732 28.0714 23.4124 23.7014 23.9819 24.2543 24.5187 24.7754 25.0247 25.2667 25.5017 19.9931 20.1856 20.3708 20.5488 20.7200 20.8847 21.0429 21.1951 21.3415 18.5661 18.7236 18.8742 19.0184 19.1563 19.2884 19.4147 19.5356 19.6513 17.2944 17.4232 17.5459 17.6628 17.7741 17.8801 17.9810 18.0772 18.1687 13.3941 13.4524 13.5070 13.5579 13.6055 13.6500 13.6910 13.7305 13.7668 31.4236 28.3623 25.7298 23.4556 21.4822 19.7620 18.2559 15.7619 13.8007 132 Table XII e — Logarithms for Interest Computations r 1-^r log {1 + r) \% 1.005 00216 60617 56508 1% 1.010 00432 13737 82643 U% 1.015 00646 60422 49232 2% 1.020 00860 01717 61918 2^% 1.025 01072 38653 91773 3% 3^% 1.030 01283 72247 05172 1.035 01494 03497 92937 4% 1.040 01703 33392 98780 4^% 1.045 01911 62904 47073 6% 1.050 02118 92990 69938 r 1 +r log (1 + r) 5h% 1.055 02325 24596 33711 6% 1.060 02530 58652 64770 6^% 1.065 02734 96077 74757 7% 1.070 02938 37776 85210 7i% 1.075 03140 84642 51624 8% 1.080 03342 37554 86950 Sh% 1.085 03542 97381 84548 9% 1.090 03742 64979 40624 91% 1.095 03941 41191 76137 10% 1.100 04139 26851 58225 For Amount, A, of any principal, P, after n years : A = P (l-\- r)n For present worth, P, of any amount. A, at the end of n years: P = A-i- (l-\-r)n To find logarithms and antilogarithms of A and P to many significant figures, use Table XI, p. 126, and Table I a, p. 20. TABLE XII /—AMERICAN EXPERIENCE MORTALITY TABLE Based on 100,000 living at age 10 At Age 10 Number Surviving Deaths At Age Number Surviving Deaths At Age Number Surviving Deaths At Age Number Surviving Deaths 100,000 749 35 81,822 732 60 57,917 1,546 85 5,485 1,292 11 99,251 746 36 81,090 737 61 56,371 1,628 86 4,193 1,114 12 98,505 743 37 80,353. 742 62 54,743 1,713 87 3,079 933 13 97,762 740 38 79,611 749 63 53,030 1,800 88 2,146 744 14 97,022 737 39 78,862 756 64 51,230 1,889 89 1,402 555 15 96,285 735 40 78,106 765 65 49,341 1,980 90 847 385 16 95,550 732 41 77,341 774 66 47,361 2,070 91 462 246 17 94,818 729 42 76,567 785 67 45,291 2,158 92 216 137 18 94,089 727 43 75,782 797 68 43,133 2,243 93 79 58 19 93,362 725 44 74,985 812 69 40,890 2,321 94 21 18 20 92,637 723 45 74,173 828 70 38,569 2,391 95 3 3 21 91,914 722 46 73,345 848 71 36,178 2,448 22 91,192 721 47 72,497 870 72 33,730 2.487 23 90,471 720 48 71,627 896 73 31,243 2,505 24 89,751 719 49 70,731 927 74 28,738 2,501 25 89,032 718 50 69,804 962 75 26,237 2,476 26 88,314 718 51 68,842 1,001 76 23,761 2,431 27 87,596 718 52 67,841 1,044 77 21,330 2,369 28 86,878 718 53 66,797 1,091 78 18,961 2,291 29 86,160 719 54 65,706 1,143 79 16,670 2,196 30 85,441 720 55 64,563 1,199 80 14,474 2,091 31 84,721 721 56 63,364 1,260 81 12,383 1,964 32 84,000 723 57 62,104 1,325 82 10,419 1,816 33 83,277 726 58 60,779 1,394 83 8,603 1,648 34 82,551 729 59 59,385 1,468 84 6,955 1,470 XIII] Table XIII — Important Constants Logarithms of Important Constants 133 n — NUMBER Value of n LoGio n IT 3.14159265 0.49714987 l-^TT 0.31830989 9.50285013 7r2 9.86960440 0.99429975 v^ 1.77245385 0.24857494 e = Naperian Base 2.71828183 0.43429448 M = logio e 0.43429448 9.63778431 l^.¥=logelO 2.30258509 0.36221569 180 -f- TT = degrees in 1 radian 57.2957795 1 75812263 TT ^ 180 = radians in 1° 0.01745329 8.24187737 w -f- 10800 = radians in 1' 0.0002908882 6.46372612 TT -^ 648000 = radians in 1" 0.000004848136811095 4.68557487 sin 1" 0.000004848136811076 4.68557487 tan 1" 0.000004848136811152 4.68557487 centimeters in 1 ft. 30.480 1.4840158 feet in 1 cm. 0.032808 8.5159842 inches in 1 m. 39.37 (exact legal value) 1.5951654 pounds in 1 kg. 2.20462 0.3433340 kilograms in 1 lb. 0.453593 9.6566660 g (average value) 32.16 ft./sec./sec. = 981 cm. /sec. /sec. 1.5073 2.9916690 weight of 1 cu. ft. of water 62.425 lb. (max. density) 1.7953586 weight of 1 cu. ft. of air 0.0807 lb. (at 32° F.) 8.907 cu. in. in 1 (U. S.) gallon 231 (exact legal value) 2.3636120 ft. lb. per sec. in 1 H. P. 550 (exact legal value) 2.7403627 kg. m. per sec. in 1 H. P. 76.0404 1.8810445 watts in 1 H. P. 745.957 2,8727135 Several Numbers Very Accurately TT = 3.14159 26535 89793 23846 26433 83280 e = 2.71828 18284 59045 23536 02874 71353 3/ =0.43429 44819 03251 82765 11289 18917 1 --3/ =2.30258 50929 94045 68401 79914 54684 logio TT = 0.49714 98726 94133 85435 12682 88291 logio M = 9.63778 43113 00536 78912 Certain Convenient Values for 1 TO n = 10 n \/n "nAJ' %fn n\ \/n\ LoGio n 1 1.000000 1.00000 1.00000 1 1.0000000 0.000000000 2 0.500000 1.41421 1.25992 2 0.5000000 0.301029996 3 0.333333 1.73205 1.44225 6 0.1666667 0.477121255 4 0.250000 2.00000 1.58740 24 0.0416667 0.602059991 5 0.200000 2.23607 1.70998 120 0.0083333 0.698970004 6 0.166667 2.44949 1.81712 720 0.0013889 0.778151250 7 9.142857 2.64575 1.91293 5040 0.0001984 0.845098040 8 3.125000 2.82843 2.00000 40320 0.0000248 0.903089987 9 0.111111 3.00000 2.08008 362880 0.0000028 0.954242509 10 0.100000 3.16228 2.15443 3628800 0.0000003 1.000000000 134 Table XIV a- - Four Place Logarithms [XIV N 1 2 3 4 5 6 7 8 9 12 3 4 6 6 1 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 4 8 12 17 2125 29 33 37 11 12 13 14 15 16 17 18 19 0414 0792 1139 1461 1761 2041 2304 2553 2788 0453 0828 1173 1492 1790 2068 2330 2577 2810 0492 0864 1206 1523 1818 2095 2355 2601 2833 0531 0899 1239 1553 1847 2122 2380 2625 2856 0569 0934 1271 1584 1875 2148 2405 2648 2878 0607 0969 1303 1614 1903 2175 2430 2672 2900 0645 1004 1335 1644 1931 2201 2465 2695 2923 0682 1038 1367 1673 1959 2227 2480 2718 2945 0719 1072 1399 1703 1987 2253 2504 2742 2967 0755 1106 1430 1732 2014 2279 2529 2765 2989 4 8 11 3 7 10 3 6 10 3 6 9 3 6 8 3 5 8 2 6 7 2 5 7 2 4 7 15 19 23 14 17 21 13 16 19 12 15 18 11 14 17 11 13 16 10 12 15 9 12 14 9 11 13 26 30 34 24 28 31 23 26 29 21 24 27 20 22 25 18 21 24 17 20 22 16 19 21 \i^ 18 20 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 2 4 6 8 1113 15 17 19 21 22 23 24 25 26 27 28 29 3222 3424 3617 3802 3979 4150 4314 4472 4624 3243 3444 3636 3820 3997 4166 4330 4487 4639 3263 3464 3655 3838 4014 4183 4346 4502 4654 3284 3483 3674 3856 4031 4200 4362 4518 4669 3304 3502 3692 3874 4048 4216 4378 4533 4683 3324 3522 3711 3892 4065 4232 4393 4548 4698 3345 3541 3729 3909 4082 4249 4409 4564 4713 3365 3560 3747 3927 4099 4265 4425 4579 4728 3385 3579 3766 3946 4116 4281 4440 4594 4742 3404 3598 3784 3962 4133 4298 4456 4609 4757 2 4 6 2 4 6 2 4 6 2 4 6 2 4 5 2 3 5 2 3 6 2 3 5 13 4 8 10 12 8 10 12 7 9 11 7 9 11 7 9 10 7 8 10 6 8 9 6 8 9 6 7 9 14 16 18 14 16 17 13 15 17 12 14 16 12 14 16 11 13 15 11 12 14 11 12 14 10 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 41^)00 13 4 6 7 9 10 11 13 31 32 33 34 35 36 37 38 39 4914 5051 5185 5315 5441 5563 5682 5798 5911 4928 5065 5198 5328 5453 5575 5694 5809 6922 4942 5079 5211 5340 5465 5587 6705 5821 5933 4955 5092 5224 5353 5478 5599 5717 6832 5944 4969 5105 6237 6366 6490 5611 5729 5843 5955 4983 5119 6260 6378 6502 6623 6740 5855 5966 4997 5132 6263 5391 5514 5635 6762 5866 5977 5011 5145 5276 6403 5527 6647 6763 5877 6988 6024 5159 6289 6416 5539 6658 5776 6888 6999 5038 5172 5302 5428 5551 6670 6786 5899 6010 13 4 13 4 13 4 12 4 12 4 12 4 12 4 1 2 3 1 2 3 5 7 8 5 7 8 5 7 8 6 6 8 5 6 7 5 6 7 6 6 7 6 6 7 4 5 7 10 11 12 91112 9 1112 9 10 11 9 10 11 8 10 11 8 9 11 8 9 10 8 9 10 40 6021 6031 6042 6053 6064 6076 6085 6096 6107 6117 12 3 4 5 6 8 9 10 41 42 43 44 45 46 47 48 49 6128 6232 6335 6435 6532 6628 6721 6812 6902 6138 6243 6345 6444 6542 6637 6730 6821 6911 6149 6253 6355 6454 6551 6646 6739 6830 6920 6160 6263 6365 6464 6561 6656 6749 6839 6928 6170 6274 6375 6474 6571 6665 6758 6848 6937 6180 6284 6385 6484 6580 6675 6767 6857 6946 6191 6294 6395 6493 6590 6684 6776 6866 6955 6201 6304 6405 6503 6599 6693 6785 6875 6964 6212 6314 6415 6513 6609 6702 6794 6884 6972 62?? 6325 6425 6522 6618 6712 6803 6893 6981 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 12 3 4 5 6 4 6 6 4 5 6 4 6 6 4 5 6 4 5 6 4 5 6 4 6 6 4 4 6 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 7 8 7 7 8 7 7 8 6 7 8 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 12 3 3 4 5 6 7 8 51 52 63 54 7076 7160 7243 7324 7084 7168 7251 7332 7093 7177 7259 7340 7101 7185 7267 7348 7110 7193 7275 7356 7118 7202 7284 7364 7126 7210 7292 7372 7135 7218 7300 7380 7143 7226 7308 7388 7152 7235 7316 7396 1 2 3 12 3 12 2 12 2 3 4 5 3 4 6 3 4 5 3 4 6 6 7 8 6 7 7 6 6 7 6 6 7 N 1 2 3 4 5 6 7 8 9 1 2 2 4 5 6 7 8 9 The proportional parts are stated in full lor every tenth at the right-hand side. Xh% logarithm of any number of four significant figures can be read directly by add- XIVJ Table XIV a- - Four Place Logarithms 135 N 1 2 3 4 5 6 7 8 » 12 3 4 5 6 7 8 9 55 56 57 58 59 60 ()1 62 (53 (;4 65 6(i 67 69 7404 7482 7559 7634 7709 7782 7412 7490 7566 7642 7716 7789 7419 7497 7574 7649 7723 7796 7427 7505 7582 7657 7731 7435 7513 7589 7664 7738 7443 7520 7597 7672 7745 7451 7528 7604 7679 7752 7459 7536 7612 7686 7760 7466 7543 7619 7694 7767 7474 7551 7627 7701 7774 1 2 2 12 2 112 1 1 2 112 3 4 5 3, 4 5 3 4 5 3 4 4 3 4 4 5 6 7 5 6 7 5 6 7 5 6 7 5 6 7 7803 7810 7818 7825 7832 7839 7846 1 1 2 3 4 4 5 6 6 7853 7924 7993 8062 8129 8195 8261 8325 8388 7860 7931 8000 8069 8136 8202 8267 8331 8395 7868 7938 8007 8075 8142 8209 8274 8338 8401 7875 7945 8014 8082 8149 8215 8280 8344 8407 7882 7952 8021 8089 8156 8222 8287 8351 8414 7889 7959 8028 8096 8162 8228 S29S 8357 8420 7896 7966 8035 8102 8169 8235 8299 8363 8426 7903 7973 8041 8109 8176 8241 8306 8370 8432 7910 7980 8048 8116 8182 8248 8312 8376 8439 7917 7987 8055 8122 8189 8254 8319 8382 8445 112 112 112 112 112 112 112 112 112 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 5 6 6 5 5 6 6 6 6 5 5 6 5 5 6 5 5 6 5 5 6 4 5 6 4 5 6 70 71 72 73 74 75 76 77 78 79 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 112 3 3 4 4 5 6 8513 8573 8633 8692 8751 8808 8865 8921 8976 8519 8579 8639 8698 8756 8814 8871 8927 8982 8525 8585 8645 8704 8762 8820 8876 8932 8987 8531 8591 8651 8710 8768 8825 8882 8938 8993 8537 8597 8657 8716 8774 8831 8887 8943 8998 8543 8603 8663 8722 8779 8837 8893 8949 9004 8549 8(309 8669 8727 8785 8842 8899 8954 9009 8555 8615 8675 8733 8791 8848 8904 89()0 9015 8561 8621 8681 8739 8797 8854 8910 8965 9020 8567 8627 8686 8745 8802 8859 8915 8971 9025 112 112 112 112 112 112 1 1 2 112 112 3 3 4 3 3 4 2 3 4 2 3 4 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 4 5 6 4 5 6 4 5 5 4 5 5 4 5 6 4 4 5 4 4 5 4 4 5 4 4 5 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 1 1 2 2 3 3 4 4 5 81 82 83 84 85 86 87 88 89 9085 9138 9191 9243 9294 9345 9395 9445 9494 9090 9143 9196 9248 9299 9350 9400 9450 9499 9096 9149 9201 9253 9304 9355 9405 9455 9504 9101 9154 9206 9258 9309 9360 9410 9460 9509 9106 9159 9212 9263 9315 9365 9415 9465 9513 9112 9165 9217 9269 9320 9370 9420 9469 9518 9117 9170 9222 9274 9325 9375 9425 9474 9523 9122 9175 9227 9279 9330 9380 9430 9479 9528 9128 9180 9232 9284 9335 9385 9435 9484 9533 9133 9186 9238 9289 9340 9390 9440 9489 9538 112 112 112 112 112 112 112 Oil 1 1 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 2 3 2 2 3 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 3 4 4 3 4 4 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 Oil 2 2 3 3 4 4 91 92 93 94 95 97 98 99 9590 9638 9685 9731 9777 9823 9868 9912 9956 9595 9643 9689 9736 9782 9827 9872 9917 9961 9600 9647 9694 9741 9786 9832 9877 9921 9965 9605 9652 9699 9745 9791 9836 9881 9926 9969 9609 9657 9703 9750 9795 9841 9886 9930 9974 9614 9661 9708 9754 9800 9845 9890 9934 9978 9619 9666 9713 9759 9805 9850 9894 9939 9983 9624 9671 9717 9763 9809 9854 9899 9943 91^87 9628 9675 9722 9768 9814 9859 9903 9948 9991 9633 9680 9727 9773 9818 9863 9908 9952 9996 1 1 Oil Oil Oil Oil Oil Oil 1 1 Oil 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 3 4 4* 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 1 4 3 3 4 3 3 4 N 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 mg the proportional part corresponding to the fourth figure to the tahular numhei corresponding to the first three figures. There may be an error of 1 in the last place. 136 Table XIV &- ■ Antilogarithms to Four Places [xiy 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 .00 1000 1002 1005 1007 1009 1012 1014 1016 1019 1021 1 111 2 2 2 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 1023 1047 1072 1096 1122 1148 1175 1202 1230 1026 1050 1074 1099 1125 1151 1178 1205 1233 1028 1052 1076 1102 1127 1153 1180 1208 1236 10:30 1054 1079 1104 1130 1156 1183 1211 1239 1033 1057 1081 1107 1132 1159 1186 1213 1242 1035 1059 1084 1109 1135 1161 1189 1216 1245 1038 1062 1086 1112 1138 1164 1191 1219 1247 1040 1064 1089 1114 1140 1167 1194 1222 1250 1042 1067 1091 1117 1143 1169 1197 1225 1253 1045 1069 1094 1119 1146 1172 1199 1227 125(> 1 1 1 Oil oil oil oil oil oil 111 111 1 1 1 112 112 112 112 112 112 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 3 1259 1288 1318 1349 1380 1413 1445 1479 1514 1549 1262 1291 1321 1352 1384 1416 1449 1483 1517 1552 1265 1294 1324 1355 1387 1419 1452 1486 1521 1556 1268 1297 1327 1358 1390 1422 1455 1489 1524 1560 1271 1274 1276 1279 1282 1285 1 1 1 1 2 2 2 3 1300 1330 1361 1393 1426 1459 1493 1528 1563 1303 1334 1365 1396 1429 1462 1496 1531 1567 1306 1337 1368 1400 1432 1466 1500 1535 1570 1309 1340 1371 1403 1435 1469 1503 1538 1574 1312 1343 1374 1406 1439 1472 1507 1542 1578 1315 1346 1377 1409 1442 1476 1510 1545 1581 oil 1 1 oil oil 1 1 oil oil oil 1 1 12 2 12 2 12 2 12 2 12 2 12 2 1 2 2 12 2 12 2 2 2 3 2 2 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 1585 1589 1626 1663 1702 1742 1782 1824 1866 1910 1954 1592 1596 1600 1603 1607 1611 1614 1618 oil 1 2 2 3 3 3 1622 1660 1698 1738 1778 1820 1862 1905 1950 1629 1667 1706 1746 1786 1828 1871 1914 1959 1633 1671 1710 1750 1791 1832 1875 1919 1963 1637 1675 1714 1754 1795 1837 1879 1923 1968 1641 1679 1718 1758 1799 1841 1884 1928 1972 1644 1683 1722 1762 1803 1845 1888 1932 1977 1648 1687 1726 1766 1807 1849 1892 1936 1982 1652 1690 1730 1770 1811 1854 1897 1941 1986 1656 1694 1734 1774 1816 1858 1901 1945 1991 1 1 oil oil oil oil oil oil oil oil 12 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 4 3 3 4 3 3 4 3 4 4 3 4 4 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 1995 2042 2089 2138 2188 2239 2291 2344 2399 2455 2000 2046 2094 2143 2193 2244 2296 2350 2404 2460 2004 2051 2099 2148 2198 2249 2301 2355 2410 2466 2009 2014 2018 2023 2028 2032 2037 oil 2 2 3 3 4 4 2056 2104 2153 2203 2254 2307 2360 2415 2472 2061 2109 2158 2208 2259 2312 2366 2421 2477 2065 2113 2163 2213 2265 2317 2371 2427 2483 2070 2118 2168 2218 2270 2323 2377 2432 2489 2075 2123 2173 2223 2275 2328 2382 2438 2495 2080 2128 2178 2228 2280 2333 2388 2443 2500 2084 2133 2183 2234 2286 2339 2393 2449 2506 oil oil oil 112 1 1 2 1 1 2 112 1 1 2 112 2 2 3 2 2 3 2 2 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 3 4 4 3 4 4 3 4 4 4 4 5 4 4 5 4 4 5 4 4 5 4 5 5 4 5 5 .40 2512 2518 2523 2529 2535 2541 2547 2553 2559 2564 112 2 3 4 4 5 5 .41 .42 .43 .44 .45 .46 .47 .48 .49 2570 2630 2692 2754 2818 2884 2951 3020 3090 2576 2636 2698 2761 2825 2891 2958 3027 3097 2582 2642 2704 2767 2831 2897 2965 3034 3105 2588 2649 2710 2773 2838 2904 2972 3041 3112 2594 2655 2716 2780 2844 2911 2979 3048 3119 2600 2661 2723 2786 2851 2917 2985 3055 3126 2606 2667 2729 2793 2858 2924 2992 3062 3133 2612 2673 2735 2799 2864 2931 2999 3069 3141 2618 2679 2742 2805 2871 2938 3006 3076 3148 2624 2685 2748 2812 2877 2944 3013 3083 3155 112 1 1 2 112 112 1 1 2 1 1 2 1 1 2 112 1 1 2 2 3 4 2 3 4 2 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 4 4 4 5 6 4 5 6 4 5 6* 4 5 6 5 5 6 5 5 6 5 6 6 5 6 6 5 6 6 XIV] Table XIV &- -Antilogarithms to Four Places 137 1 2 3 4 5 6 7 8 9 12 3 4 5 6 7 8 9 .50 3162 3170 3177 3184 3192 3199 3206 3214 3221 3228 112 3 4 4 5 6 7 .51 .52 .53 .54 .55 .56 .57 .58 .59 3236 3311 3388 3467 3548 3631 3715 3802 3890 3243 3319 3396 3475 3556 3639 3724 3811 3899 3251 3327 3404 3483 3565 3648 3733 3819 3VK)8 3258 3334 3412 3491 3573 3656 3741 3828 3917 3266 3342 3420 3499 3581 3664 3750 3837 3926 3273 3350 3428 3508 3589 3673 3758 3846 3936 3281 3357 3436 3516 3597 3681 3767 3855 3945 3289 3365 3443 3524 3606 3690 3776 3864 3954 3296 3373 3451 a532 3614 3698 3784 3873 3963 3304 3381 3459 3540 3622 3707 3793 3882 3972 1 1 2 1 1 2 1 2 2 12 2 1 2 2 12 2 12 3 12 3 12 3 3 4 4 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 3 4 5 4 5 5 5 6 7 5 6 7 6 6 7 6 6 7 6 7 7 6 7 8 6 7 8 6 7 8 6 7 8 .60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 12 3 4 5 6 7 8 8 .61 .62 .63 .64 .65 .66 .67 .68 .69 4074 4169 4266 4365 4467 4571 4677 4786 4898 4083 4178 4276 4375 4477 4581 4688 4797 4909 4093 4188 4285 4385 4487 4592 4699 4808 4920 4102 4198 4295 4395 4498 4603 4710 4819 4932 4111 4207 4305 4406 4508 4613 4721 4831 4943 4121 4217 4315 4416 4519 4624 4732 4842 4955 4130 4227 4325 4426 4529 4634 4742 4853 4966 4140 4236 4335 4436 4539 4645 4753 4864 4977 4150 4246 4345 4446 4550 4656 4764 4875 4989 4159 4256 4355 4457 45(i0 4667 4775 4887 5000 12 3 12 3 12 3 12 3 1 2 3 12 3 12 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 6 4 5 7 5 6 7 5 6 7 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 9 10 8 9 10 8 9 10 8 910 .70 .71 .72 .73 .74 .75 .76 .77 .78 .79 .80 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 12 3 5 6 7 8 910 5129 5248 5370 5495 5623 5754 588'8 6026 6166 6310 5140 5260 5383 5508 5636 5768 5902 6039 6180 5152 5272 5395 5521 5649 5781 5916 6053 6194 5164 5284 5408 5534 5662 5794 5929 6067 6209 5176 5297 5420 5546 5675 5808 5943 6081 6223 5188 5309 5433 5559 5689 5821 5957 6095 6237 5200 5321 5445 5572 5702 5834 5970 6109 6252 5212 5333 5458 5585 5715 5848 5984 6124 626() 5224 5346 5470 5598 5728 5861 5908 6138 6281 5236 5358 5483 5610 5741 5875 6012 6152 6295 1 2 4 1 2 4 13 4 1 3 4 13 4 1 3 4 1 3 4 1 3 4 1 3 4 5 6 7 5 6 7 5 6 7 5 6 8 5 7 8 5 7 8 5 7 8 6 7 8 6 7 9 8 10 11 9 10 11 910 11 910 12 9 11 12 9 11 12 10 11 12 10 11 13 10 11 13 6324 6339 6353 6368 6383 6397 6412 6427 6442 13 4 6 7 9 10 12 13 .81 .82 .83 .84 .85 .86 .87 .88 .89 6457 6607 6761 6918 7079 7244 7413 7586 7762 6471 6622 6776 6934 7096 7261 7430 7603 7780 6486 6637 6792 6950 7112 7278 7447 7621 7798 6501 6653 6808 6966 7129 7295 7464 7638 7816 6516 6668 6823 6982 7145 7311 7482 7656 7834 6531 6683 6839 6998 7161 7328 7499 7674 7852 6546 6699 6855 7015 7178 7345 7516 7691 7870 6561 6714 6871 7031 7194 7362 7534 7709 7889 6577 6730 6887 7047 7211 7379 7551 7727 7907 6592 ()745 6902 7063 7228 7396 7568 7745 7925 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 3 5 2 4 5 2 4 5 2 4 6 6 8 9 6 8 9 6 8 9 7 8 10 7 8 10 7 8 10 7 910 7 9n 7 9 11 11 12 14 11 12 14 11 13 14 11 13 15 12 13 15 12 14 15 12 14 16 12 14 16 13 15 K) .90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 2 4 6 7 9 11 13 15 17 .91 .92 .93 .94 .95 .96 .97 .98 .99 8128 8318 8511 8710 8913 9120 9333 9550 9772 8147 8337 8531 8730 8933 9141 9354 9572 9795 8166 8356 8551 8750 8954 9162 9376 9594 9817 8185 8375 8570 8770 8974 9183 9397 9616 9840 8204 8395 8590 8790 8995 9204 9419 9638 9863 8222 8414 8610 8810 9016 9226 9441 9661 9886 8241 8433 8630 8831 9036 9247 9462 9683 9908 8260 8453 8650 8851 9057 9268 9484 9705 9931 8279 8472 8670 8872 9078 9290 9506 9727 9954 8299 8492 8690 8892 9099 9311 9528 9750 9977 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 7 2 6 7 8 9 11 8 10 12 8 10 12 8 10 12 8 10 12 9 1113 9 1113 9 1113 9 1114 13 15 17 14 15 17 14 16 18 14 16 18 15 17 19 15 17 19 15 17 19 16 18 20 16 18 21 138 Table XIV c — Four Place Trigonometric Functions [xiv [Characteristics of Logarithms omitted — determine by the usual rule from the value] Radtanr Degrees Sine Tangent Cotangent Cosine Xv.A.i-'X.a.jLi o .M^ JltKx 3X*UM2iiy Value Lo^io Value Logio Value Logio Value Logio .0000 .0029 0°00' 10 .0000 0000 1.0000 .0000 9G°G0' 50 1.5708 1.5679 .0029 .4637 !0029 .4637 343.77 .5363 I'.OOOO !oooo .0058 20 .0058 .7648 .0058 .7648 171.89 .2352 1.0000 .0000 40 1.5650 .0087 30 .0087 .9408 .0087 .9409 114.59 .0591 1.0000 .0000 30 1.5621 .0116 40 .0116 .0658 .0116 .0658 85.940 .9342 .9999 .0000 20 1.5592 .0145 50 .0145 .1627 .0145 .1627 68.750 .8373 .9999 .0000 10 1.5563 .0175 1°00' .0175 .2419 .0175 .2419 57.290 .7581 .9998 .9999 89° GG' 1.5533 .0204 10 .0204 .3088 .0204 .3089 49.104 .6911 .9998 .9999 50 1.5504 .0233 20 .0233 .3668 .0233 .3669 42.964 .6331 .9997 .9999 40 1.5475 .0262 30 .0262 .4179 .0262 .4181 38.188 .5819 .9997 .9999 30 1.5446 .0291 40 .0291 .4637 .0291 .4638 34.368 .5362 .9996 .9998 20 1.5417 .0320 50 .0320 .5050 .0320 .5053 31.242 .4947 .9995 .9998 10 1.5388 .0349 2° 00' .0349 .5428 .0349 .5431 28.636 .4569 .9994 .9997 88° GG' 1.5359 .0378 10 .0378 .5776 .0378 .5779 26.432 .4221 .9993 .9997 50 1.5330 .0407 20 .0407 .6097 .0407 .6101 24.542 .3899 .9992 .9996 40 1.5301 .0436 30 .0436 .6397 .0437 .6401 22.904 .3599 .9990 .9996 30 1.5272 .0465 40 .0465 .6677 .0466 .6682 21.470 .3318 .9989 .9995 20 1.5243 .0495 50 .0494 .6940 .0495 .6945 20.206 .3055 .9988 .9995 10 1.5213 .0524 3° GO' .0523 .7188 .0524 .7194 19.081 .2806 .9986 .9994 87° GG' 1.5184 .0553 10 .0552 .7423 .0553 .7429 18.075 .2571 .9985 .9993 50 1.5155 .0582 20 .0581 .7645 .0582 .7652 17.169 .2348 .9983 .9993 40 1.5126 .0611 30 .0610 .7857 .0612 .7865 16.350 .2135 .9981 .9992 30 1.5097 .0640 40 .0640 .8059 .0641 .8067 15.605 .1933 .9980 .9991 20 1.5068 .0669 50 .0669 .8251 .0670 .8261 14.924 .1739 .9978 .9990 10 1.5039 .0698 4° 00' .0698 .8436 .0699 .8446 14.301 .1554 .9976 .9989 86° GG' 1.5010 .0727 10 .0727 .8613 .0729 .8624 13.727 .1376 .9974 .9989 50 1.4981 .0756 20 .0756 .8783 .0758 .8795 13.197 .1205 .9971 .9988 40 1.4952 .0785 30 .0785 .8946 .0787 .8960 12.706 .1040 .9969 .9987 30 1.4923 .0814 40 .0814 .9104 .0816 .9118 12.251 .0882 .9967 .9986 20 1.4893 .0844 50 .0843 .9256 .0846 .9272 11.826 .0728 .9964 .9985 10 1.4864 .0873 5° GO' .0872 .9403 .0875 .9420 11.430 .0580 .9962 .9983 85° GG' 1.4835 .0902 10 .0901 .9545 .0904 .9563 11.059 .0437 .9959 .9982 50 1.4806 .0931 20 .0929 .f)682 .0934 .9701 10.712 .0299 .9957 .9981 40 1.4777 .0960 30 .0958 .9816 .0963 .9836 10.385 .0164 .9954 .9980 30 1.4748 .0989 40 .0987 .9945 .0992 .9966 10.078 .0034 .9951 .9979 20 1.4719 .1018 50 .1016 .0070 .1022 .0093 9.7882 .9907 .9948 .9977 10 1.4690 .1047 6°GG' .1045 .0192 .1051 .0216 9.5144 .9784 .9945 .9976 84° GG' 1.4661 .1076 10 .1074 .0311 .1080 .0336 9.2553 .9664 .9942 .9975 50 1.4632 .1105 20 .1103 .0426 .1110 .0453 9.0098 .9547 .9939 .9973 40 1.4603 .1134 30 .1132 .0539 .1139 .0567 8.7769 .9433 .9936 .9972 30 1.4573 .1164 40 .1161 .0648 .1169 .0678 8.5555 .9322 .9932 .9971 20 1.4544 .1193 50 .1190 .0755 .1198 .0786 8.3450 .9214 .9929 .9969 10 1.4515 .1222 7° GO' .1219 .0859 .1228 .0891 8.1443 .9109 .9925 .9968 83° GG' 1.4486 .1251 10 .1248 .0961 .1257 .0995 7.9530 .9005 .9922 .9966 ^0 1.4457 .1280 20 .1276 .1060 .1287 .109() 7.7704 .8904 .9918 .9964 lo 1.4428 .1309 30 .1305 .1157 .1317 .1194 7.5958 .8806 .9914 .9963 30 1.4399 .1338 40 .1334 .1252 .1346 .1291 7.4287 .8709 .9911 .9961 20 1.4370 .1367 50 .1363 .1345 .1376 .1385 7.2687 .8615 .9907 .9959 10 1.4341 .1396 8°GG' .1392 .1436 .1405 .1478 7.1154 .8522 .9903 .9958 82° GG' 1.4312 .1425 10 .1421 .1525 .1435 .1569 6.9682 .8431 .9899 .9956 50 1.4283 .1454 20 .1449 .1612 .1465 .1658 6.8269 .8342 .9894 .9954 40 1.4f^54 l.i224 .148^1 30 .1478 .1697 .1495 .1745 6.6912 .8255 .9890 .9952 30 .1513 40 .1507 .1781 .1524 .1831 6.5606 .8169 .9886 .9950 20 1.4195 .1542 50 .1536 .1863 .1554 .1915 6.4348 .8085 .9881 .9948 10 1.4166 .1571 9°GG' .1564 .1943 .1584 .1997 6.3138 .8003 .9877 .9946 81° GG' 1.4137 Value Logio Value Login Value Logio Value Logio Degbbes Radians Cosine Cotangent Tangent Sine XIV] Four Place Trigonometric Functions [Characteristics of Log-arith ms omitted — determine by the usual rule from the value] T? A T»T A VR Dkg-rees Sine Tangent Cotangent Cosine XW.d.l^i>-^^.i.^ O 1^ Xm \T Xk XU A.' S^alue Logio Value Loi^io Value Lo?io Value I^Offio .1571 9° 00 .1564 .1^)43 .15^ .1997 6.3138 .8003 .9877 .91>4() 81° 00' 1.4137 .1600 10 .1593 .2022 .1614 .2078 6.1970 .7922 .9872 .9944 50 1.4108 .1029 20 .1622 .2100 .1644 .2158 6.0844 .7842 .9868 .9942 40 1.4079 .1658 30 .1650 .217() .1673 .2236 5.9758 .7764 .98()3 .9940 30 1.4050 .1687 40 .1679 .2251 .1703 .2313 5.8708 .7687 .9858 .9938 20 1.4021 .1716 50 .1708 .2324 .1733 .2389 5.76V)4 .7611 .9853 .9^):36 10 1.3992 .1745 10^00' .1736 .2397 .1763 .2463 5.6713 .7537 .9848 .9934 80° 00' 1..3963 .1774 10 .1765 .2468 .1793 .2536 5.5764 .7464 .9843 .9931 50 1.3934 .1804 20 .1794 .2538 .1823 .2609 5.4845 .7391 .9838 .9929 40 1.3904 .1833 30 .1822 .260<3 .1853 .2680 5.3955 .7320 .9833 .9927 30 1.3875 .1862 40 .1851 .2674 .1883 .2750 5.3093 .7250 .9827 .9924 20 1.3846 .1891 50 .1880 .2740 .1914 .2819 5.2257 .7181 .9822 .9922 10 1.3817 .1920 11°00' .1^)08 .2806 .1944 .2887 5.1446 .7113 .9816 .9919 79° 00' 1.3788 .1949 10 .1937 .2870 .1974 .2953 5.0658 .7047 .9811 .9917 50 1.3759 .1978 20 .1965 .2934 .2004 .3020 4.98i)4 .6980 .9805 .9914 40 1.3730 .2007 30 .1994 .2997 .2035 .3085 4.9152 .6915 .9799 .9<)12 30 1.3701 .2036 40 .2022 .3058 .20(>5 .3149 4.8430 .6851 .9793 .9909 20 1.3672 .2065 50 .2051 .3119 .2095 .3212 4.7729 .6788 .9787 .9907 10 1.3&43 .2094 12° 00' .2079 .3179 .2126 .3275 4.7046 .6725 .9781 .9904 78° 00' 1.3614 .2123 10 .2108 .3238 .2156 .3336 4.6382 .6664 .9775 .9i)01 50 1.3584 .21.53 20 .2136 .3296 .2186 .3397 4.5736 .6603 .9769 .9899 40 1.3555 .2182 30 .21f>4 .3353 .2217 .^3458 4.5107 .6542 .9763 .9896 30 1.3526 .2211 40 .2193 .3410 .2247 .3517 4.4494 .6483 .9757 .9893 20 1.3497 .2240 50 .2221 .3466 .2278 .3576 4.3897 .6424 .9750 .9890 10 1.3468 .2269 13° 00' .2250 .3521 .2309 .3634 4.3315 .6366 .9744 .9887 77° 00' 1.3439 .2298 10 .2278 .3575 .2339 .3691 4.2747 .6309 .9737 .9884 50 IMIO .2327 20 .2306 .3629 .2370 .3748 4.2193 .6252 .9730 .9881 40 1.3381 .2356 30 .23;^ .3682 .2401 .3804 4.1653 .6196 .9724 .9878 30 l.e3352 .2385 40 .2:3(53 .3734 .2432 .3859 4.1126 .6141 .9717 .9875 20 1.3323 .2414 50 .2391 .3786 .2462 .3914 4.0611 .6086 .9710 .9872 10 1.3294 .2443 14° 00' .2419 .3837 .2493 .3968 4.0108 .6032 .9703 .9869 76° 00' 1.3265 .2473 10 .2447 .3887 .2524 .4021 3.9fn7 .5979 .9696 .9866 50 1.3235 .2502 20 .2476 .3937 .2555 .4074 3.9136 .5926 .9689 .9863 40 1.3206 .2531 30 .2504 .3986 .2586 .4127 3.8667 .5873 .9681 .9859 30 1.3177 .2;560 40 .2532 .4035 .2617 .4178 3.8208 .5822 .9674 .985() 20 1.3148 .2589 50 .2560 .4083 .2648 .4230 3.7760 .5770 .9(367 .9853 10 1.3119 .2618 15°00' .2588 .4130 .2679 .4281 3.7321 .5719 .9659 .9849 75° 00' 1.3090 .2647 10 .2616 .4177 .2711 .4331 3.6891 .5669 .9652 .984(5 50 1.3061 .2676 20 .2644 .4223 .2742 .4381 3.6470 .5619 .9644 .9843 40 1.3032 .2705 30 .2672 .4269 .2773 .4430 3.6059 .5570 .9{)36 .9839 30 1.3003 .27:34 40 .2700 .4314 .2805 .4479 3.5656 .5521 .9628 .9836 20 1.2974 .2763 50 .2728 .4359 .2836 .4527 3.5261 .5473 .9621 .9832 10 1.2945 ,2793 16° 00' .2756 .4403 .2867 .4575 3.4874 .5425 .9613 .9828 74° 00' 1.2915 .2822 10 .2784 .4447 .2899 .4622 3.4495 .5378 .9605 .9825 50 1.2886 .2851 20 .2812 .4491 .2931 .4669 3.4124 .5331 .9596 .9821 40 1.2857 .2880 30 .2840 .4533 .2962 .4716 3.3759 .5284 .9588 .9817 30 1.2828 .2()09 40 .2868 .4576 .2994 .4762 3.3402 .5238 .9580 .9814 20 1.2799 .2938 50 .2896 .4618 .3026 .4808 3.3052 .5192 .9572 .9810 10 1.2770 .2f)67 17° 00' .2924 .4659 .3057 .4^53 3.2709 .5147 .9563 .9806 73° 00' 1.2741 .29m 10 .2952 .4700 .3089 .4898 3.2371 .5102 .9555 .9802 50 1.2712 , m5 20 .2979 .4741 .3121 .4^)43 3.2041 .5057 .9546 .9798 40 1.2683 .3054 30 .3007 .4781 I .3153 .4987 3.1716 .5013 .9537 .9794 30 1.2654 .3083 40 .3035 .4821 .3185 .5031 3.1397 .4969 .9528 .9790 20 1.2625 .3113 50 .3062 .4861 .3217 .5075 3.1084 .4925 .9520 .9786 10 1.2595 .3142 18° 00' .3090 .4900 .3249 .5118 3.0777 .4882 .9511 .9782 72° 00' 1.2566 Value Logio Value Logio Value Lo^io Value Logio Degrees Radians COSIXE Cotangent Tangent Sine 140 Four Place Trigonometric Functions [xiv [Characteristics of Logarithms omitted — determine by the usual rule from the value] RADIAJNfi Degrees Sine Tangent Cotangent | Cosine ^%)JX.X/±f^~i.^ o Value Logio Value Logio Value Logio Value Logio .3142 18° 00' .3090 .4900 .3249 .5118 3.0777 .4882 i .9511 .9782 72° 00' 1.2566 .3171 10 .3118 .4939 .3281 .5161 3.0475 .4839 i .9502 .9778 50 1.2537 .3200 20 .3145 .4977 .3314 .5203 3.0178 .4797 .9492 .9774 40 1.2508 .3229 30 .3173 .5015 .3346 .5245 2.9887 .4755 .9483 .9770 30 1.2479 .3258 40 .3201 .5052 .3378 .5287 2.9600 .4713 .9474 .9765 20 1.2450 .3287 50 .3228 .50^)0 .;3411 .5329 2.9319 .4671 I .9465 .9761 10 1.2421 .3316 19° 00' .3256 .5126 .3443 .5370 2.f)042 .4630 .9455 .9757 71° 00' 1.2392 .3345 10 .3283 .5163 .3476 .5411 2.8770 .4589 .9446 .9752 50 1.2363 .3374 20 .3311 .5199 .3508 .5451 2.8.502 .4549 .9436 .9748 40 1.2334 .3403 30 .3338 .5235 .3541 .5491 2.8239 .4509 .9426 .9743 30 1.2:305 .3432 40 .3365 .5270 .3574 .5531 2.7980 .4469 .9417 .9739 20 1.2275 .3462 50 .3393 .5;306 .3607 .5571 2.7725 .4429 .9407 .9734 10 1.2246 .3491 20° 00' .3420 ..5341 .3640 .5611 2.7475 .4389 .9397 .9730 70° 00' 1.2217 .3520 10 .3448 .5375 .3673 .5650 2.7228 .43,50 .9387 .9725 50 1.2188 .3549 20 .3475 .5409 .'Sim .5689 2.6985 .4311 .9377 .9721 40 1.2159 .3578 30 .3502 .5443 .3739 .5727 2.6746 .4273 .9367 .9716 30 1.2130 .3607 40 .3529 .5477 .3772 .5766 2.6511 .42:34 .9356 .9711 20 1.2101 .3636 50 .3557 .5510 .3805 .5804 2.6279 .4196 .9346 .9706 10 1.2072 .3665 21° 00' .3584 .5543 .3839 .5842 2.6051 .4158 .9336 ,9702 69° 00' 1.2043 .3694 10 .3611 .5576 .3872 .5879 2.5826 .4121 .9325 .9697 50 1.2014 .3723 20 .3638 .5609 .3906 .5917 2.5605 .4083 .9315 .9692 40 1.1985 .3752 30 .3665 .5641 .3939 .5954 2.5386 .4046 .9304 .9687 30 1.1956 .3782 40 .3692 .5673 .3973 .5f)91 2.5172 .4009 .9293 .9682 20 1.1926 .3811 50 .3719 .5704 .4006 .6028 2.4960 .3972 .9283 .9677 10 1.1897 .3840 22° 00' .3746 .5736 .4040 .6064 2.4751 .3936 .9272 .9672 68° 00' 1.1868 .:3869 10 .3773 .5767 .4074 .6100 2.4545 .3900 .9261 .9667 50 1.1839 .3898 20 .3800 .5798 .4108 .6136 2.4342 .3864 .9250 .9661 40 1.1810 .3927 30 .3827 .5828 .4142 .6172 2.4142 .3828 .9239 .9656 30 1.1781 .3956 40 .3854 .5859 .4176 .6208 2.3945 .3792 .9228 .9651 20 1.1752 .3985 50 .3881 .5889 .4210 .6243 2.3750 .3757 .9216 .9646 10 1.1723 .4014 23° 00' .3907 .5919 .4245 .6279 2.3559 .3721 .9205 .9640 67° 00' 1.1694 .4043 10 .39:^ .5948 .4279 .6314 2.3369 .3686 .9194 .9635 50 1.1665 .4072 20 .3%1 .5978 .4314 .6348 2.3183 .3652 .9182 .9629 40 1.1636 .4102 30 .3987 .6007 .4348 .6383 2.2998 .3617 .9171 .9624 30 1.1606 .4131 40 .4014 .mm .4383 .6417 2.2817 .3583 .9159 .9618 20 1.1577 .4160 50 .4041 .6065 .4417 .6452 2.2637 .3548 .9147 .9613 10 1.1548 .4189 24° 00' .4067 .6093 .4452 .6486 2.'L.m .3514 .9135 .9607 66° 00' 1.1519 .4218 10 .4094 .6121 .4487 .6520 -2 2286 .3480 .9124 .9602 50 1.1490 .4247 20 .4120 .6149 .4522 .6553 2.2113 .3447 .9112 .9596 40 1.1461 .4276 30 .4147 .6177 .4557 .6587 2.1913 .3413 .9100 .P59() 30 1.1432 .4305 40 .4173 .6205 .4592 .6620 2.1775 .3380 .9088 .9584 20 1.1403 .4334 50 .4200 .6232 .4628 .6654 2.1609 .3346 .9075 .9579 10 1.1374 .4363 26° 00' .4226, .6259 .4663 .6687 2.1445 .3313 .9063 .9573 65° 00' 1.1345 .4392 10 .4253 .6286 .4699 .6720 2.1283 .3280 .9051 .9567 50 1.1:316 .4422 20 .4279 .6313 .47*^ .6752 2.1123 .3248 .9038 .9561 40 1.1286 .4451 30 .4305 .6340 .4770 .6785 2.0965 .3215 .9026 .9555 30 1.1257 .4480 40 .4331 .6366 .4806 .6817 2.0809 .3183 .9013 .9549 20 1.1228 .4509 50 .4358 .6392 .4841 .6850 2.0655 .3150 .9001 .9543 10 1.1199 .4538 26° 00' .4384 .6418 .4877 .6882 2.0503 .3118 .8988 .9537 64° 00' 1.1170 .4567 10 .4410 .6444 .4913 .6914 2.0a53 .3086 .8975 .9530 50 1.1141 .4596 20 .4436 .6470 .4950 .6946 2.0204 .3054 .8962 .9524 40 1.1112 .4625 30 .4462 .6495 .4986 .f]977 2.0057 .3023 .8949 .9518 30 1.1083 .4654 40 .4488 .6521 .5022 .7009 1.9912 .2991 .8936 .9512 20 1.1054 .4683 50 .4514 .6546 .5059 .7040 1.9768 .2960 .8923 .9505 10 1.1025 .4712 27° 00' .4540 .6570 .5095 .7072 1.9626 .2928 .8910 .9499 63° 00' 1.0996 Value Logio Value Logio Value Logio Value Logio Degrees Radians Cosine Cotangent Tangent | Sine | XIV] Four Place Trigonometric Functions [Characteristics of Logarithms omitted — determine by the usual rule from the vahie] Radians Degrees Sine Tangent Cotangent Cosine Value Logio Value Logio Value Logio Value Logio .4712 27° 00' .4540 .6570 .5095 .7072 1.9626 .2928 .8910 .9499 63° 00' 1.091^ .4741 10 .45(^ .6595 .5132 .7103 1.9486 .2897 .8897 .9492 50 1.09(J6 .4771 20 .4592 .6620 .5169 .7134 1.9347 .2866 .8884 .9486 40 1.0937 .4800 30 .4617 .6644 .5206 .7165 1.9210 .2835 .8870 .9479 30 1.0908 .4829 40 .4643 .6668 .5243 .7196 1.9074 .2804 .8857 .9473 20 1.0879 .4858 50 .4669 .6692 .5280 .7226 1.8940 .2774 .8843 .9466 10 1.0850 .4887 28° 00' .4695 .6716 .5317 .7257 1.8807 .2743 .8829 .9459 62° 00' 1.0821 .4916 10 .4720 .6740 .5354 .7287 1.8676 .2713 .8816 .9453 50 1.0792 AM5 20 .4746 .6763 .5392 .7317 1.8546 .2683 .8802 .9446 40 1.0763 .4974 30 .4772 .6787 .5430 .7348 1.8418 .2652 .8788 .9439 30 1.0734 .5003 40 .4797 .6810 .5467 .7378 1.8291 .2(^22 .8774 .9432 20 1.0705 .5032 50 .4823 .6833 .5505 .7408 1.8165 .2592 .8760 .9425 10 1.0676 .5061 29° 00' .4848 .6856 .5543 .7438 1.8040 .2562 .8746 .9418 61° 00' 1.0647 ..5091 10 .4874 .6878 .5581 .7467 1.7917 .2533 .8732 .9411 50 1.0617 .5120 20 .4899 .6901 .5619 .7497 1.7796 .2503 .8718 .9404 40 1.0588 .5149 30 .4924 .6923 .5658 .7526 1.7675 .2474 .8704 .9397 30 1.0559 .5178 40 .4950 .6946 .5696 .7556 1.7556 .2444 .8689 .9390 20 1.0530 .5207 50 .4975 .6968 .5735 .7585 1.7437 .2415 .8675 .9383 10 1.0501 .5236 30° 00' .5000 .6990 .5774 .7614 1.7321 .2386 .8660 .9375 60° 00' 1.0472 .5265 10 .5025 .7012 .5812 .7644 1.7205 .2356 .8646 .9368 50 1.0443 .5294 20 .5050 .7033 .5851 .7673 1.7090 .2327 i .8631 .9361 40 1.0414 .5323 30 .5075 .7055 .58^K) .7701 1.6977 .2299 : .8616 .9353 30 1.0385 .5352 40 .5100 .7076 .5930 .7730 1.6864 .2270 .8f>01 .9346 20 1.0356 .5381 50 .5125 .7097 .5969 .7759 1.6753 .2241 .8587 .9338 10 1.0327 .5411 31° 00' .5150 .7118 .6009 .7788 1.6643 .2212 .8572 .9331 59° 00' 1.0297 .5440 10 .5175 .7139 .6048 .7816 1.6534 .2184 : .8557 .9323 50 1.0268 .5469 20 .5200 .7160 .6088 .7845 1.6426 .2155 ! .8542 .9315 40 1.0239 .5498 30 .5225 .7181 .6128 .7873 1.6319 .2127 1 .8526 .9308 30 1.0210 .5527 40 .5250 .7201 .6168 .7902 1.6212 .2098 .8511 .9300 20 1.0181 .5556 50 .5275 .7222 .6208 .7930 1.6107 .2070 .8496 .9292 10 1.0152 .5585 32° 00' .5299 .7242 .6249 .7958 1.6003 .2042 .8480 .9284 58° 00' 1.0123 .5614 10 .5324 .7262 .6289 .7986 1.5900 .2014 .8465 .9276 50 1.0094 .5643 20 .5348 .7282 .6330 .8014 1.5798 .1986 .8450 .9268 40 1.0065 .5672 30 .5373 .7302 .6371 .8042 1.5697 .1958 .8434 .9260 30 1.0036 .5701 40 .5398 .7322 .6412 .8070 1.5597 .1930 .8418 .9252 20 1.0007 .5730 50 .5422 .7342 .6453 .8097 1.5497 .1903 .8403 .9244 10 .9977 .5760 33° 00' .5446 .7361 .6494 .8125 1.5399 .1875 .8387 .9236 57° 00' .9948 .5789 10 .5471 .7380 .6536 .8153 1.5301 .1847 ! .8371 .9228 50 .9^)19 .5818 20 .5495 .7400 .6577 .8180 1.5204 .1820 ! .8355 .9219 40 .9890 .5847 30 .5519 .7419 .6619 .8208 1.5108 .1792 ; .8339 .9211 30 .9861 .5876 40 .5544 .7438 .6()61 .8235 1.5013 .1765 ; .8323 .9203 20 .9832 .5905 50 .5568 .7457 .6703 .8263 1.4919 .1737 .8307 .9194 10 .9803 .5934 34° 00' .5.592 .7476 .6745 .8290 1.4826 .1710 .8290 .9186 56° 00' .9774 .5963 10 .5(516 .7494 .()787 .8317 1.4733 .1083 ! .8274 .9177 50 .9745 .5992 20 .5640 .7513 .6830 .8344 1.4641 .1656! .8258^ .9169 40 .9716 .6021 30 .5664 .7531 .6873 .8371 1.4550 .16291 .8241 .9160 30 .9687 .6050 40 .5688 .7550 .6916 .8398 1.4460 .1602 .8225 .9151 20 .9657 .6080 50 .5712 .7568 .6959 .8425 1.4370 .1575 .8208 .9142 10 .9628 .6109 35° 00' .5736 .7586 .7002 .8452 1.4281 .1548 .8192 .9134 55° 00' .9599 .6138 10 .5760 .7604 .7046 .8479 1.4193 .1521 j .8175 .9125 50 .9570 .6167 20 .5783 .7622 .7089 .850<5 1.4106 .1494 ; .8158 .911() 40 .9541 .6196 30 .5807 .7640 .7133 .8533 1.4019 .1467 .8141 .9107 30 .9512 .6225 40 .5831 .7657 .7177 .8559 1.39ri4 .1441 .8124 .9098 20 .9483 .6254 50 .5854 .7675 .7221 .8586 1.3848 .1414 .8107 .9089 10 .9454 .6283 36° 00' .5878 .7692 .7265 .8613 1.3764 .1387 .8090 .9080 54° 00' .9425 Value Logio Value Logio Value Logio Value Logio Degrees Radians Cosine Cotangent | Tangent Sine 142 Four Place Trigonometric Functions [xiv [Characteristics of Logarithms omitted — determine by the usual rule from the valuej Radians Degrees Sine Tangent Cotangent Cosine ^alue Logio Value Logic Value Logio Value Logio .6283 36^00' .5878 .7692 .7265 .8613 1.3764 .1387 .8090 .9080 54° 00' .9425 .63] 2 10 .5901 .7710 .7310 .8639 1.3680 .1361 .8073 .9070 50 .9396 .6341 20 .5925 .7727 .7355 .8666 1.3597 .1334 .8056 smi 40 .9367 .6370 30 .5948 .7744 .7400 .8692 1..3514 .1308 .8039 .9052 30 .9338 .6400 40 .5972 .7761 .7445 .8718 1.3432 .1282 .8021 .9042 20 .9308 .6429 50 .5995 .7778 .7490 .8745 1.3351 .1255 .8004 .9033 10 .9279 .6458 37° 00' .6018 .7795 .75.36 .8771 1.3270 .1229 .7986 .9023 53° 00' .9250 .6487 10 .6041 .7811 .7581 .8797 1.3190 .1203 .7969 .9014 50 .9221 .6516 20 .6065 .7828 .7627 .8824 1.3111 .1176 .7951 .9004 40 .9192 .6545 30 .6088 .7844 .7673 .8850 1.3032 .1150 .7934 .8995 30 .9163 .6574 40 .6111 .7861 .7720 .8876 1.2954 .1124 .7916 .8985 20 .9134 .6603 50 .6134 .7877 .7766 .8902 1.2876 .1098 .7898 .8975 10 .9105 .6632 38° 00' .6157 .7893 .7813 .8928 1.2799 .1072 .7880 .8^)65 52° 00' .9076 .6661 10 .6180 .7910 .7860 .8954 1.2723 .1046 .7862 .8955 50 .9047 .6690 20 .6202 .7926 .7907 .8980 1.2647 .1020 .7844 .8945 40 .9018 .6720 30 .6225 .7941 .7954 Sm6 1.2572 .0994 .7826 .8935 ■SO .8988 .6749 40 .6248 .7957 .8002 .9032 1.2497 .0968 .7808 .8925 20 .8959 .6778 50 .6271 .7973 .8050 .9058 1.2423 .0942 .7790 .8915 10 .8930 .6807 39° 00' .6293 .7989 .8098 .9084 1.2349 .0916 .7771 .8905 51°00' .8901 .68.36 10 .631(5 .8004 .8146 .9110 1.2276 .0890 .7753 .8895 50 .8872 .6865 20 .6338 .8020 .8195 .9135 1.2203 .0865 .7735 .8884 40 .8843 .6894 30 .6.361 .8035 .8243 .9161 1.2131 .0839 .7716 .8874 30 .8814 .6923 40 .6383 .8050 .8292 .9187 1.2059 .0813 .7698 .8864 20 .8785 .6952 50 .6406 .8066 .8342 .9212 1.1988 .0788 .7679 .8853 10 .8756 .6981 40° 00' .6428 .8081 .8391 .9238 1.1918 .0762 .7660 .8843 50° 00' .8727 .7010 10 .6450 .8096 .8441 .9264 1.1847 .0736 .7642 .8832 50 .8698 .7039 20 .6472 .8111 .8491 .9289 1.1778 .0711 .7623 .8821 40 .8668 .7069 30 .6494 .8125 .8541 .9315 1.1708 .0685 .7604 .8810 30 .8639 .7098 40 .6517 .8140 .8591 .9.341 1.1640 .0659 .7585 .8800 20 .8610 .7127 50 .6539 .8155 .8642 .9366 1.1571 .06.34 .7566 .8789 10 .8581 .7156 41°00' .6561 .8169 .8693 .9392 1.1504 .0608 .7547 .8778 49° 00' .8552 .7185 10 .6583 .8184 .8744 .9417 1.1436 .0583 .7528 .8767 50 .8523 .7214 20 .6604 .8198 .8796 .9443 1.1369 .0557 .7509 .8756 40 .8494 .7243 30 .6626 .8213 .8847 .9468 1.1303 .0532 .7490 .8745 30 .8465 .7272 40 .6648 .8227 .8899 .9494 1.1237 .0506 .7470 .8733 20 .8436 .7301 50 .6670 .8241 .8952 .9519 1.1171 .0481 .7451 .8722 10 .8407 .7330 42° 00' .6691 .8255 .9001 .9544 1.1106 .0456 .7431 .8711 48° 00' .8378 .7359 10 .6713 .8269 .9057 .9570 1.1041 .0430 .7412 .8699 50 .8348 .7389 20 .6734 .8283 .9110 .9595 1.0977 .0405 .7392 .8688 40 .8319 .7418 30 .6756 .8297 .9163 .9621 1.0913 .0379 .7373 .8676 30 .8290 .7447 40 .6777 .8311 .9217 .9646 1.0850 .0354 .7353 .8665 20 .8261 .7476 50 .6799 .8324 .9271 .9671 1.0786 .0329 .7333 .8653 10 .8232 .7505 43° 00' .6820 .8338 .9325 .9697 1.0724 .0303 .7314 .8641 47° 00' .8203 .7534 10 .6841 .8351 .9380 .9722 1.0661 .0278 .7294 .8629 50 .8174 .7563 20 .6862 .8365 .9435 .9747 1.0599 .0253 .7274 .8618 40 .8145 .7592 30 .6884 .8378 .9490 .9772 1.0538 .0228 .7254 .8606 30 .8116 .7621 40 .6905 .8391 .9545 .9798 1.0477 .0202 .7234 .8594 20 .8087 .7650 50 .6926 .8405 .9601 .9823 1.0416 .0177 .7214 .8582 10 .8058 .7679 44° 00' .6947 .8418 .9657 .9848 1.0.355 .0152 .7193 .8569 46° 00' .8029 .7709 10 .6%7 .8131 .9713 .9874 1.0295 .0126 .7173 .8557 50 .7999 .7738 20 .6988 .8444 .9770 .9899 1.02a5 .0101 .7153 .8545 40 .7970 .7767 30 .7009 .84.57 .9827 .9924 1.0176 .0076 .7133 .8532 30 .7941 .7796 40 .7030 .8469 .9884 .9949 1.0117 .0051 .7112 .8520 20 .7912 .7825 50 .7050 .8482 .9942 .9975 1.0058 .0025 .7092 .8507 10 .7883 .7854 46° 00' .7071 .8495 1.0000 .0000 1.0000 .0000 .7071 .8495 45° 00' .7854 Value Logio Value Logjo Value Logjo Value Logio Degrees Radians Cosine Cotangent Tangent Sine SLIDE-RULE I II (J) (S) (S) -■I I i iiii Directions A reasonably accurate slide-rule may be made by the student, for temporary practice, as follows. Take three strips of heavy stiff cardboard l'^3 wide by &' long; these are shown in cross-section in (1), (2), (8) above. On (3) paste or glue the adjoining cut of the slide rule. Then cut strips (2) and (3) accurately along the lines marked. Paste or glue the pieces together as shown in (4) and (5). Then (5) forms the slide of the slide-rule, and it will fit in the groove in (4) if the work has been carefully done. Trim off the ends as shown in the large cut. < CO o Q 00 I> CD CO 03 O) 00 t> CD CO = = m ^:^ = — Oi = = m = = = = tr- E E = E E E = = 1 io t> = = -o = = = = E ~ = E lO 1 1 :_ E = :i 1 -~ = j-iO -i = ZEE = - = = 1 1 " ~ E — = 1 - CO - = zz - "i i -= = -co = EE E E — — - ;| E^ CO (M = = = E — Z E E 1 1 - 1 = - t CO i = E E E E E E = — _j — -^ — = — < CD O a ^ _y ; iiiii mm mm Mm m m lit iiillPi !i ill I! ill i ||!|j|!jifl||i| lii! iilliiiiiil i ill iiif I P illf!!itJljij!|jj|i||!!i!|jl{S ,._ ll I ill I ill^i lii i If i iiiffi iiiliii mmm II I! I |i|l!!ii!|IJij Iniliiiilii ^mmm I