_ _, {)O' THE |J THE GIFT OF jj| HE^1~d a.it d id ^i~~l~~u~~iilgnljJ M^O _ — 1N 'i: -- -,_1~ 1..>. TIRE GIANT OF l } n 0xj is: i o v C A l ot2 ffi ~limi I3.t1a ' Iil% l fi _ 7 _ _ _r _ U~g$ _ -s ~ PLANE AND SPHERICAL TRIGONOMETRY BY ELMER A. LYMAN MICHIGAN STATE NORMAL COLLEGE AND EDWIN C. GODDARD UNIVERSITY OF MICHIGAN ALLYN AND BACON 33oston ant C)icalgo COPYRIGHT, 1899, 1900, BY ELMER A. LYMAN A.ND EDWIN C. GODDARD. Norbyaob jorfoo J. S. Cushing & Co. - Berwick & Smith Norwood Masi. U.S.A. PREFACE. MANY American text-books on trigonometry treat the solution of triangles quite fully; English text-books elaborate analytical trigonometry; but no book available seems to meet both needs adequately. To do that is the first aim of the present work, in the preparation of which nearly everything has been worked out and tested by the authors in their classes. The work entered upon, other features demanded attention. For some unaccountable reason nearly all books, in proving the formulae for functions of a ~ 1i, treat the same line as both positive and negative, thus vitiating the proof; and proofs given for acute angles are (without further discussion) supposed to apply to all angles, or it is suggested that the student can draw other figures and show that the formulae hold in all cases. As a matter of fact the average student cannot show anything of the kind; and if he could, the proof would still apply only to combinations of conditions the same as those in the figures actually drawn. These difficulties are avoided by so wording the proofs that the language applies to figures involving any angles, and to avoid drawing the indefinite number of figures necessary fully to establish the formulae geometrically, the general case is proved algebraically (see page 58). Inverse functions are introduced early, and used constantly. Wherever computations are introduced they are made by means of logarithms. The average student, using logarithms for a short time and only at the end of the subject, straightway forgets what manner of things they are. It is hoped, by dint of much,1(ctice, extended over as long a time as possible, to give the student a command of logarithms that will stay. The fundamental formule of trigonometry must be memorized. There is no substitute for this. For this purpose oral work is introduced, and there are frequent lists of review problems involving all principles and formulae previously developed. These lists serve the iii iv PREFACE. further purpose of throwing the student on his own resources, and compelling him to find in the problem itself, and not in any model solution, the key to its solution, thus developing power, instead of ability to imitate. To the same end, in the solution of triangles, divisions and subdivisions into cases are abandoned, and the student is thrown on his own judgment to determine which of the three possible sets of formulae will lead to the solutions with the data given. Long experience justifies this as clearer and simpler. The use of checks is insisted upon in all computations. For the usual course in plane trigonometry Chapters I-VII, omitting Arts. 26, 27, contain enough. Articles marked * (as Art. * 26) may be omitted unless the teacher finds time for them without neglecting the rest of the work. Classes that can accomplish more will find a most interesting field opened in the other chapters. More problems are provided than any student is expected to solve, in order that different selections may be assigned to different students, or to classes in different years. Do not assign work too fast. Make sure the student has memorized and can use each preceding formu la, before taking up new ones. No complete acknowledgment of help received could here be made. The authors are under obligation to many for general hints, and to several who, after going over the proof with care, have given valuable suggestions. The standard works of Levett and Davison, Hobson, Henrici and Treutlein, and others have *been freely consulted, and while many of the problems have been prepared by the authors in their class-room work, they have not hesitated to take, from such standard collections as writers generally have drawn upon, any problems that seemed better adapted than others to the work. Quality has not been knowingly sacrificed to originality. Corrections and suggestions will be gladly received at any time. E. A. L., YPSILANTI. E. C. G., ANN ARBOR. October, 1900. CONTENTS. CHAPTER I. ANGLES MEASUREMENT OF ANGLES. PAGE Angles; magnitude of angles... 1 Rectangular axes; direction.. 2 Measurement; sexagesimal and circular systems of measurement; the radian...... 3 Examples...... 6 CHAPTER II. THE TRIGONOMETRIC FUNCTIONS. Function defined.......... 8 The trigonometric functions........ 9 Fundamental relations..... 11 Examples... 14 Functions of 0~, )3~, 45,, 90~...... 15 Examples..18 Variations in the trigonometric functions.. 19 Graphic representation of functions.... 22 Examples.......27 CHAPTER III. FUNCTIONS OF ANY ANGLE-INVERSE FUNCTIONS. Relations of functions of - 0, 90~ ~, 180~ ~ 0, 270~ ~ 0 to the functions of 0.......... 29 Inverse functions. 35 Examples............ 36 Review......... 38 CHAPTER IV. COMPUTATION TABLES. Natural functions..... 40 Logarithms.......40 Laws of logarithms...... 42 Use of tables....... 45 Cologarithms....... 49 Examples. 50 vi CONTENTS. CHAPTER V. APPLICATIONS. Measurements of heights and distances Common problems in measurement. Examples CHAPTER VI. GENERAL FORMULAE TRIGONOMETRIC EQUATIONS AND IDENTITIES. Sine, cosine, tangent of a. Examples Sin 0 + sin q, 0 + cos........ Examples Functions of the double angle Functions of the half angle Examples Trigonometric equations and identities Method of attack... Examples Simultaneous trigonometric equations Examples CHAPTER VII. TRIANGLES. PAGE 51 52 54 56 59 61 62 63 64 64 66 66 67 69 70 Laws of sines, tangents, and cosines....72 Area of the triangle. 76 Solution of triangles.......... 76 Ambiguous case........... 78 Model solutions........... 80 Examples..........83 Applications........... 84 Review........ 86 CHAPTER VIII. MISCELLANEOUS. Incircle, circumcircle, escribed circle...... 92 Orthocentre, centroid, medians....... 94 Examples............ 96 CHAPTER IX. SERIES. Exponential series........97 Logarithmic series...... 99 Computation of logarithms....... 100 I)e Moivre's theorem......... 103 Computation of natural functions.......104 Hyperbolic functions..........109 Examples.........110 CONTENTS. vii CHAPTER X. SPHERICAL TRIGONOMETRY. Spherical triangles General formulae Right spherical triangles. Area of spherical triangles Examples. PAGE. 112. 114. 123. 125. 128 CHAPTER XI. SOLUTION OF SPHERICAL TRIANGLES. General principles Fornlulae for solution. Model solutions.. 129.130.131 Ambiguous cases.......... 132 Right triangles.... 134 Species............ 135 Examples........... 137 Applications to Geodesy and Astronomy...... 138 PLANE TRIGONOMETRY. CHAPTER I. ANGLES-MEASUREMENT OF ANGLES. 1. Angles. It is difficult, if not impossible, to define an angle. This difficulty may be avoided by telling how it is formed. If a line revolve about one of its points, an angle is generated, the magnitude of the angle depending on the amount of the rotation. Thus, if one side of the angle 0, as OR, be originally in the position OX, and be revolved about the point 0 to the position in the figure, the angle XOR is generated. R OX is called the initial line,: and aly position of OR the terminal line of the angle x formed. The angle 0 is,considered positive if gener- FIG.. ated by a counter-clockwise rotation of OR, and hence negative if generated by a clockwise rotation. The magnitude of 0 depends on the amount of rotation of OR, and since the amount of such rotation may be unlimited, there is no limit to the possible magnitude of angles, for, evidently, the revolving line may reach the position OR by rotation through an acute angle 0, and, likewise, by rotation through once, twice,..., n times 360~, plus the acute angle 0. So that XOR may mean the acute angle 0, 0 + 360~, 0 + 720~,, 0 + n 360~. 1 2 PLANE TRIGONOMETRY. In reading an angle, read first the initial line, then the terminal line. Thus in the figure the acute angle XOR, or xr, is a positive angle, and ROX, or rx, an equal negative angle. Ex. 1. Show that if the initial lines for 2, -, -2, - -, right angles are the same, the terminal lines may coincide. 2. Name four other angles having the same initial and terminal lines as 1 of a right angle; as 5 of a right angle; as 3 of a right angle. 2. Rectangular axes. Any plane surface may be divided by two perpendicular straight lines XX' and YY' into four portions, or quadrants. Y XX' is known as the x-axis, YY' as the y-axis, and the two together are called axes of referx-'.... - x ence. Their intersection 0 is the origin, and the four portions of the plane surface, XOY, YOX', Y' X' OY, Y' OX, are called respecFIG. 2. tively the first, second, third, and fourth quadrants. The position of any point in the plane is determined when we know its distances and directions from the axes. 3. Any direction may be considered positive. Then the opposite direction must be negative. Thus, if AB represents any positive line, BA is an equal nega- A B tive line. Mathematicians usually consider lines measured in the same direction as OX or OY (Fig. 2) as positive. Then lines measured in the same direction as OX' or OY' must be negative. The distance of any point from the y-axis is called the abscissa, its distance from the x-axis the ordinate, of that point; the two together are the coordinates of the point, usually denoted by the letters x and y respectively, and written (x, y). ANGLES- MEASUREMENT. 3 When taken with their proper signs, the coordinates define completely the position of the point. Thus, if the point P is + a units from YY', and + b units from XX', any convenient unit of length being chosen, the position of Y P is known. For we have only to measure P' a distance ON equal to a units along OX, and then from N measure a distance b units parallel to OY, and we arrive at the O N position of the point P, (a, b). In like manner we may locate P', (- a, b), in the p" p"I second quadrant, P", (- a, - b), in the Y third quadrant, and P"', (a, -b), in FIG. 3. the fourth quadrant. Ex. Locate (2, -2); (0,0); (-8, -7); (0, 5); (-2, 0); (2, 2); (m, n). 4. If OX is the initial line, 0 is said to be an angle of the first, second, third, or fourth quadrant, according as its terminal line is in the first, second, third, or fourth quadrant. It is clear that as OR rotates its quality is in no way affected, and hence it is in all positions considered positive, and its extension through 0, OR', negative. The student should notice that the initial line may take any position and revolve in either direction. While it is customary to consider the counter-clockwise rotation as forming a positive angle, yet the condiR X ' R tions of a figure may be such -/ \ i / that a positive angle may be ~X%...x x \ generated by a clockwise rotation. Thus the angle XOR in ~' R'R ' i each figure may be traced as FIG. 4. a positive angle by revolving the initial line OX to the position OR. No confusion can result if the fact is clear that when an angle is read XOR, OX is considered a positive line revolving to the position OR. OX' and OR' then are negative lines in whatever directions drawn. These conceptions are mere matters of agreement, and the agreement may be determined in a particular case by the conditions of the problem quite as well as by such general agreements of mathematicians as those referred to in Arts. 3 and 4 above. 5. Measurement. All measurements are made in terms of some fixed standard adopted as a unit. This unit must 4 PLANE TRIGONOMETRY. be of the same kind as the quantity measured. Thus, length is measured in terms of a unit length, surface in terms of a unit surface, weight in terms of a unit weight, value in terms of a unit value, an angle in terms of a unit angle. The measure of a given quantity is the number of times it contains the unit selected. Thus the area of a given surface in square feet is the number of times it contains the unit surface 1 sq. ft.; the length of a road in miles, the number of times it contains the unit length 1 mi.; the weight of a cargo of iron ore in tons, the number of times it contains the unit weight 1 ton; the value of an estate, the number of times it contains the unit value $1. The same quantity may have different measures, according to the unit chosen. So the measure of 80 acres, when the unit surface is 1 acre, is 80, when the unit surface is 1 sq. rd., is 12,800, when the unit surface is 1 sq. yd., is 387,200. What is its measure in square feet? 6. The essentials of a good unit of measure are: 1. That it be invariable, i.e. under all conditions bearing the same ratio to equal magnitudes. 2. That it be convenient for practical or theoretical purposes. 3. That it be of the same kind as the quantity measured. 7. Two systems of measuring angles are in use, the sexagesimal and the circular. The sexagesimal system is used in most practical applications. The right angle, the unit of measure in geometry, though it is invariable, as a measure is too large for convenience. Accordingly it is divided into 90 equal parts, called degrees. The degree is divided into 60 minutes, and the minute into 60 seconds. Degrees, minutes, seconds, are indicated by the marks ~ ' ", as 36~ 20' 15". The division of a right angle into hundredths, with subdivisions into hundredths, would be more convenient. The French have proposed such MEASUREMENT OF ANGLES. 5 a centesimal system, dividing the right angle into 100 grades, the grade into 100 minutes, and the minute into 100 seconds, marked g' ", as 50s 70' 28". The great labor involved in changing mathematical tables, instruments, and records of observation to the new system has prevented its adoption. 8. The circular system is important in theoretical considerations. It is based on the fact that for a given angle the ratio of the length of its arc to the length of the radius of that arc is constant, i.e. for a fixed angle the ratio ar: radius is the same CoR no matter what the length of the A B radius. In the figure, for the angle 0, 0 ABC 'X OA OB 00' -— OA ~ ~ _ O B ~_ - - -....FIG. 5. AA' BB' CC' That this ratio of arc to radius for a fixed angle is constant follows from the established geometrical principles: 1. The circumference of any circle is 2 7r times its radius. 2. Angles at the centre are in the same ratio as their arcs. The Radian. It follows that an angle whose arc is equal in length to the radius is a constant angle for all circles, since in four right angles, or the perigon, there are always 27r such angles. This constant angle, whose are is equal in length to the radius, is taken as the unit angle of circular / ~/^ mneasure, and is called the radian. From / the definition we have 4 right angles = 360~ = 2 7r radians, 2 right angles = 180~ = 7 radians, FIG. 6. 1 right angle = 90~= 7 radians. 7r is a numerical quantity, 3.14159+, and not an angle. When we speak of 180~ as 7r, 90~ as 7, etc., we always mean 7r radians, r radians, etc. 2 2 6 PLANE TRIGONOMETRY. 9. To change from one system of measurement to the other we use the relation, 2 7r radians = 360~. 180~ 1. radian = -- 57~.2958-; 7r i.e. the radian is 57~.3, approximately. Ex. 1. Express in radians 75~ 30'. 75~ 30' = 75~.5; 1 radian = 57~.3..-. 75~ 30' = 75 1.317 radians. 57.3 2. Express in degree measure 3.6 radians. 1 radian = 57~.3.... 3.6 radians = 3.6 x 57~.3 = 206~ 16' 48". EXAMPLES. 1. Construct, approximately, the following angles: 50~, - 20~, 90~, 179~, -135~, 400~, -380~, 1140 radians, n radians, - radians, 4 3 6 3 7r radians, - 37r radians, -7 radians. Of which quadrant is each angie? 2. What is the measure of: (a) X of a right angle, when 30~ is the unit of measure? (b) an acre, when a square whose side is 10 rds. is the unit? (c) m miles, when y yards is the unit? 3. What is the unit of measure, when the measure of 21 miles is 50? 4. The Michigan Central R.R. is 535 miles long, and the Ann Arbor R.R. is 292 miles long. Express the length of the first in terms of the second as a unit. 5. What will be the measure of the radian wvhen the right angle is taken for the unit? Of the right angle when the radian is the unit? 6. In which quadrant is 45~? 10~? - 60~? 145~? 1145~? -725~? Express each in right angles; in radians. 7. Express in sexagesimal measure r 1 7r 4 radians. -, ~, 1, 6.28, 1,-, -i-, radians. 3 71-2 3 3 EXAMPLES. 7 8. Express in each system an interior angle of a regular hexagon; an exterior angle. 9. Find the distance in miles between two places on the earth's equator which are 11~ 15' apart. (The earth's radius is about 3963 miles.) 10. Find the length of an arc which subtends an angle of 4 radians at the centre of a circle of radius 12 ft. 3 in. 11. An arc 15 yds. long contains 3 radians. Find the radius of the circle. 12. Show that the hour and minute hands of a watch turn through angles of 30' and 6~ respectively per minute; also find in degrees and in radians the angle turned through by the minute hand in 3 hrs. 20 mins. 13. Find the number of seconds in an arc of 1 mile on the equator; also the length in miles of an arc of 1' (1 knot). 14. Find to three decimal places the radius of a circle in which the arc of 71~ 36' 3'.6 is 15 in. long. 15. Find the ratio of 7r to 5~. 6 16. What is the shortest distance measured on the earth's surface from the equator to Ann Arbor, latitude + 42~ 16' 48"? 17. The difference of two angles is 10~, and the circular measure of their sum is 2. Find the circular measure of each angle. 18. A water wheel of radius 6 ft. makes 30 revolutions per minute. Find the number of miles per hour travelled by a point on the rim. CHAPTER II. THE TRIGONOMETRIC FUNCTIONS. 10. Trigonometry, as the word indicates, was originally concerned with the measurement of triangles. It now includes the analytical treatment of certain functions of angles, as well as the solution of triangles by means of certain relations between the functions of the angles of those triangles. 11. Function. If one quantity depends upon another for its value, the first is called a function of the second. It always follows that the second quantity is also a function of the first; and, in general, functions are so related that if one is constant the other is constant, and if either varies in value, the other varies. This relation may be extended to any number of mutually dependent quantities. Illustration. If a train moves at a rate of 30 miles per hour, the distance travelled is a function of the rate and time, the time is a function of the rate and distance, and the rate is a function of the time and distance. Again, the circumference of a circle is a function of the radius, and the radius of the circumference, for so long as either is constant the other is constant, and if either changes in value, the other changes, since circumference and radius are connected by the relation C = 2 7rR. N R Once more, in the right triangle NOP, the ratio of any two sides is a function of the angle a, because o - L N' N x all the right triangles of which a is FIG. 7. one angle are similar, i.e. the ratio 8 THE TRIGONOMETRIC FUNCTIONS. 9 of two corresponding sides is constant so long as a is constant, and varies if a varies. Thus, the ratios NP N'P' N'P" OP OP' OP" ON ON' ON"t and NP N'P' = N7 p"' etc., depend on a for their values, i.e. are functions of a. 12. The trigonometric functions. In trigonometry six functions of angles are usually employed, called the trigonometric functions. By definition these functions are the six ratios between the sides of the triangle of reference of the given angle. The triangle of reference is formed by drawing from some point in the initial line, or the initial line produced, a perpendicular to that line meeting the terminal line of the angle. aY XY. rR Xr J? x N y' Y. /-r _ Ir N X 0 y' Y XR~' OX X Y P P 7?1? Y/ Y' FIG. 8. Let a be an angle of any quadrant. Each triangle of reference of a, NOP, is formed by drawing a perpendicular to OX, or OX produced, meeting the terminal line OR in P. 10 PLANE TRIGONOMETRY. If a is greater than 360~, its triangle of reference would not differ from one of the above triangles. It is perhaps worthy of notice that the triangle of reference might be defined to be the triangle formed by drawing a perpendicular to either side of the angle, or that side produced, meetP/ ing the other side or the other side produced. N In the figure, NOP is in all cases the triangle N P a -c of reference of a. The principles of the fol0 P... lowing pages are the same no matter which.~..-'J;~ of the triangles is considered the triangle of FIG. 9. reference. It will, however, be as well, and perhaps clearer, to use the triangle defined under Fig. 8, and we shall always draw the triangle as there described. 13. The trigonometric functions of a (Fig. 8) are called the sine, cosine, tangent, cotangent, secant, and cosecant of a. These are abbreviated in writing to sin a, cos a, tan a, cot a, sec a, csc a, and are defined as follows: sin a = perp whence = r sin a; hyp. r base x cosa = s=- whence x= r cos a; hyp. r tan a = prp =, whence y = xtan a; base; base x cot a = pe = whence x = y cot a; perp. y see a = hyp = whence r = x seca; base x csca = phyp. =, whence r ycsca. perp. y 1 - cos a and 1 - sin a, called versed-sine a and coversed-sine a, respectively, are sometimes used. Ex. 1. Write the trigonometric functions of /, NPO (Fig. 8), and compare with those of a above. The meaning of the prefix co in cosine, cotangent, and cosecant appears from the relations of Ex. 1. For the sine of an angle equals the cosine, i.e. the complement-sine, of the complement of that angle; the tangent THE TRIGONOMETRIC FUNCTIONS. 11 of an angle equals the cotangent of its complementary angle, and the secant of an angle equals the cosecant of its complement- ary angle. B 2. Express each side of triangle ABC in b\ a terms of another side, and some function of an angle in all possible ways, as a = b tan A, etc. FIG. 10. 14. Constancy of the trigonometric functions. It is ilnportant to notice why these ratios are functions of the angle, i.e. are the same for equal angles and different for unequal angles. This is shown by the principles of similar triangles. a ' a FIG. 11. In each figure show that in all possible triangles of reference for a the ratios are the same, but in the triangles of reference for a and a', respectively, the ratios are different. The student must notice that sin a is a single symbol. It is the name of a number, or fraction, belonging to the angle a; and if it be at any time convenient, we may denote sin a by a single letter, such as o, or x. Also, sin2 a is an abbreviation for (sin a)2, i.e. for (sin a) x (sin a). Such abbreviations are used because they are convenient. Lock, Elementary Trigonometry. 15. Fundamental relations. From the definitions of Art. 13 the following reciprocal relations are apparent: 1 1 sin a = -- c, sc oa — = cos a = 1 -- sec — ' = see a cos a tan a = 1-, cot a = ~cot a ~tan a Also from the definitions, tan a = si a cot a =. Cos a sin, t VVu v 12 PLANE TRIGONOMETRY. From the right triangle NOP, page 9, y2 + x2 = r2; whence (1) y + 1, r2 r2 (2) + X2 X2 x2 r2 (3) y2 y2 From (1) sin2 a + cos2 a= 1; sin a = V1- cos2 c; cos =? (2) tan2a+1=sec2a; tan a = Vsec2 a-; seca=? (3) 1+cot2a=csc2 a; cota= Vcsc2 a-1; csca=? The foregoing definitions, and fundamental relations are of the highest importance, and must be mastered at once. The student of trigonometry is helpless without perfect familiarity with them. These relations are true for all values of a, positive or negative, but the signs of the functions are not in all cases positive, as appears from the fact that in the triangles of reference in Fig. 8 x and y are sometimes negative. The equations sin a = ~ V/ - cos2 a, tan a = ~ sec2 a - 1, cot a = ~+ csc2 a - 1, have the double sign ~. Which sign is to be used in a given case depends on the quadrant in which a lies. 16. The relations of Art. 15 enable us to express any function in terms of any other, or when one function is given, to find all the others. Ex. 1. To express the other functions in terms of tangent: sin _ 1 1 tana; cota= 1 csc a v' + cot2a 1 + tan2a tan a 1 1 cos a = - - =; sec a = V' + tan2 a; sec a /l + tan2 a '/1 + tan2 a tan a = tan a; csc a = tan a THE TRIGONOMETRIC FUNCTIONS. 13 In like manner determine the relations to complete the following table: sin a cos a tan a cot a sec a csc a tan a sin a sin + tan2 a 1 cos a VI + tan2 a tan a tan a cot a --- tan a sec x V/1 + tan2 a /1 + tan2 a CSC a tan a 2. Given sin a = j; find the other functions. cos ( =1 1- X = ~V7; tan a = - = ' 7; iV7 V7 cot 1= 1 =-}/7; sec a 1 4 =4/7; cscx =1 4 cota= a-= v'/; sec a s - 3V7 V//7 V7 3 3. Given tan > + cot q = 2; find sin <. tan + = 2, tan2 - 2tan + 1 = 0, tan = 1. tnq+ tan t5.'. sin (- = tan 2. V/1 + tan2 < Or, expressing in terms of sine directly, si- + cos = 2, cos 4 sin 4 sin2 q + cos2 4 = 2 sin < cos 4, sin2 ( - 2 sin cos 4b + cos2 =_ 0; whence sin < -cos 4 = 0, sin 4 = cos b..'. sin ( = v2. 4. Prove sec4 x - sec2 x = tan2 x + tan4 x. sec4x - sec2x = sec2x(sec2x - 1)=(l + tan2 x)tan2 x = tan2 x + tan4 x. 5. Prove sin6 y + cos6 y = 1 - 3 sin2 y cos2 y. sin6 y + cos6 y = (sin2 y + cos2 y) (sin4 y - sin2 y cos2 y + cos4 y) = (sin2 y + cos2 y)2 _ 3 sin2 y cos2 y = 1 - 3 sin2 y cos2 y. 14 PLANE TRIGONOMETRY. tanzx cotzx 6. Prove + = secz cscz+ 1. 1 -cot z 1 - tail z sin z Cos z tan z cot z Cos z sin z 1- cot z~1 -- tanz Cosz 1 sin z sill Z COS Z si il2 z Cos2 z Si,12 + COS2 cos z (sin z - cos z) sin z (cos z - sin z) sin3 z - cos z sin2 Z + sin z cos z + cos2 Z sin z cos z (sin z - cos z) sill z cos z 1 + sin z cosz sz - - ~~~~~+ 1 =seczxcscz+3-1. sin z cos z sin z cos z In solving problems like 3, 4, 5, and 6 above, it is usually safe, if no other step suggests itself, to express all other functions of one member in terms of sine and cosine. The resulting expression may then be reduced by the principles of algebra to the expression in the other member of the equation. For further suggestions as to the solution of trigonometric equations and identities see page 66. EXAM PLES. 1. Find the values of all the functions of a, if sin a = 2; if tan a =3 L,, if Cot a = 3 c a = -%/if sec a = 2; if cos a = I / io;if csca 2. 2. Compute the functions of each acute angle in the right triangles whose sides are: (1) 3, 4, 5; (2) 8, 15, 17; (3) 480, 31, 481; (4) a, b, c; (5) 2 xy X~2 Y2 ~ Y X-Y X-Y 3. If Cos a = I find the value of in a+tanae cos a-cot a 4. If 2 cos a = 2 - sin a, find tan a. 5. If sec2 a csc2 a- 4 = 0, find cot a. 6. Solve for sin/3 in 13 sin/3 + cos2/3= 11. Prove 7. sin44 - Cos4 = -1- 2cos240. 8. (sin a + cos a)(sin a - cos a)- 2 sin2 a -1. 9. (sec a + tan a) (sec a - tana)= 1. 10. cos2, (sec2 / 3- 2 sin2 /) = cos4 / + sin4 3. 11. tan v +secv= Cos V 1 - sin v sinw -1+cosw 3.2. 1 - cos w sin w 13. (sec + 1)(1 - cos 0) = tan2 9 cos 9. FUNCTIONS OF CERTAIN ANGLES. 15 14. sin4 t - sin2 t = cos4 t - cos2 t. 15. sin_ + 1 + sin sec2 (csc/ +1). 1 - sin 3 sin /3 16. (tan A + cot A)2 = sec2 A cs2 A. 17. sec2x - sin2 x = tan2x + cos2x. In the triangle ABC, right angled at C, 18. Given cos A = 8, BC = 45, find tan B, and AB. m'2 2 19. If cos A = - n and AB = m2 + n2, find A C and BC rm2 +- n2' 20. If AC = m + n, BC = m - n, find sin A, cos B. 21. In examples 18, 19, 20, above, prove sin2A + cos2A 1; 1 + tan2 A = sec2A. 17. Functions of certain angles. The trigonometric functions are numerical quantities which may be determined for any angle. In general these values are taken from tables prepared for the purpose, but the principles already studied enable us to calculate the functions of the following angles. 18. Functions of 0~. If a value of y is very small, and decreases as a diminishes. Clearly, when a approaches 0~ as a limit, y likewise approaches 0, and x approaches r, be a very small angle, the P F. 12. N FIG. 12. so that when a = 0~, y = 0, and x = r..s. sn0=Y=0, cot 0~=, r tan 00 cos 0~ = = 1, r 1 sec 0~ = --, cos 0~ tan ~= - =, C 0~= 1 =X O. tan00 Y~0, ese O s x sin 0~ In the figure of Art. 18, by diminishing a it is clear that we can make y as small as we please, and by making a small enough, we can make the value of y less. than any assignable quantity, however small, so that sin a approaches as a limit 0. This is what we mean when we say sin 0~ = 0. In like manner, it is evident that, by sufficiently diminishing a we can make cot a greater than any assignable quantity. This we express by saving cot 0~ = o. 16 16 ~~PLANE TRIGONOMETRY. 19. Functions of 300. Let NOP be the triangle of reference for an -angle of 300. Make triangle NOP' = NOP. Then P OP' is an equilateral triangle (why?), and ON bisects PP'. Hence PP=r = 2y. Also x Vr2 y =VW ji\yJ., esec300= 2, -1-\V3 cot 300=-V3. FIG. 13. Cos 30 = - __ r 2 tan30'-"y' Y - x YV\3 20. Functions of 450. Let NOP be the triangle of reference. If angle NOP = 450, OPN= 450. Y R FIG. 14. Then Y=x, and r=Vx~y= 2x~2=xV2. sin 450 /.x _ r xV\2 cos 450 - -__9 tan 450=Y = ~1. x x cot 450, sec 450, cse 450. Find, FUNCTIONS OF CERTAIN ANGLES. 17 21. Functions of 60~. The functions of 60~ may be computed by means of the figure, or they may be written from the func- tions of the complement, or 30~. Let the student in both ways show that sin60~= 3, cos 60~= ( r -- 2 V V ~ 2- O N P tan 60 = 3. FIG. 15. Compute also the other functions of 60~. 22. Functions of 90~. If a be an angle very near 90~, the value of x is very small, and deyI~ IP creases as a increases toward 90~. Clearly when a approaches 90~ as a limit, x approaches 0, and y approaches r, so that when O lo X oa=90~, x=0, Y=r. FIG. 16... sin 90~ = 1, cos 90~ = 0, tan 90~ = o. Compute the other functions. Also find the functions of 90~ from those of its complement, 0~. 23. It is of great convenience to the student to remember the functions of these angles. They are easily found by recalling the relative values of the sides of the triangles of reference for the respective angles, or the values of the other functions may readily be computed by means of the fundamental relations, if the values of the sine and cosine are remembered, as follows: 18 PLANE TRIGONOMETRY. ORAL WORK. 1. Which is greater, sin 450 or - sin 907"? sin 600 or 2 sin 300? 2. From the functions of 600, find those of 30'; from the functions of 9900, those of 00. Why are the functions of 450 equal to the co-functions,of 450.? 3. Given sin A =,find cos A; tanA. 4. Show that sin B csc B = 1; cos C sec C = 1; cot x taix = 1. 5. Show that sec2 6 - tan2 & = csc2 6 - cot2 6 = sin2 6 + cos2 6. 6. Show that tan 300 tan 600 = cot 60' cot 30' = tan 450' 7. Show that tan 600 sin2 450 = cos 300 sin 90'. 8. Show that cos a tan a = sin a; sin/3 cot/3 = cos/3. 1 - tan2 300 = o 0 1 cos 00. 9. Show that ta230 os 600 I + tan 2300 " 10. Show that (tan y + cot y) sin y cos y = 1. EXAMPLES. 1. Show that sin 300 cos 60' + cos 300 sin 600 = sin 900. 2. Show that cos 600 cos 300 + sin 600 sin 300 = cos 300. 3. Show that sin 450 cos 00 - cos 450 sin 00 = cos 450. 4. Show that cos2450 - sin245' = cos 900. 5. Show that tan 45 + tan 0 = tan 450. 1 - tan 450 tan 00 If A = 600, verify 6. sinA=<1josA A2 2' csA 7. tanA=\ CosA. 2 A 8. cos A 2 cos2A 1=1 - 2 sin24. 2 2 If a = 00, 3 = 300, y = 450, 7 = 60', E = 900, find the values of 9. sin/3+cosS. 10. cos,/ +tan S. 11. sin 8 cos 8 + cos / sin 8 - sin E. 12. (sin/3 + sin ) (cos a + cos 8) - 4 sin a (cos y + sinec). VARIATIONS IN THE FUNCTIONS. 19 24. Variations in the trigonometric functions. Signs. Thus far no account has been taken of the signs of the functions. By the definitions it appears that these depend on the signs of x, y, and r. Now r is always positive, and from the figures it is seen that x is positive in the first Y Y Sin. + Cos. + \PI P^! Sin. + Tan. + Csc. + Cot. + y\\r N'r^Nt \y i~ Sec. + X, 1 NN3, N1, X N(r+) o(y) X3 4 4 ~y} 3 /3 r P Tan. + Cos. + Cot. + Sec. + Y Y' FIG. 17. and fourth quadrants, and y is positive in the first and second. Hence For an angle in the first quadrant all functions are positive, since x, y, r are positive. In the second quadrant x alone is negative, so that those functions whose ratios involve x, viz. cosine, tangent, cotangent, secant, are negative; the others, sine and cosecant, are positive. In the third quadrant x and y are both negative, so that those functions involving r, viz. sine, cosine, secant, cosecant, are negative; the others, tangent and cotangent, are positiue. In the fourth quadrant y is negative, so that sine, tangent, cotangent, cosecant are negative, and cosine and secant, positive. Values. In the triangle of reference of any angle, the hypotenuse r is never less than x or y. Then if r be taken of any fixed length, as the angle varies, the base and perpendicular of the triangle of reference may each vary in length from 0 to r. Hence the ratios - and Y can never be greater r r than 1, nor if x and y are negative, less than -1; and -, - x y 20 PLANE TRIGONOMETRY. cannot have values between + 1 and - 1. But the ratios Y and x may vary without limit, i.e. from + 0 to- o. x y Therefore the possible values of the functions of an angle are: sine and cosine between + 1 and - 1, i.e. sine and cosine cannot be numerically greater than 1; tangent and cotangent between + oo and- oo, i.e. tangent and cotangent may have any real value; secant and cosecant between + oo and + 1, and - 1 and - oo, i.e. secant and cosecant may have any real values, except values between + 1 and - 1. These limits are indicated in the following figures. The student should carefully verify. Sin 90~= 1 90o Cos 90 +~0 Y Tan 90~= to Sin. +1 +1 7 > ol L Cos. -0 +0 Tan. -oa +oo Sin 180to Sin 0'- ~ 00 0 X 180 j7-+ 0,_ -o + o, + +?v. oo Cos -o- 0+o0 -10 -o +0, "+1+0 0 Tan 180~ —10 X=-, 00 1 X0;1 +1, a Tan 0'=~0,~ fe ^ y Sin. -1 -1 Cos. -0 +0 Sin 27O= -1 Tan. +oo -co Cos 270 =+0 Y Y Tant 270=+o0 270 FIG. 18. 25. In tracing the changes in the values of the functions as a changes from 0~ to 360~, consider the revolving line r as of fixed length. Then x and y may have any length between O and r. Sine. At 0~, sin = y = = 0. As a increases through r r the first quadrant, y increases from 0 to r, whence Y increases from 0 to 1. In passing to 180~ sin a decreases from 1 to 0, VARIATIONS IN THE FUNCTIONS. 21 since y decreases from r to 0. As a passes through 180~, y changes sign, and in the third quadrant decreases to negative r, so that sin a decreases from 0 to - 1. In the fourth quadrant y increases from negative r to 0, and hence sin a increases from - 1 to 0. Cosine depends on changing values of x. Show that, as a increases from 0~ to 360~, cos a varies in the four quadrants as follows: 1 to 0, 0 to - 1, - 1 to 0, 0 to 1. Tangent depends on changing values of both y and x. At 0~, = 0, x= r, at 180, y = 0, x = -r, at 90~, x = 0, y = r, at 270, x = 0, y =- r. Hence tan 0~ = = -0 = 0. As a passes to 90~, y increases x T to r, and x decreases to 0, so that tan a increases from 0 to oo. As a passes through 90~, x changes sign, so that tan a changes from positive to negative by passing through oo. In the second quadrant x decreases to negative r, y to 0, and tan a passes from - o to 0. As a passes through 180~, tan changes from minus to plus by passing through 0, because at 180~ y changes to minus. In the third quadrant tan a passes from 0 to oo, changing sign at 270~ by passing through oo, because at 270~ x changes to plus. In the fourth quadrant tan a passes from - oo to 0. Cotangent. In like manner show that cot a passes through the values oo to 0, 0 to - o, oo to 0, 0 to - so, as a passes from 0~ to 360~. Secant depends on x for its value. Noting the change in x as under cosine, we see that secant passes from 1 to oo, - oo to - 1, -1 to - o, oo to 1. Cosecant passes through the values oo to 1, 1 to oo, - oo to -1, - 1 to - co. The student should trace the changes in each function fully, as has been done for sine and tangent, giving the reasons at each step. 22 PLANE TRIGONOMETRY. a 0~ to 90~ 90~ to 180~ 180~ to 2700 270~ to 360~ sin 0 to 1 1 to 0 -0 to -1 - to -0 cos I to 0 - 0 to - 1 -1 to - 0 to 1 tan 0 to co -co to-O O to oo -oo to-0 cot oo to 0 - to - c o to 0 - 0 to - co sec ito o - - 1 - to -1 coto 1 csc s o to 1 1 to oo - o to - 1 - 1 to - oo * 26. Graphic representation of functions. These variations are clearly brought out by graphic representations of the functions. Two cases will be considered: I, when a is a constant angle; II, when a is a variable angle. I. When a is a constant angle. The trigonometric functions are ratios, pure numbers. By so choosing the triangle of reference that the denominator of the ratio is a side of unit length, the side forming the numerator of that ratio will be a geometrical representation of the value of that function, e.g. if in Fig. 19 r = 1, then sin a =Y== y. This may be done by making a a r 1 central angle in a circle of radius 1, and drawing triangles of reference as follows: E C E,A E /E C_ _ E3 0 A BA P FIG. 19. GRAPHIC REPRESENTATION OF FUNCTIONS. 23 In all the figures A OP = a, and BP BP sin a = = BP, OP 1 OB OB = COS a... OB OP 1 BP AD AD tan a.= = = AD, OBOA 1 OA EC EC, cot a....= EC, AD OE 1 OP OD OD sec a= =OA= = OD OB OA 1 OP OC OC n csc a = = = = BP OE 1 It appears then that, by taking a radius 1, sine is represented by the perpendicular to the initial line, drawn from that line to the terminus of the arc subtending the given angle; cosine is represented by the line from the vertex of the angle to the foot of the sine; tangent is represented by the geometrical tangent drawn from the origin of the arc to the terminal line, produced if necessary; cotangent is represented by the geometrical tangent drawn from a point 90~ from the origin of the arc to the terminal line, produced if necessary; secant is represented by the terminal line, or the terminal line produced, from the origin to its intersection with the tangent line; cosecant is represented by the terminal line, or the terminal line produced, from the origin to its intersection with the cotangent line. 24 PLANE TRIGONOMETRY. These lines are not the functions, but in triangles drawn as explained their lengths are equal to the numerical values of the functions, and in this sense the lines may be said to represent the functions. It will be noticed also that their directions indicate the signs of the functions. Let the student by means of these representations verify the results of Arts. 24 and 25. II. When a is a variable angle. Take XX' and YYT as axes of reference, and let angle units be measured along the x-axis, and values of the functions parallel to the y-axis, as in Art. 3. We may write corresponding values of the angle and the functions thus: a=o0, 30~, 45~, 60~, 90~, 120~, 135~, 150~, 180~, 210~, 225~, sin a== 0, 2 12 2 3, 1, -I3, 2, 0, -1, - 1 2, a= 240~, 270~, 300~, 315~, 330~,3, 360~,0, -450, -60, -90~, etc., sin a=- V/3, -1,-2 V3, -~/2-, - 02, — i, --, -1x/3, -1, etc. These values will be sufficient to determine the form of the curve representing the function. By taking angles between those above, and computing the values of the function, as ~\, ~\ 0; ~given in mathematical tables, /,,/ the form of the curve can be x^^^\ I \,d determined to any required 0 ~~^ degree of accuracy. Reduc/ \ / \ ing the above fractions to i i I t decimals, it will be convenient ' I i to make the y-units large in Curves of Sine and Cosecant. comparison with the x-units. Sinev Cosecant - In the figure one x-unit repreFIG. 20. sents 15~, and one y-unit 0.25. Measuring the angle values along the x-axis, and from these points of division measuring the corresponding values of sin a parallel to the y-axis, as in Art. 3, we have, approximately, GRAPHIC REPRESENTATION OF FUNCTIONS. 25 OX1 = 30~ = 2 units, OX = 450 = 3 units, X1Y1 = 1 = 2 units, X2Y2 = 0.71 = 2.84 units, OX3 = 60~ = 4 units, etc., X3Y3= 0.86 = 3.44 units, etc. We have now only to draw through the points Y1, Y2, Y3. etc., thus determined, a continuous curve, and we have the sine-curve or sinusoid. The dotted curve in the figure is the cosecant curve. Let the student compute values, as above, and draw the curve. In like manner draw the cosine and secant curves, as follows: m ' i. i i i I \ I X Curves of Cosine and Secant. Cosine Secant --- —-- FIG. 21. Tangent curve. Compute values for the angle a and for tan a, as before: a = 0~, 30~, 45~, 60~, 90~, 120~, 135~, 150~, 180~, 210~, 225~, 240~, 270~, tan a = 0, 3/3, 1, v3, ~, -x/3, -1, — /3, 0, V/3, I, /3, ~o, a = - 30~, - 45~, - 60~, - 90~, etc., tan a= —1x/3, — 1, - -V, oo, etc. Then lay off the values of a and of tan a along the x, and parallel to the y-axis, respectively. It will be noted that, 26 PLANE TRIGONOMETRY. as a approaches 90~, tan a increases to o, and when a passes 90~, tan a is negative. Hence the value is measured parallel Curves of Tangent and Cotangent. Tangent Cotangent --- —-- FIG. 22. to the y-axis downward, thus giving a discontinuous curve, as in the figure. * 27. The following principles are illustrated by the curves: 1. The sine and cosine are continuous for varying values of the angle, and lie within the limits + 1 and - 1. Sine changes sign as the angle passes through 180~, 360~,...*, n 180~, while cosine changes sign as the angle passes through 90~, 270~,..*, (2n + 1) 90~. Tangent and cotangent are discontinuous, the one as the angle approaches 90~, 270~,..., (2n+1) 90~, the other as the angle approaches 180~, 360~, *.., n 180~, and each changes sign as the angle passes through these values. The limiting values of tangent and cotangent are + o and - o. 2. A line parallel to the y-axis cuts any of the curves in but one point, showing that for any value of a there is but one value of any function of a. But a line parallel to the x-axis cuts any of the curves in an indefinite number of points, if at all, showing that for any value of the function there are an indefinite number of values, if any, of a. GRAPHIC REPRESENTATION OF FUNCTIONS. 27 3. The curves afford an excellent illustration of the variations in sign and value of the functions, as a varies from 0 to 360~, as discussed in Art. 25. Let the student trace these changes. 4. From the curves it is evident that the functions are periodic; i.e. each increase of the angle through 360~ il the case of the sine and cosine, or through 180~ in the case of the tangent and cotangent, produces a portion of the curve like that produced by the first variation of the angle within those limits. 5. The difference in rapidity of change of the functions at different values of a is important, and reference will be made to this in computations of triangles. (See Art. 64, Case III.) A glance at the curves shows that sine is changing in value rapidly at 0~, 180~, etc., while near 90~, 270~, etc., the rate of change is slow. But cosine has a slow rate of change at 0~, 180~, etc., and a rapid rate at 90~, 270~, etc. Tangent and cotangent change rapidly throughout. Ex. Let the student discuss secant and cosecant curves. ORAL WORK. 1. Express in radians 180~, 120~, 45~; in degrees, ly radians, 2 r, ITr,. 7r 2. If I of a right angle be the unit, what is the measure of ~ of a right angle? of 90~? of 135~? 3. Which is greater, cos 30~ or ~ cos 60~? tan7r or cotT? sin 7 or cos 6 3 4 4 4. Express sin a in terms of sec a; of tan a; tan a in terms of cos a; of sec a. 5. Given sin a = a, find tan a. If tan a = 1, find sin a, csc a, cot a; also tan 2 a, sin 2 a, cos 2 a. 6. If cos a = ~, find sin ' tan a 7. In what quadrant is angle t, if both sin t and cos t are minus? if sin t is plus and cos t minus? if tan t and cot t are both minus? if sin t and csc t are of the same sign? Why? 8. Of the numbers 3, a, - b, OW, 0, which may be a value of sinp? of secp? of tan p? Why? 28 PLANE TRIGONOMETRY. EXAMPLES. 1. If sin 26040' = 0.44880, find, correct to 0.00001, the cosine and tangent. 2. If tan a -V3 and cot13 - lv3, find sin cacos/3 - cos a sin. 3. Evaluate sin 300 cot 30' - cos 600 tan 600 sin 900 cos 00 Prove the identities: 4. tan A (1- Cot2A) +cot A (1 - tan2 A) = 0. 5. (sin A + sec A)2 + (CosA + csc A)2 =( 1+ sec A csc A)2. 6. sin2 x cos x cscx - cos3 X csc x sin2 x + cos4 X sec x sin x = sin3X cos x + cos3 x sin x. 7. tan2W + cot2w = se2 W CSC2 W - 2. 8. sec2 v + cos2 V = 2 + tan 2v sin2 V. 9. cos2t + 1 = 2 cos3t sec t + sin2 t. 10. csc2 t - sec2 t - cos2 t csc2 t - sin2 t sec2 t. isn2 n2 11. The sine of an angle is -; find the other functions. m2 + n2 12. If tan A + sinA = m, tan A - sinA =n, prove m2- n2= 4Vmn. Solve for one function of the angle involved the equations: 13. sinO0 + 2cos9 = 1. 16. 2sin2x + cos x - 1 = 0. cosax 3 17. sec2 x - 7tanx - 9 = 0. 14. tan (t 2 18. 3cscy+10coty-35=0. 15. Vj csc2O = 4 cotO0. 19. sin2v - I cosv - 1 = 0. 20. a sec 2w + b tan w + c - a =0. 21. If sinA tanA -"/3, findA and B. sinB tanB 22. Find to five decimal places the arc which subtends the angle of 1' at the centre of a circle whose radius is 4000 miles. 23. If csc A - 2v'3, find the other functions, when A lies between 7r and ir. 2 24. In each of two triangles the angles are in G. P. The least angle of one of them is three times the least angle of the other, and the sum of the greatest angles is 2400. Find the circular measure of each of the angles. CHAPTER III. FUNCTIONS OF ANY ANGLE -INVERSE FUNCTIONS. 28. By an examination of the figure of Art. 24 it is seen that all the fundamental relations between the functions hold true for any value of a. The table of Art. 16 expresses the functions of a, whatever be its magnitude, in terms of each of the other functions of that angle if the ~ sign be prefixed to the radicals. The definitions of the trigonometric functions (Art. 12) apply to angles of any size and sign, but it is always possible to express the functions of any angle in terms of the functions of a positive acute angle. The functions of any angle 0, greater than 360~, are the same as those of 0 ~ n - 360~, since 9 and 8 ~ n 360~ have the same triangle of reference. Thus the functions of 390~, or of 750~, are the same as the functions of 390~- 360~, or of 750~- 2-360~, i.e. of 30~, as is at once seen by drawing a figure. So also the functions of -315~, or of - 675~ are the same as those of - 315~ + 360~, or of - 675~ + 2-3600, i.e. of 45~. For functions of angles less than 360~ the relations of this chapter are important. 29. To find the relations of the functions of - 0, 90~ ~ 9, 180~ ~ 0, and 270~ ~ 8 to the functions of 0, 0 being any angle. Four sets of figures are drawn, I for 9 an acute angle, II for 9 obtuse, III for 8 an angle of the third quadrant, and IV for 0 an angle of the fourth quadrant. In every case generate the angles forming the compound angles separately, i.e. turn the revolving line first through 29 30 (a) PLANE TRIGONOMETRY. (b) -0 (c) 1804o T' x I r 0 y X' r0 I x I X' V\X II I II II V 0 XQ rI -, > ' f Yy III III III x \ OrTV X U 'p y - IV IV IV FIG. 23. FUNCTIONS OF ANY ANGLE. 31 90oi 270~o 0 x \T' r/\ ^Ik. 'r X' X' XI y Y I I X X y c y XI X1Y ' III I I 9 Y rI *^of0 y a:' x'^ U xl it + Ix- IV. FIG. 23. 32 PLANE TRIGONOMETRY. 0~, 90~, 180~, or 270~, and then from this position through 0, or - 0, as the case may be. Form the triangles of reference for (a) the angle 0, (b) - 8, () 180~ ~0, (d) 90~ ~ 0, (e) 270~ ~ 9. The triangles of reference (a), (6), (c), (d), and (e), in each of the four sets of figures, I, II, III, IV, are similar, being mutually equiangular, since all have a right angle and one acute angle equal each to each. Hence the sides x, y, r of the triangles (a) are homologous to x', yt, rf of the corresponding triangles (b) and (c), but to y', x', r', of the corresponding triangles (d) and (e). For the sides x of triangle (a) and x' of the triangles (b) and (c) are opposite equal angles, and hence are homologous, but the sides y' are opposite this same angle in triangles (d) and (e), and therefore sides y' of (d) and (e) are homologous to x of (a). Attending to the signs of x and x', y and y' in the similar triangles (a) and (6), sin(- 0) ' - =-sin 0, x x cos(-0)== = - = os, tan (-0) =y Y — -tan 9. X X Also in the similar triangles (a) and (e), sin (180~ - ) = = sin0, r r X! X cos (180 - 0) = - - - = - cos 0, r Iy _ r tan (180~ - 0)= =Y = = tan S. In like manner show that sin (180~ + ) =- sin 0, cos (180~ + ) =- cos 0, tan (180~ + 0)= tan 0. FUNCTIONS OF ANY ANGLE. Again, in the similar triangles (a) and (d), sin (90~ + ) - - = cos 0, os (90~ + 9)- =-Y = - sin 8, cos (90~ + O) = r = S cos (90~ - 0) = sin, tan (90~ - ) = c ot 9. Finally, from the similar triangles (a) and (e), show that sin (270~ ~ 0)=-cos 9, cos (270~ - 0)= sin 0, tan (270~ ~ 0)= F cot 9. From the reciprocal relations the student can at once write the corresponding relations for secant, cosecant, and cotangent. 30. Since in each of the four cases x', yI of triangles (b) and (c) are homologous to x, y of triangle (a), while x', y' of the triangles (d) and (e) are homologous to y, x of triangle (a), we may express the relations of the last article thus: The functions of { 180~ f correspond to the same functions 90 9 0 of 0, while those of 9270~0 correspond to the co-functions of 0, due attention being paid to the signs. The student can readily determine the sign in any given case, whether 0 be acute or obtuse, by considering in what quadrant the compound angle, 90~ ~ 0, 180~ ~ 0, etc., would 34 PLANE TRIGONOMETRY. lie if 0 were an acute angle, and prefixing to the corresponding functions of 0 the signs of the respective functions for an angle in that quadrant. Thus 90~ + 0, if 0 be acute, is an angle of the second quadrant, so that sine and cosecant are plus, the other functions minus. It will be seen that sin (90~ + 0)= +cost9, cos (90~ + 0)= - sin 9, etc., and this will be true whatever be the magnitude of 0. It will assist in fixing in the memory these important relations to notice that when in the compound angle 0 is measured from the y-axis, as in 90~ ~ 0, 270~ ~ 0, the functions of one angle correspond to the co-functions of the other, but when in the compound angle 0 is measured from the x-axis, as in ~ 0, 180~ ~ 0, then the functions of one angle correspond to the same functions of the other. These relations, as has been noted in Art. 28, can be extended to angles greater than 360~, and it may be stated generally that function 0 = i function (2 n * 90~ ~ 0), function 0 = ~ co-function [(2 n + 1) 90~ ~ 9]. Computation tables contain angles less than 90~ only. The chief utility of the above relations will be the reduction of functions of angles greater than 90~ to functions of acute angles. Thus, to find tan 130~ 20', look in the tables for cot 40~ 20', or for tan 49~ 40'. Why? Ex. 1. What angles less than 360~ have the same numerical cosine as 20~? cos 20 =- cos (180~ + 20) = cos (360~ - 20~)..'. 200~, 160~, 340~ have the same cosine numerically as 20~. 2. Find the functions of 135~; of 210~. sin 135~ = sin (90~ + 45~) cos 45~ = 0 /2, cos 135~ = cos (180~ - 45) = - cos 45~ = - V2, etc. sin 210~ = sin (180~ + 30~) = - sin 30~ = - Let the student give the other functions for each angle. INVERSE FUNCTIONS. 35 ORAL WORK. 1. Determine the sine and tangent of each of the following angles: 30~, 120~, - 30~, - 60~, 7r, 22 r,- - 135~, - 7r. 2. Which is the greater, sin 30~ or sin(- 30~)? tan 1350 or tan 45~? cos 60~ or cos(- 60~)? sin 22~ 301 or cos 67~ 30'? 3. What positive angle has the same tangent as -? the same sine as 50~? 4. If tan =- 1, find sin 0. 5. Find sin 510~, cos(- 60~), tan 150~. 6. Reduce in two ways to functions of a positive acute angle, cos 122~ tan 140~ 30', sin(- 60~). 7. Find all positive values of x, less than 360~, satisfying the following equations: cos x = cos 45~, sin 2 x = sin 10~, tan 3 x = tan 60~, sin x = sin 30~, tan x = tan 135~. 8. What angles are determined when (a) sine and cosine are +? (b) cotangent and sine are -? (c) sine + and cosine -? (d) cosine - and cotangent +? INVERSE FUNCTIONS. 31. That a is the sine of an angle 0 may be expressed in two ways, viz., sin 0 = a, or, inversely, 0 = sin- a, the latter being read, 0 equals an angle whose sine is a, or, more briefly, 8 is the anti-sine of a. The notation sin-' a, cos-1 a, tan-1 a, etc., is not a fortunate one, but is so generally accepted that a change is not probable. The symbol may have been suggested from the fact that if ax = b, then x = a-1 b, whence, by analogy, if sin 0 = a, 0 = sin-1 a. But the likeness is an analogy only, for there is no similarity in meaning. Sin-1 a is an angle 0, where sin 0 = a, 1 and is entirely different from (sin a)-1 =-. In Europe the symbols sill a arc sin a, arc cos a, etc., are employed. 32. Principal value. We have found that in sin 0 = a, for any value of 0, a can have but one value; but in 0 = sin-l a, for any value of a there are an indefinite number of values of 0 (Art. 27, 2). Thus, when sin 0 = a, if a =, 0 may be 30~, 150~, 390~, 510~, - 330~, etc., or, in general, nwr + (- 1) 30~. In the solution of problems involving inverse functions, 36 36 ~~PLANE TRIGONOMETRY. the numerically least of these angles, called the principal value, is always used; i.e. we understand that sin-1 a, tan-' a, are angles between + 900 and - 90', while the limits of cos-1a are 00 and 1800. Thus, sin-' 1 = 30', sin'l( - -1) = - 30', cos-1 1= 600, 2T 2_ cos'l(- 1) = 1200. How many degrees in radians? 1.Co- 3 2. tan-' 1? 5. cos-' ( - 14x2)? 6. sn1-V ORAL WORK. each of the following angles? How many 7. tan-' V3? 8. Cos-'O0? 9. sin-'1? 10. tan-' 0? 12. sin-'( - 1)? Find the values of tl 14. tan (cos-' 1). 15. tan(cot-'[18. tan (tan'I x) ie functions: '3). ~~~19. cos(sin'10). 20. sin(cos-'[- 1]). 21. cos(cot-'V-3). 2). ~~~22. ta-n(sin'1[- 1]). 23. sin(tarc'[- 1]). Ex. 1. Construct cot-' ~. Construct the right triangle xyr, so that x =4, y =3, whence angle xr =cot-' 4. 2. Find cos(tan-1 ITLet 0 = tanv' I, whence x=4 ~~~tan90= 8, and cos 0-=I4. FIG. 24. ~~~.. cos 0 =cos(tan-1 Tl~) -,1 3. If 0 = csc-1 a, prove 0 = cos-' Vra'2 - 1 a csc 0= a;... sin 0=!I, a Cos 0= 1 -, or 0=coor a a a and EXAMPLES. 37 EXAM PLES. 1. Construct sin-12, tan-'-A, cos-'(- D. 2. Find tan(sin'-1 5 ) sin(tan-' 5 3. If 0 = sin-' a, prove 0 = tan-' a ~VI - a2 4. Show that sin-' a - 900 - cos-' a. 5. Prove tan-' V + cot-1v', = 7. 2 6. Prove tan-'(sin c)s=1cos-'. 7. What angles, less than 360', have the same tangent numerically as 100? 8. Given tan 1430 22' - 0.74357; find, correct to 0.00001, sine and cosine. 9. If cot2(900 + /) + csc(900 - /)- 1 = 0, find tan /. 10. Find all positive values of x, less than 3600, when sin x =sin 220 30'; when tan 2 x = tan 600. 11. When is sin x a2 + 12 possible, and when impossible? 2 ab 12. Verify sin-' 1 + cos-1 + tan-lv'3 = sin-' 2 2 13. What values of x will satisfy sini'(x2 - x) = 300? 14. If tan20 - sec2a = 1, prove sec 0~+ tan8 0 csc 0 = (3 ~ tan2 a). 15. Prove sin A (1 + tan A)+ cos A (1 + cotA)= sec A + csc A. 16. Solve the simultaneous equations: sin-'(2 x + 3 y)= 30' and 3 x + 2 y=2. 17. Verify (a) tan 600 = <1 - cos 1200 1 + cos 1201 1 - tan2 300 (b) cos 600 tn 0 (c) 2 sin2 600 - 1 - cos 1200. 18. Show that the cosine of the complement of L equals the sine of the supplement of Z6 6 38 PLANE TRIGONOMETRY. REVIEW. Before leaving a problem the student should review and master all principles involved. 1. Construct cos-1' 8; sin-l(- 3D; tan-12. 2. Find cos (sin-' 3); tan (cos-1 3. Prove cot-' a = cos' a 1 +a2 4. Given a = cot-1, find tan a ~ sin (90' + a). 5. Find tall (sin-' I + cos-' 6. State the fundarnental relations between the trigonometric functions in terms of the inverse functions. Thus, 12 sin-1a = csc-1 sin-1 a = cos-1V - a2, etc. a' 7. Find all the angles, less than 3600, whose cosine equals sin 1200. 8. Given cot- 2.8449, find the sine and cosine of the angle, correct to 0.0001. 9. If tan2 (1SOl - 9) - sec (l80 + 9) = 5, find cos 9. 10. If sin9- = 23 find tan29~cos29 tain2 9 - cos2 9 11. Is sin x - 2 cos x + 3 sin x - 6 = 0 a possible equation? 2 tan ~30'12. Verify (a) sin 60' 2 1 + tan2 300 (b) 2 cos2 600 = 1 + cos 1200. (c) cos 600 - cos 900 = 2 cos2 300 - 2 cos2 450 13. If sin x = a(a + 2b) find secx and tan x. a2 + 2 ab + 2 b2 1 + sill n - cos 0 1 ~ sinll90+ cos 0 2 esc 92 14. Prove + 1 + sin 0 + cos 9 1 +sind0 - cos 0 15. Prove cos 450 + cos 1350 + cos 300 + cos 1500 - cos 2100 + cos 270' = sin 600. 16. If tan, prove that ~a2- b2 sin 0(1 + tan90) ~ cos 0(1 + cot90) - sec = _. 17. Solve sin2 X + Siii2 (x + 901) + sin2 (X + 1800) = 1. EXAMPLES. 3 39 18. Given cos2 a = M Sin a - n, find sin a. 19. If sin12 /3find / 2 see/3' 20. Given tan 2380 = 1.6, find sin 1480. 21. Prove tan-' m + cot-1 m =990*. 22. Find sin (sin-1p + cos-1p). 23. Solve cot2O0(2 csc60 - 3) +3 (csc90 - 1) = 0. 24. Pr-ove sin2 a seC2 /3 + tan2 /3cos2 a =sin2a+tn 3 25. Prove cos6 V ~ sin6 V = 1 - 3 sin2 V ~ 3 sin4 V. 26. What values of A satisfy sin 2 A =cos 3 A? VI -m2 <I Co C of27. If tan C m,adtan D ~,find tan D in terms 28. If sin x - cos x + 4COS2x -=2 find tan x; sec x. 29. Does the value of sec x, derived from sec2x= l2cos2X 2 give a possible value of x?1-COX 30. Prove [cot (900 - A)- tan (900 + A)] [sin (180' - A) sin (90' + A)] = 1. 31. Prove (1 + sin A) 2[cot A + 2secA (I -cescA) ]+cescA cos3A =0. 32. Given sin x = m sin y, and tan x = n tan y, find cos x and cos y. 33. Given cot 201' - 2.6, find cos 11 10. 34. Find the value of cos'1 1 + sin-'1 N/2 + csc-1( - 1) + tan-' 1 - 2 cot-' V3. 35. Solve 2COS20 1sinO0- 7=0. 36. Prove cos2 B +COS2 (B + 900) + cos2 (B + 180') + cos2 (B + 270') = 2. CHAPTER IV. COMPUTATION TABLES. 33. Natural functions. It has been noted that the trigonometric functions of angles are numbers, but the values were found for only a few angles, viz. 0~, 30~, 45~, 60~, 90~, etc. In computations, however, it is necessary to know the values of the functions of any angle, and tables have been prepared giving the numerical values of the functions of all angles between 0~ and 90~ to every minute. In these tables the functions of any given angle, and conversely the angle corresponding to any given function, can be found to any required degree of accuracy; e.g. by looking in the tables we find sin24~ 26'= 0.41363, and also 1.6415 = tan 58~ 39'. These numbers are called the natural functions, as distinguished from their logarithms, which are called the logarithmic functions of the angles. Ex. 1. Find from the tables of natural functions: sin 35~14'; cos 54~ 46'; tan 78~ 29'; cos 112~ 58'; sin 135~. 2. Find the angles less than 180~ corresponding to: sin-10.37865; cos-10.37865; tan-10.58670; cos-10.00291; sin-10.99999. 34. Logarithms. The arithmetical processes of multiplication, division, involution, and evolution, are greatly abridged by the use of tables of logarithms of numbers and of the trigonometric ratios, which are numbers. The principles involved are illustrated in the following table: Write in parallel columns a geometrical progression having the ratio 2, and an arithmetical progression having the difference 1, as follows: 40 LOGARITHMS. 41 G. P. A. P. It will be perceived that the numbers in 1 0 the second column are the indices of the 2 1 powers of 2 producing the corresponding 4 2 numbers in the first column, thus: 26 = 64, 8 211= 2048, 218s= 262144, etc. The use of 16 4 such a table will be illustrated by examples. 32 5 Ex. 1. Multiply 8192 by 128. 64 6 From the table, 8192 = 213, 128 = 27. Then by 128 7 actual multiplication, 8192 x 128 = 1048576, or by the 256 8 law of indices, 213 x 27 = 220 1048576 (from table). 512 9 Notice that the simple operation of addition is sub1024:10 stituted for multiplication by adding the numbers in the second column opposite the given factors in the 2048 d1 first column. This sum corresponds to the number 4096 12 in the first column which is the required product. 8192 13 2. Divide 16384 by 512. 16384 14 16384 - 512 = 32, which corresponds to the result 32768 15 obtained by use of the table, or 214 - 29 = 25 = 32. 65536 16 The operation of subtraction takes the place of 131072 17 division. 262144 18 3. Find /262144. 524288 19 = 262144 = 1 = 2 = 23= 8. 1048576 20 In the table, 262144 is opposite 18. 18 - 6 = 3, which is opposite 8, the required root; i.e. simple division takes the place of the tedious process of evolution. 4. Cube 64. 5. Multiply 256 by 4096. 6. Find 32768. 7. Divide 1048576 by 32768. 35. The above table can be made as complete as desired by continually inserting between successive numbers in the first column the geometrical mean, and between the opposite numbers in the second, the arithmetical mean, but in practice logarithms are computed by other methods. The numbers in the second column are called the logarithms of the numbers opposite in the first column. 2 is called the base of this system, so that the logarithm of a number is the exponent by which the base is affected to produce the number. 42 PLANE TRIGONOMETRY. Thus, the logarithm of 512 to the base 2 is 9, since 29 = 512. Logarithms were invented by a Scotchman, John Napier, early in the seventeenth century, but his method of constructing tables was different from the above. See Encyc. Brit., art. "Logarithms," for an exceedingly interesting account. De Morgan says that by the aid of logarithms the. labor of computing has been reduced for the mathematician to about one-tenth part of the previous expense of time and labor, while Laplace has said that John Napier, by the invention of logarithms, lengthened the life of the astronomer by one-half. Columns similar to those above might be formed with any other number as base. For practical purposes, however, 10 is always taken as the base of the system, called the common system, in distinction from the natural system, of which the base is 2.71828..., the value of the exponential series (Higher Algebra). The natural system is used in theoretical discussions. It follows that common logarithms are indices, positive or negative, of the powers of 10. Thus, 103 = 1000; i.e. log 1000 = 3; 10-2 = = 0.01; i.e. log 0.01 = -2. 102 36. Characteristic and mantissa. Clearly most numbers are not integral powers of 10. Thus 300 is more than the second and less than the third power of 10, so that log 300 = 2 plus a decimal. Evidently the logarithms of numbers generally consist of an integral and a decimal part, called respectively the characteristic and the mantissa of the logarithms. 37. Characteristic law. The characteristic of the logarithm 'of a number is independent of the digits composing the number, but depends on the position of the decimal point, and is found by counting the number of places the first significant figure in the number is removed from the units' place, being positive or negative according as the first significant LOGARITHMS. 43 figure is at the left or the right of units' place. This follows fromi the fact that common logarithms are indices of powers of 10, and that 10o, n being a positive integer, contains n + 1 places, while 10-" contains n - 1 zeros at the right of units' place. Thus in 146.043 the first significant figure is two places at the left of units' place; the characteristic of log 146.043 is therefore 2. In 0.00379 the first significant digit is three places at the right of units' place, and the characteristic of log 0.00379 is - 3. To avoid the use of negative characteristics, such characteristics are increased by 10, and - 10 is written after the logarithm. Thus, instead of log 0.00811 = 3.90902, write 7.90902 - 10. In practice the - 10 is generally not written, but it must always be remembered and accounted for in the result. Ex. Determine the characteristic of the logarithm of: 1; 46; 0.009; 14796.4; 230.001; 105 x 76; 0.525; 1.03; 0.000426. 38. Mantissa law. The mantissa of the logarithm of a number is independent of the position of the decimal point, but depends on the digits composing the number, is always positive, and is found in the tables. For, moving the decimal point multiplies or divides a number by an integral power of 10, i.e. adds to or subtracts from the logarithm an integer, and hence does not affect the mantissa. Thus, log 225.67 = log 225.67, log 2256.7 = log 225.67 x 101 = log 225.67 + 1, log 22567.0 = log 225.67 x 102 = log 225.67 + 2, log 22.567 = log 225.67 x 10-1 = log 225.67 +(-1), log 0.22567 = log 225.67 x 10-3 = log 225.67 +(- 3), so that the mantissae of the logarithms of all numbers composed of the digits 22567 in that order are the same,.35347. Moving the decimal point affects the characteristic only. The student must remember that the mantissa is always positive. 44 PLANE TRIGONOMETRY. Log 0.0022567 is never written - 3 +.35347, but 3.35347, the minus sign being written above to indicate that the characteristic alone is negative. In computations negative characteristics are avoided by adding and subtracting 10, as has been explained. 39. We may now define the logarithm of a number as the index of the power to which a fixed number, called the base, must be raised to produce the given number. Thus, ax = b, and x = logab (where logab is read logarithm of b to the base a) are equivalent expressions. The relation between base, logarithm, and number is always (base)l~g = number. To illustrate: log28= 3 is the same as 23= 8; log381= 4 and 34= 81 are equivalent expressions; and so are log1l1000 = 3 and 103 = 1000, and log10.001= -3 and 10-3= 0.001. Find the value of: log464; log5125; log3243; loga(a); log27 3; logxl. 40. From the definition it follows that the laws of indices apply to logarithms, and we have: 1. The logarithm of a product equals the sum of the logarithms of the factors. II. The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. III. The logarithm of a power equals the index of the power times the logarithm of the number. IV. The logarithm of a root equals the logarithm of the number divided by the index of the root. For if ax = n and ay = m, then n x m = ax+,.'. log nm= x + y = log n + log m; and n - m = ax,.. log =x-y=log n-log m; also nr= (a)r= ar,.'. log nr = rx = r x log n; finally, n = ax = a,.*. log /n = x log n. r r LOGARITHMS. 45 EXAM PLES. Given log 2 = 0.30103, log 3 = 0.47712, log 5 = 0.69897, find: 1. log4. 4. log9. 7. log 153. 10. log V. 2. log 6. 5. log 25. 8. log 1. l 92 x 53 3. log 10. 6. logV3. 9. log 15 x 9. 11 x 10 USE OF TABLES. 41. To find the logarithm of a number. First. Find the characteristic, as in Art. 37. Second. Find the mantissa in the tables, thus: (a) When the number consists of not more than four figures. In the column N of the tables find the first three figures, and in the row N the fourth figure of the number. The mantissa of the logarithm will be found in the row opposite the first three figures and in the column of the fourth figure. Illustration. Find log 42.38. The characteristic is 1. (Why?) In the table in column N find the figures 423, and on the same page in row N the figure 8. The last three figures of the mantissa, 716, lie at the intersection of column 8 and row 423. To make the tables more compact the first two figures of the mantissa, 62, are printed in column 0 only. Then log 42.38 = 1.62716. Find log 0.8734 = 1.94121, log 3.5 = log 3.500 = 0.54407, log 36350 = 4.56050. (b) When the number consists of more than four figures. Find the mantissa of the logarithm of the number composed of the first four figures as above. To correct for the remaining figures we interpolate by means of the principle of proportional parts, according to which it is assumed that, for differences small as compared with the numbers, the differences 46 PLANE TRIGONOMETRY. between several numbers are proportional to the differences between their logarithms. The theorem is only approximately correct, but its use leads to results accurate enough for ordinary computations. Ex. 1. To find log 89.4562. As above, mantissa of log 894500 = 0.95158, mantissa of log 894600 = 0.95163,.'. log 894600 - log 894500 = 0.00005, called the tabular difference. Let log 894562 - log 894500 = x hundred-thousandths. Now, by the principle of proportional parts, log 894562 - log 894500 _ 894562 - 894500 log 894600 - log 894500 894600 - 894500' x 62 or x-= 6, whence x =.62 of 5 = 3.1 5 100.. log 89.4562 = 1.95158 + 0.00003 = 1.95161, all figures after the fifth place being rejected in five-place tables. If, however, the sixth place be 5 or more, it is the practice to add 1 to the figure in the fifth place. Thus, if x = 0.0000456, we should call it 0.00005, and add 5 to the mantissa. 2. Find log 537.0643. To interpolate we have x: 9 = 643: 1000, i.e. x = 5.787;.'. log 537.0643 = 2.72997 + 0.00006. 3. Find log 0.0168342 = 2.22619. 4. Find log 39642.7 = 4.59816. 42. To find the number corresponding to a given logarithm. The characteristic of the logarithm determines the position of the decimal point (Art. 37). (a) If the mantissa is in the tables, the required number is found at once. Ex. 1. Find log-1 1.94621 (read, the number whose logarithm is 1.94621). The mantissa is found in the tables at the intersection of row 883 and column 5.... log-11.94621 = 88.35, the characteristic 1 showing that there are two integral places. LOGARITHMS. 47 (b) If the exact mantissa of the given logarithm is not in the tables, the first four figures of the corresponding number are found, and to these are annexed figures found by interpolating by means of the principle of proportional parts, as follows: Find the two successive mantissae between which the given mantissa lies. Then, by the principle of proportional parts, the amount to be added to the four figures already found is such a part of 1 as the difference between the successive mantissae is of the difference between the smaller of them and the given mantissa. 2. Find log-l 1.43764. Mantissa of log 2740 = 0.43775 of log 2739 - 0.43759 Differences 1 16 Mantissa of log required number = 0.43764 of log 2739 = 0.43759 Differences x 5 By p. p. x: 1 = 5: 16 and x = - = 0.3125. Annexing these figures, log-1 1.43764 = 27.3931+. 3. Find log-1 1.48762. The differences in logarithms are 14, 6. 6.. x= - =.428+, and log-' 1.48762 = 0.307343+. 4. Find log 891.59; log 0.023; log ~; log 0.1867; log V2. 5. Find log-1 2.21042; log-' 0.55115; log-' 1.89003. 43. Logarithms of trigonometric functions. These might be found by first taking from the tables the natural functions of the given angle, and then the logarithms of these numbers. It is more expeditious, however, to use tables showing directly the logarithms of the functions of angles less than 90~ to every minute. Functions of angles greater than 90~ are reduced to functions of angles less than 90~ by 48 PLANE TRIGONOMETRY. the formula of Art. 29. To make the work correct for seconds, or any fractional part of a minute, interpolation is necessary by the principle of proportional parts, thus: Ex. 1. Find log sin 28~ 32t 21". In the table of logarithms of trigonometric functions, find 28~ at the top of the page, and in the minute column at the left find 32'. Then under log sin column find log sin 28~ 32' = 9.67913 - 10 log sin 28~ 33 = 9.67936 - 10 Differences 1' 23 By p. p. x: 23 = 21": 60", i.e. x = 21 x 23 = 8.05. 60.'. log sin 28~ 32' 21" = 9.67913 + 0.00008 - 10 = 9.67921 - 10. Whenever functions of angles are less than unity, i.e. are decimals (as sine and cosine always are, except when equal to unity, and as tangent is for angles less than 45~), the characteristic of the logarithm will be negative, and, accordingly, 10 is always added in the tables, and it must be remembered that 10 is to be subtracted. Thus, in the example above, the characteristic of the logarithm is not 9, but 1, and the logarithm is not 9.67913, as written in the tables, but 9.67913 - 10. 2. Find log cos 67~ 27' 50". In the table of logarithms at the foot of the page, find 67~, and in the minute column at the right, 27'. Then computing the difference as above, x = 25. But it must be noted that cosine decreases as the angle increases toward 90~. Hence, log cos 67~27' 50" is less than log cos 67~ 27', i.e. the difference 25 must be subtracted, so that log cos 67~ 27' 50" = 9.58375 - 0.00025 - 10 = 9.58350 - 10. 44. To find the angle when the logarithm is given, find the successive logarithms between which the given logarithm lies, compute by the principle of proportional parts the seconds, and add them to the less of the two angles corresponding to the successive logarithms. This will not necessarily be the angle corresponding to the less of the two logarithms; for, as has been seen, the number, and, therefore, the logarithm, may decrease as the angle increases. LOGARITHMS. Ex. l. Find the angle whose log tan is 9.88091. log tan 37~ 14' = 9.88079 - 10 log tan 37~ 15' = 9.88105 - 10 Differences 60" 26 log tan 37~ 14' = 9.88079 - 10 log tan angle required = 9.88091 - 10 Differences x" 12.. x':60 = 12:26, or x1 = i2 x 60" = 28", approximately, and the angle is 37~ 14' 28". 2. Find the angle whose log cos = 9.82348. We find x = - x 60" = 26", and the angle is 48 14' 26". 3. Show that log cos 25~ 31' 20" = 9.95541; log sin 110~ 25' 20" = 9.97181; log tan 49~ 52' 10" = 0.07417. 4. Show that the angle whose log tan is 9.92501 is 40~4' 39"; whose log sin is 9.88365 is 49~ 54' 18"; whose log cos is 9.50828 is 71~ 11' 49". 45. Cologarithms. In examples involving multiplications and divisions it is more convenient, if n is any divisor, to add log than to subtract log n. The logarithm of - is n n called the cologarithm of n. Since 1 log - = log 1 - log n = 0 - log n, it follows that cologn =- log n, i.e. log n subtracted from zero. To avoid negative results, add and subtract 10. Ex. 1. Find colog 2963. log 1 = 10.00000 - 10 log 2963 = 3.47173... colog 2963 = 6.52827 - 10 2. Find colog tan 16~ 17'. log 1 = 10.00000 - 10 log tan 16~ 17' = 9.46554 - 10.'. colog tan 16~ 17' = 0.53446 50 PLANE TRIGONOMETRY. By means of the definitions of the trigonometric functions, the parts,of a right triangle may be computed if any two parts, one of them being a side, are given. Thus, B given a and A in the rt. triangle ABC. Then c = a - sin A, b = a - tan A, and B = 90~ -A. A b C Again, if a and b are given, then FIG. 25. tan A =, c = a sinA, andB = 90 - A. b 3. Given c = 25.643, B = 37~ 25' 20", compute the other parts. A = 90~ - 37~ 25' 20" = 52~ 34' 40". a = c cos B. b = a tan B. log c = 1.40897 log a = 1.30889 log cos B = 9.89992 log tan B = 9.88376 log a = 1.30889 log b = 1.19265.-. a = 20.365..'. b = 15.583. Check: c2 = a2 + b2 = 20.3652 + 15.5832 = 657.57 = 25.6432. 4. Given b = 0.356, B = 63~ 28' 40", compute the other parts. A = 26~ 31' 20". b b c = ' a= sin B tan B log b = 9.55145 log b = 9.55145 colog sin B = 9.04829 colog tan B = 9.69816 log c = 9.59974 log a = 9.24961 c = 0.3979 a = 0.1777 Check: c2 - a2 = 0.1583 - 0.03157 = 0.12673 = b2. EXAMPLES. Compute the other parts: 1. Given a = 9.325, A = 43~ 22' 35". 2. Given c = 240.32, a = 174.6. 3. Given B = 76~ 14' 23", a = 147.53. 4. Given a = 2789.42, b = 4632.19. 5. Given c = 0.0213, A = 23~ 14". 6. Given b = 2, c = 3. CHAPTER V. APPLICATIONS. 46. Many problems in measurements of heights and distances may be solved by applying the preceding principles. By means of instruments certain distances and angles may be measured, and from the data thus determined other distances and angles computed. The most common instruments are the chain, the transit, and the compass. The chain is used to measure distances. Two kinds are in use, the engineer's chain and the Gunter's chain. They each contain 100 links, each link in the engineer's chain being 12 inches long, and in the Gunter's 7.92 inches. FIG. 26. The transit is the instrument most used to measure horizontal angles, and with certain attachments to measure vertical angles. The figure shows the form of the instrument. 51 52 PLANE TRIGONOMETRY. The mariner's compass is used to determine the directions, or bearings, of objects at sea. Each quadrant is divided into 8 parts, making the 32 points of the compass, so that each point contains 11~ 15'. W -W b- FIG. 27. 47. The angle between the horizontal plane and the line of vision from the eye to the object is called the angle of B Deprssio elevation, or of depression, according ^^^)o;sp as the object is above or below the Elevation observer. A It is evident that the elevation FIG. 28. angle of B, as seen from A, is equal to the depression angle of A, as seen from B, so that in the solution of examples the two angles are interchangeable. PROBLEMS. 48. Some of the more common problems met with in practice are illustrated by the following: To find the height of an object A when the foot is accessible. The distance BC, and the eleva- x tion angle B are measured, and x is determined from the relation B x = BC tan B. FIG. 29. APPLICATIONS. 53 Ex. 1. The elevation angle of a cliff measured from a point 300 ft. from its base is found to be 30~. How high is the cliff? BC = 300, B = 30~. Then x = 300 tan =30~ = 300 3 = 100V/3. 2. From a point 175 ft. from the foot of a tree the elevation of the top is found to be 27~19'. Find the height of the tree. The problem may be solved by the use of natural functions, or of logarithms. The work should be arranged for the solution before the tables are opened. Let the student complete. BC = 175. B = 27 19'. Then x = BC tan B. Or by natural functions, logBC = BC = 175 log tan B = tan B = 0.5165 log x =.'. x = 90.3875..'. x = 90.39. A To find the height of an object when the foot is inaccessible. x Measure BB', 0 and 0f. Then Bx=B BB +B'C B B' y C Then x -- -- cot 0 cot 0 FIG. 30. But B' C = x cot 0', whence substituting, BB' x = cot 0 - cot 0' which is best solved by the use of the natural functions of 0 and 0'. 3. Measured from a certain point at its base the elevation of the peak of a mountain is 60~. At a distance of one mile directly from this point the elevation is 30~. Find the height of the mountain. BB' = 5280 ft., 0 = 30~, 0' = 60~. x y + 5280 But y = x cot 60~. cot 30~ 5280.'. x = 5280 4572.48 ft. cot 30~ - cot 60~ 54 PLANE TRIGONOMETRY. In surveying it is often necessary to make measurements across a stream or other obstacle too wide to be spanned by a single chain. 1\3~ ~ To find the distance from C to a point B on the opposite side of a stream. At C measure a right angle, and take CA a convenient distance. Fl IG. A Measure angle A, then FIG. 31. BC = CA.tanA. 4. Find CB when angle A = 47~ 16', and CA = 250 ft. 5. From a point due south of a kite its elevation is found to be 42~30'; from a point 20 yds. due west D from this point the elevation is 36~ 24'. A How high is the kite above the ground? AB = x. cot 42~ 30', 7/ / A C = x. cot 36~ 24', / 360 24' AC2-AB2 BC2 = 400.. x2 (cot2 360 4' - cot2 429 30') = 400, 20 ' / whence E -, and x= = 20 24.84 yds..6489'.805 FIG. 32. EXAMPLES. 1. What is the altitude of the sun when a tree 71.5 ft. high casts a shadow 37.75 ft. long? 2. What is the height of a balloon directly over Ann Arbor when its elevation at Ypsilanti, 8 miles away, is 10~ 15'? 3. The Washington monument is 555 ft. high. How far apart are two observers who, from points due east, see the top of the monument at elevations of 23~ 20' and 47~ 30', respectively? 4. A mountain peak is observed from the base and top of a tower 200 ft. high. The elevation angles being 25~ 30' and 23~ 15', respectively, compute the height of the mountain above the base of the tower. 5. From a point in the street between two buildings the elevation angles of the tops of the buildings are 30~ and 60~. On moving across APPLICATIONS. 55 the street 20 ft. toward the first building the elevation angles are found to be each 45~. Find the width of the street and the height of each building. 6. From the peak of a mountain two towns are observed due south. The first is seen at a depression of 48~ 40', and the second, 8 miles farther away and in the same horizontal plane, at a depression of 20~ 50'. What is the height of the mountain above the plane? 7. A building 145 ft. long is observed from a point directly in front of one corner. The length of the building subtends tan-' 3, and the height tan-1 2. Find the height. 8. An inaccessible object is observed to lie due N.E. After the observer has moved S.E. 2 miles, the object lies N.N.E. Find the distance of the object from each point of observation. 9. Assuming the earth to be a sphere with a radius of 3963 miles, find the height of a lighthouse just visible from a point 15 miles distant at sea. 10. The angle of elevation of a tower 120 ft. high due north of an observer was 35~; what will be its angle of elevation from a point due west from the first point of observation 250 ft.? Also the distance of the observer from the base of the tower in each position? 11. A railway 5 miles long has a uniform grade of 2~ 30'; find the rise per mile. What is the grade when the road rises 70 ft. in one mile? (The grade depends on the tangent of the angle.) 12. The foot of a ladder is in the street at a point 30 ft. from the line of a building, and just reaches a window 221 ft. above the ground. By turning the ladder over it just reaches a window 36 ft. above the ground on the other side of the street. Find the breadth of the street. 13. From a point 200 ft. from the base of the Forefathers' monument at Plymouth, the base and summit of the statue of Faith are at an elevation of 12~ 40' 48" and 22~ 2' 53", respectively; find the height of the statue and of the pedestal on which it stands. 14. At a distance of 100 ft. measured in a horizontal plane from the foot of a tower, a flagstaff standing on the top of the tower subtends an angle of 8~, while the tower subtends an angle of 42~20'. Find the length of the flagstaff. 15. The length of a string attached to a kite is 300 ft. The kite's elevation is 56~ 6'. Find the height of the kite. 16. From two rocks at sea level, 50 ft. apart, the top of a cliff is observed in the same vertical plane with the rocks. The angles of elevation of the cliff from the two rocks are 24~ 40' and 32~ 30'. What is the height of the cliff above the sea? CHAPTER VI. GENERAL FORMULAE -TRIGONOMETRIC EQUATIONS AND IDENTITIES. 49. Thus far functions of single angles only have been considered. Relations will now be developed to express functions of angles which are sums, differences, multiples, or sub-multiples of single angles in terms of the functions of the single angles from which they are formed. First it will be shown that, sin (a ~ p) = sin a cos p t cos a sin p, cos (a ~ p) = cos a cos p T sin a sin p tan (cL~ p)= tan a ~ tan p 1:F tatan a tan The following cases must be considered: 1. a, 13, a + 13 acute angles. 2. a, /3, acute, but a + /, an obtuse angle. 3. Either a, or I, or both, of any magnitude, positive or negative. The figures apply to cases 1 and 2. B B o A A 900: N C Q N / _9 0 D M X D 0 M X FIG. 33. Let the terminal line revolve through the angle a, and then through the angle /S, to the position OB, sb that angle 56 GENERAL FORMULAE. 57 XOB = a + /3. Through any point P in OB draw perpendiculars to the sides of a, DP and CP, and through C draw a perpendicular and a parallel to OX, MC and NC. Then the angle QCA = a (why?), and CNP is the triangle of reference for angle QCP = 90~ + a. CNP is sometimes treated as the triangle of reference for angle CPN. The fallacy of this appears when we develop cos (a + /l), in which PC would be treated as both plus and minus. DP MCO NP Now sin(a+/3)=sinXOB=DP = + N-P OP OP OP' or expressing in trigonometric ratios, MC OC NP CP 0C OP CP OP = sin a cos / + sil (90~ + a) sin /3. Hence, since sin (90~ + a) = cos a, we have sin (a + /3) = sin a cos /3 + cos a sin /. In like manner OD OM CUN cos ( +,) = cos XOB = OP = - + or expressing in trigonometric ratios, OM O C CN CP OC OP CP OP = cos a cos /3 + cos (90~ + a) sin /3. And since cos (90~ + a) = - sin a, we have cos (a + /) = cos a cos /3 - sin a sin /. It will be noted that the wording of the demonstration applies to both figures, the only difference being that when a + / is obtuse OD is negative. CN is negative in each figure. 50. In the case, when a, or /, or both, are of any magnitude, positive or negative, figures may be constructed as before described by drawing through any point in the terminal line of 38 a perpendicular to each side of a, and through the Joot of the perpendicular on the terminal line of a a perpendicular and a parallel to the initial line of a. Noting negative lines, 58 PLANE TRIGONOMETRY. the demonstrations already given will be found to apply for all values of a and 83. To make the proof complete by this method would require an unlimited number of figures, e.g. we might take a obtuse, both a and / obtuse, either or both greater than 180~, or than 360~, or negative angles, etc. Instead of this, however, the generality of the proposition is more readily shown algebraically, as follows: Let a' = 90~ + a be any obtuse angle, and a, /, acute angles. Then sin (a' + /3) = sin (90~ + a + /) = cos (a + /3) = cos a cos /3 - sin a sin /3 = sin (90~ + a) cos, + cos (90~ + a) sinll (why?) = sin a' cos /3 + cos a' sin 3. In like manner, considering any obtuse angle 3' = 90 + /3, it can be shown that sin (a' + /') = sin a' cos /' + cos a' sin f'. Show that cos (a' + /3i) = cos a' cos /3 - sin a' sin /'. By further substitutions, e.g. ax" = 90~ ~ a', /" = 90~ ~ /', etc., it is clear that the above relations hold for all values, positive or negative, of the angles a and /3. Since a and /3 may have any values, we may put -,/ for /3, and sin (a+ [-/3]) = sill (a - 3) = sin a cos (- /) + cos a sin ( - /) = sin a cos /3 - cos a sin / (why?). Also cos (a - 8) = cos a cos(- /3) - sin a sin (- /) = cos a cos /3 + sin a sin /3. Finally, tan (a f /) - sin (a ~ /3) sin a cos/3 ~ cosa sin / cos (a ~/) cos a cos /3 sin a sin / sin a cos,8 cos a sin / cos a cos/3 cos a cos/3 tan a ~ tan/3 cos a cos/3 sin a sin / 1 T tan a tan/3 cos a cos/3 cos a cos/3 .EXAMPLES. 5 59 ORAL WORK. By the above formulke develop: 1. sin (2 A + 3 B). 7. sin 90' = sin (450 + 450). 2. cos (900 - B). 8. Cos 90g. 3. tan (450 + p). 9. tan 900. 4. sin 2A =sin (A +A). 10. sin (900 +1 +y) 5. cos 2 0. 11. Cos (2700 - m - n). 6. tanl (1800 + 0). 12. tan (901 + m + n). Ex. 1. Find sin 750. sin 750 sin (450 + 300) = sin 450 cos 30' + cos 450 sin 300 1 \v'3 +1.1 1~\3=0.9659. - _ 2 /2 2V_/2 2. Find tan 15'. tan 150 tan (450 300) tan 450 - tanl 300 1 + tan 450 tan 300. V_+ j \ 20Y0 2679. sin3A cos3A Combninsin3AcosA-cos3AsinA sin(3A -A) Cmiig, sin A cos A sin A cos A sin 2A _sin (A + A) sin A cos A ~ cos Asin A.2. sin A cos A -sin A cos A sin A cos A 4. Prove tan-' a ~ tan-' b = tan'- a ~ b* 1I ab Let cc = tan-'a, /3 = tan-'b, -y=tan-' a + b* 1 - ab Hence, tan a = a, tanfl = b, tanly c ab Then a~/ 3= and hence tan (a+/3) = tany Expandintang,+tan = tan 'y. Expanding, 1 - tan a tan p Substituting, a +b - a +b* 1 -ab 1- ab 60 PLANE TRIGONOMETRY. EXAMPLES. 1. Find cos 150, tan 75'. cot a co I 2. Prove cot(a 3) =t — cot fl 1 cot/3 ~ cot a 3. Prove geometrically sin (ae + /3) - sin a cos / + cos a sin l, and cos (a + /3) = cos a cos / - sin a sin 3, given (a) a acute, f3 obtuse; (b) a, /3, obtuse; (c) a, /3, either, or both, negative angles. 4. Prove geometrically tan (a + /) tan a + tan 1 - tan a tan /3 Verify the formula by assigning values to a and /3, and finding the values of the functions from the tables of natural tangents. 5. Prove cos (a + 3) cos (a - /3) = cos2 a - sin2 /3 6. Show that tan a +- tan sin (a + /) Cos a cos / 7. Given tan a = 4, tan/3= 3, find sin (a +/) 8. Given sin 2800 = s, find sin 170'. 9. If a = 67' 22', / = 128' 40', by use of the tables of natural functions verify the formuke on page 56. 10. Prove tan-' = tanl x + tan-x/a. 1-x~2ax 11. Prove tan'-' + tan1 b - xtan-'. bx/3 xv' - 12. Prove sec1 a = sin/a2 - X2 a 13. If a + /3 = o, prove cos2 a ~ cos2/3 -2 cos a cos13 cos w = sin2 W. 14. Solve 1 sinuO - I - cosO0. 15. Prove sin (A + B) cos A - cos (A ~ B) sin A = sin B. 16. Prove cos (A + B) cos (A -B) + sin(A + B) sin(A - B) = cos 2 B. 17. Prove sin (2 a - 3) cos (a - 2/3) - cos (2 a - /3) sin (a - 2 /3 = sin (a + /). 18. Prove sin(n - 1)a cos(n+ 1)a + cos(n - 1)a sin(n + 1) a = sin 2 na. 19. Prove sin (1350 - 0) + cos (1351 + 0) = 0. ADDITION- SUBTRACTION FORMULA]. 61 20. Prove 1 - tan2 a tan2 / = cos2 c- s2 a cos2 a COS2P 21. Prove tan a +tan tan a tan. cot a + cot / 22. tan2 ( a) = 1-2 sin a cos a \4 /1 + 2 sin a cos a 51. The following formula are very important and should be carefully memorized. They enable us to change sums and differences to products, i.e. to displace terms by factors. sin e + sin 4 = 2 sin +2 cos0 2 2 sin e - sin4 = 2 cos0 2- sin 2, cos e + cos 4 = 2 cose c 2 osecos 0 - cos 4 = - 2 sin +2- sine - 2 2 Since sin (( +,8) = sil a cos /3 + cos a sin /, and sin (a -, ) = sin a cos, - cos a sin 3, then sin (a + 3) + sin (a - 3) = 2 sin a cos i, (1) and sin (a + 38)- sin (a - 3) = 2 cos a sin 3. (2) Also since cos (a + 8) = cos a cos, - sin a sin 3, and cos (a - 3) = cos a cos 8 + sin a sin 3, then cos (a + 83) + cos (a - ) = 2 cos a cos /, (3) and cos (a + 8) - cos (a - 3) = - 2 sin a sin/3. (4) Put a + 3 = and a- 8 = 2a = +, and a = 2 2/3= -b, and =0-. Substituting in (1), (2), (3), (4), we have the above formule. 62 PLANE TRIGONOMETRY. EXAM PLES, 1. Prove sin 2 0+ si0 = tan 3 cos 20 + cos 0 2 By formulae of last article the first member becomes 2 sin 30 cos 0 2 2 30 = tan 2 cos cos 2 2 2 2. Prove sin a + 2 sin 3 a + sin 5 a sin3 ac 2. Prove v.3m7 -sin 3 a + 2 sin 5 a + sin 7 a sin 5 a (sin + sin 5 a) 2 sin 3 2sin 3 acos 2 a+ 2 sin3 (sin 3 a + sin 7 a)+ 2 sin5a 2sin 5acos 2a+2 sin 5 a _ (cos 2 a + 1) sin 3 a _ sin 3 a (cos 2 a + 1) sin 5 a sin 5 a 3. Prove sin (4A - 2B)+ sin (4B - 2 A) = tan (A + B). cos (4 A -2 B) + cos (4 B - 2 A) 2 sin4A -2BB + 4 B- 2AosA - 2B - 4 B 2A 2 2 2cos4 + 4 - 2 A 4 A - -2 B - 4 B 2+ 2A 2 2 sin (A +B)= tan (A +B). cos (A + B) 4. Prove sin 50~ - sin 70~ + sin 10~ = 0. 2 cos 50~ 700 sin 50 - 70 = 2 cos 60~ sin (- 10~) - sin 10~. 2 2 5. Prove cos 2 a cos 3 a-cos 2 a cos 7 a +cos a cos 10 acot 6 a cot 5 a sin 4 a sin 3 a - sin 2 a sin 5 a + sin 4 a sin 7 a By (3) and (4), p. 61, cos 5 a + cos a - cos 9 a - cos 5 a + cos 11 a + cos 9 (x cos a - cos 7 a - cos 3 a + cos 7 a - cos 3 a - cos 11 cos a +cos11 a 2 cos 6 a cos 5 a cos a - cos 11 a 2 sin 6 a sin 5 a ORAL WORK. By the formulae of Art. 51 transform: 6. cos 5 ( +- cos a. 8. 2 sin 3 0 cos 0. 7. cos a -cos 5 a. 9. sin 2 a -sin 4 a. FUNCTIONS OF THE DOUBLE ANGLE. 63 10. cos 9 0 cos 2 0. 16. cos (30~+ 2 4)) sin (30~- 4). 11. sin 0 + sin 0. 17. sin (2 r + s) + sin (2 r - s). 12. sin 75~ sin 15~. 18. cos (2 - a) - cos 3 a. 13. cos 7p - cos 2 p. 19. sin 36~ + sin 54~. 14. cos (2p + 3 q) sin (2p - 3 q). 20. cos 60 + cos20. 15. sin3t sin t 1 2 2-sl 21. sin 30~ + cos 30~. Prove: 22. sin a + s = tanll cot. sill a - sinl 2 2 23. COs ( + CoS cota + cot a - cos 3 - os a 2 2 24. sinx+ inny = tal + Y cos x + cosy 2 25. sin x- sinl y _ cot x +y cos x - cos y 2 26. cos 55~ + sin 25~ = sin 85~. Simplify: 27. sin B + sin 2 3 + sin 3 B cosB + cos2 B +cos3 B 28 sin C - sin 4 C + sin 7 C - sin 10 C cos C - cos 4 C + cos 7 C - cos 10 C 52. Functions of an angle in terms of those of the half angle. If in sin (a + /) = sin a cos,3 + cos a sill /, a = 3, then sin (a + a) = sin 2 a = 2 sin a cos a. In like manner cos (a + a) = cos 2 a = cos2 a - sin2 a = 2 cos2 a - 1 = 1 - 2 sin a; 2 tan a and tan 2a a tan 1 - tan2 a 64 PLANE TRIGONOMETRY, ORAL WORK. Ex. Express in terms of functions of half the given angles: 1. sin 4 a. 4. cos x. 6. sin (2p - q). 2. cos 3p.. sin 7. cos (30 + 2 ). 3. tan5t. 2 8. sin(x y). 9. From the functions of 30~ find those of 60~; from the functions of 45~, those of 90~. 53. Functions of an angle in terms of those of twice the angle. By Art. 52, cos a= 1-2 sin2 a = 2 cos2 -1. 2 2.. 2 sin2 = 1 - cos a, and 2 cos2=1 + cos a. 2 2 -cos Ca - Cos sin-= - -; COS — 41~. 2 2 2 2 sin - a 2 1 - cos a.. tan-= = 2 a'1~os a 2 C x 1 + cos cos2 Explain the significance of the ~ sign before the radicals. Express in terms of the double angle the functions of 120~; 50~; 90~, with proper signs prefixed. Ex. 1. Express in terms of functions of twice the given angles each of the functions in Examples 1-8 above. 2. From the functions of 45~ find those of 22~ 30'; from the functions of 36~, those of 18~ (see tables of natural functions). 3. Find the corresponding functions of twice and of half each of the following angles, and verify results by the tables of natural functions: Given sin 26~ 42' = 0.4493, tan 62~ 24' = 1.9128, cos 21~ 34' = 0.9300. 4. Prove tan-il1 -cosx = x. 5. 2 tan- x= tan-I 2x 1 4- cos x 2 1 - X2 EXAMPLES. 65 6. If A, B, C are angles of a triangle, prove sinA + sin C + sinB = 4 cos Aft B AN C 2 2 2 7. If cos2 a + cos2 2 a + cos2 3 a = 1, then cos a cos 2 a cos 3 a = 0. 8. Prove cot A - cot 2 A = csc 2 A. tan (-7 ) _ l-tan 2 4 22 9. Prove /= [. tan (4-+) 1 +tanJ 10. tan a 1 2 sin tan (a + >) sin (2 a + ) + sin q 11. If y = tan-1 + 2 + 2, prove x2 = sin 2y. N/1 + X2 - -/1 - x2 12. Prove tan-1 V' + x2- 1 tan- 2x 5 tan-1 x. x 1 - x2 2 13. If y = sin-1 x —, prove x = tan y. V/1 + x2 14. Prove cos2 a + cos2 / - 1 = cos (a + /3) cos (a - 3). 15. Prove V'(cos a - cos f)2 + (sin a - sin /)2 = 2 sin - 2 16. Prove sin-1 - =tan- = cos-1. a+x a 2 ax 17. Prove cos2 0 - cos2 q = sin (4 + 0) sin (4 - 0). 18. Prove tan A + tan (A + 120~) + tan (A - 120~) = 3 tan 3 A. a a 19. Prove tan a - tan a = tan a sec a. 2 2 20. 3 tan-1 a = tan-13 a - a 1 - 3a2 21. cos2 3 A (tan2 3 A - tan2 A) = 8 sin2A cos 2 A. 22. 1 + cos 2 (A -B) cos coc2 A os 2 B A (A -2B). 23. cot2 ( + 2 csc 2 0 -sec 0 k42 2cs c20+-sec0 66 PLANE TRIGONOMETRY. TRIGONOMETRIC EQUATIONS AND IDENTITIES. 54. Identities. It was shown in Chapter I that sin2 0 + cos2 0 = 1 is true for all values of 0, and in Chapter VI, that sin (a + 3) = sin a cos, + cos a sin, is true for all values of a and 3. It may be shown that sin 2 A — 2 -tanA 1 + cos 2 A is true for all values of A, thus: sin 2 A 2 sin A cos A (by trigonometric transforma. 1 + cos 2 A 1 + 2cos2A-1 tion) _sin A = s A (by algebraic transformation) cos A = tan A (by trigonometric transformation). Such expressions are called trigonometric identities. They are true for all values of the angles involved. 55. Equations. The expression 2 cos2 a- 3 cos a + 1 = 0 is true for but two values of cos a, viz. cos a= 1 and 1, i.e. the expression is true for a = 0~, 60~, 300~, and for no other positive angles less than 360~. Such expressions are called trigonometric equations. They are true only for particular values of the angles involved. 56. Method of attack. The transformations necessary at any step in the proof of identities, or the solution of equations, are either trigonometric, or algebraic; i.e. in proving an identity, or solving an equation, the student must choose at each step to apply either some principles of algebra, or some trigonometric relations. If at any step no algebraic operation seems advantageous, then usually the expression METHOD OF ATTACK. 67 should be simplified by endeavoring to state the different functions involved in terms of a single function of the angle, or if there are multiple angles, to reduce all to functions of a single angle. [ Algebraic Transformations! Trigonometric, f Single function [ to change to a Single angle No other transformations are needed, and the student will be greatly assisted by remembering that the ready solution of a trigonometric problem consists in wisely choosing at each step between the possible algebraic and trigonometric transformations. Problems involving trigonometric functions will in general be simplified by expressing them entirely in terms of sine and cosine. EXAMPLES, 1. Prove sin 3 A cos 3A 2 sin A cos A By alg, sin 3 A cos 3 A sin 3 cosA -cos 3 A sinA sin A cos A sin A cos A by trigonometry, sin (3 A-A) - sin 2 A by trigonometry, = sin A cos A sin A cos A 2 sin A cosA _ 2 sin A cos A Or, by trigonometry, sin 3 A cos 3 A 3 sin A - 4 sin3 A 4 cos' A - 3 cos A sin A cos A sin A cos A by algebra, = 3 -4 sin2 A - 4 cos2 A + 3 = 6 - 4(sin2A + cos2A) = 2. 2. Prove sec 8 - 1 tan 80 sec4 - 1 tan 2 0 No algebraic operation simplifies. Two trigonometric changes are needed. 1. To change the functions to a single function, sine or cosine. 2. To change the angles to a single angle, 8 A, 4 A, or 2 A. 68 PLANE TRIGONOMETRY. By trigonometry and algebra, 1-cos89 sinS9 cos 8 9 cos 8 9 1 - cos 4 sin 2 9' cos 4 9 cos 2 9 by algebra, Cos4Ol-cos89)sin89cos29. 1 - cos 4 0 sin 2 9 by trigonometry, cos 4 9(1 - 1 + 2 sin2 4 9) 2sin49 cos49cos29. 1 - 1+$ 2 sin22 sin 2 0 by algebra, sin 4= 2 cos 2 9; siii 29 and sin 4 9 = 2 sin 2 0 cos 2 9, which is a trigonometric identity. 3. Solve 2cos290 +3 sin90 = 0. By trigonometry, 2(1 - sin2 9) + 3 sin 9 = 0, a quadratic equation in sin 9. By algebra, 29sin290- 3 sin90 - 2 = 0, and (sin90 - 2)(2 sin9~ + 1)= 0..'. sin9-2, or -. Verify. The value 2 must be rejected. Why? 9-. 0 = 2100, and 330' are the only positive values less than 3600 that satisfy the equation. 4. Solve sec9 - tan 0 = 2. Here tan 9 = - 0.75,.-. from the tables of natural functions, 9 = 1430 7' 48"f, or 323' 7' 48". Find sec 9, and verify. 5. Solve 2 sin 0 sin 3 9 - sin2 290 = 0. By trigonometry, cos 2 9 - cos 490 - sin2 29 = 0, also cos29 - cos229 + sin229 - sin220= 0. By algebra, cos 290(1 - cos 290) = 0. cos 290 = 0 or 1, and 2 9 = 900, 2700, 01, or 3600, whence 9 = 4502 1350, 00, or 1800. Verify. TRIGONOMETRIC EQUATIONS. 6 69 Or, by trigonometry, 2 sin 0(3 sinG0 - 4 sin3 0) - 4 siri2 0 cos2 0 =0; by trigonometry and algebra, 6 sin2 0 - 8 sin4 0 - 4 sin2 0 + 4 sin4 0 = 0; by algebra, 2 sin2 0 - 4 sin4 0 = 0, and 2 sin2O0(1 - 2si12 0) = 0.. sin 0= 0, or +-/I and 0 = 00, 18O0, 4050, 135', 225', or 3150. The last two values do not appear in the first solution, because only angles less than 360' are considered, and the solution there gave values of 2 0, which in the last two cases would be 4500 and 6300. Solve: 6..-tan 0= cot 0. 8. 2-cos2O-2sin0= 1. 7. sin2 0+ COS0=1 9. sin 2 0 COS0=sin 0. Prove: 1.0. 2 cot 2A = cot A -tan A. 11. cos2 x +cos.2 y 2 cos(x +y) cos(x-y) 12. (cos a+ sin a)2=+ sin 2 a. 57. Simultaneous trigonometric equations. 13. Solve cos (x + y) + cos (x - y) = 2, sin - + sin "Y' 0. 2 2 so ali By trigonometry, cos x cosy sin x sin y + cos x cos y + sin x sin y~ = 2, that COS x COSy=l1; so, - COS ~+ csy= 01 50, ~~~~~2 2 Substituting, COS 2x = 1, COS x = + 1.. x = 00, or 1800, id ~~~~~Y = x = 00, or 1800. Verify. ar ar 70 70 ~~PLANE TRIGONOMETRY 14. Solve for R and F. W - Psin i - R cos i = 0, W + Fcos i - R sin i = 0. To eliminate F, Wcos i - Fsin i cos i - R cs2 i = 0, W sin i + Fcos i sin i - R sin2 i = 0. Adding, W(sin i + cos i) - R(sin2 i + COS2 i) = 0. R = W(sin i + cosi). Substituting, W -Fsin i -W(sin i +cos i)cos i= 0 F w W-W(sini1+ cosi) Cos i sini If W = 3 tons, and i -220 30', compute F and R. R = 3 (0.3827 + 0.9239) = 3.9198. - 3 - 3(0.3827 + 0.9239)0.9239 -_1.624. 0.3827 Solve: 15. 472 cot60- 263 cot4=490, 307'cotO0- 379 cot k= 0. 16. sin2x + 1= Cos x +2sin x. 17. cos2O0+sin90= 1. 18. If 2h(cos2O0 -sin2O0)-2asin~cos6+2bsin.0cosO=0, prove O-tan-' a2 Prove: 19. tan y (+seey) tan Y2 20. 2 cot-'x = csc-11 + X2 2 x 21. sin (0 ~ 450) + sin (4 + 1350) = cos 4os 22. cos v +cos 3v = 1 Cos3 v+ cos 5v 2 cos 2 v-sec2 v 23. cos 3 x-sin3 x= (cos x +sin x) ( - 2sin 2x). Solve: 24. sin29+sinO=cos29+cosO. 25. 4 cos (O+ 600) -F2_ = V6 - 4cos (&+ 300). 26. cot 20=tanO-1. 27. cosO~cos2O~cos3O=0. TRIGONOMETRIC EQUATIONS. 71 28. sin 2 x + -v/'cos 2 x = 1. 29. 3 tan2p + 8 cos2p = 7. 30. Determine for what relative values of P and W the following equation is true: Cos - - 0COS - = 0. 2 TV 2 2 31. Compute N from the equation N 1 cos a- - sin a-Wcos a=0, 3 3 when W = 2000 pounds and a satisfies the equation 2 sin a = 1 + cos a. 32. sill 0O- tan40(cosO0 + sin0)= cos60, sin 9 - tan40 cosO = 1. Prove: 33. cot(t + 15')- tan(t - 15)- 4cos 2t 2 sin 2 t + 1 34. sill-1 - sin-' JL = sin-' 16 1 +_sill 35. tan(~ +) + -Y4 = $ All - sill (0 36. 2 sin-1 = cos-1I. 37. If sin A is a geometric mean between sin B and cos B, prove cos 2 A = 2 sin(45' - B) cos(45 + B). 38. Prove sin(a + / 8+ y)=sinacos/ cosy + cosa sin/cos-y + cos a Cos 3 sin y - sin a sin /3 sin y. Also find cos(a + / + y). 39. Prove tan(a + /3~ ) = tan a + tan 3 + tan -y - tan a tan / tan y 1 -tanila tan/3 -tan/3tan y - tany tain a If a, /3, and y are angles of a triangle, prove 40. tan a + tan P + tan y =-tan a tan/3 tan -y. 41. cot + cot +cot IY = cotcot cotY. 2 2 2 2 2 2 If a + / + -y 900, prove 42. tan a tan/3+tan/3 tan y + tan y tan a=l Prove: 43. sin na = 2sin(n - 1) a cos a - sin (n - 2)a. 44. cosna = 2cos (n - 1) accos a - cos(n - 2)a. 45. tanna= tan (n - 1) a + tan a 1 - tan(n - 1) a tan a CHAPTER VII. TRIANGLES. 58. In geometry it has been shown that a triangle is determined, except in the ambiguous case, if there are given any three independent parts, as follows: I. Two angles and a side. II. Two sides and an angle, (a) the angle being included by the given sides, (b) the angle being opposite one of the given sides (ambiguous case). III. Three sides. The angles of a triangle are not three independent parts, since they are connected by the relation A + B + C = 180~. The three angles of a triangle will be.designated A, B, C, the sides opposite, a, b, c. But the principles of geometry do not enable us to compute the unknown parts. This is accomplished by the following laws of trigonometry: sin A sin B sin C I. Law of Sines, = s -—. a b c II. Law of Tangents, tan2 (A B) = a-, etc. tan 1 (A + B) a + 62 + c2 a2 III. Law of Cosines, cos A = + 2, etc. 2 be 59. Law of Sines. In any triangle the sides are proportional to the sines of the angles opposite. Let ABC be any triangle, p the perpendicular from B on b. In I (Fig. 34), C is an acute, in II, an obtuse, in III, 72 LAW OF SINES - OF TANGENTS. 73 a right angle. The demonstration applies to each triangle, but in II, sinACB=sinDCB (why?); in III, sinC= 1 (why?). B B B C C p a aa a A bDC A C DA b 0 I. II. III. FIG. 34. Now sin A = PI p = c'sin A. C sin C=.. p = a sin C. a Equating values of p, c sin A = a sin C, sin A sinU. or, a e By dropping a perpendicular from A, or C, on a, or c, show that sin B sin C sin A sin B ~, or 6' b C a b sin A sin.B sin C whence a b c 60. Law of Tangents. The tangent of half the d/ference of two angles of a triangle is to the tangent of half their sum, as the difference of the sides opposite is to their sum. a sin A By Art. 59, b sill B By composition and division, a-b sinA-sinB 2cos j(A~B)sin1(A-B) a+b sinA+siunB 2sin -(A+B)cos~(A-B) 2 21 jVL 1.U tan(A - B) tan 1 (A + B) tan - (A - B) a-b Ar,B2 tan 1 (A + B) ~- R a +- b 74 PLANE TRIGONOMETRY. 61. Law of Cosines. The cosine of any angle of a triangle is equal to the quotient of the sum of the squares of the adjacent sides less the square of the opposite side, divided by twice the product of the adjacent sides. B C pa A 6DC i. B 1 aP A b C D II. FIG. 34. B C a A b C IlI. In each figure a2 =p2 ~ DC2 _AD- A02 + (b - AD)2 (in Fig. 34, II, DDC is negative; in III, zero) - C2 - AD2 h 62 2 b 6.AD + AD2 But AD = c cos A, = 62 ~ C2 - 2 b. AD..'. a2=b2~c2-2bccosA; A 62 ~ c2 - a2 cosA= a2 + C2 62 Prove that cos B a 2 ac and cos C2 -2 ab 62. Though these formuhae may be used for the solution of the triangle, they are not adapted to the use of logarithms (why?). Hence we derive the following: Since cos A = 2cos2A 1 = 1 - 2 sin2A 2 2' we have 2 cosA = 1 + cos A, and 2 sin24 = 1 - cos A..2 2 LAW OF COSINES. 75 From the latter 2A 1 2 +2 - a2 2 b - b2 — 2 + a2 2 2 be 2 be a2-(b - )2 (a- b + c)(a + b- c) 2 be 2 be Let a+b+c=2s,then a+b-ec=a+b+c-2c=2 s-2c; i.e. a + b - c = 2 (s - c). In like manner, a- b + c = 2(s - b). - a + b + c = 2 (s- a). Substituting, 2 sin2A = 2 (s - b) 2 (s -) 2 2be.. sin ( b) (2 ' 6be Show that sin -? 2 also sin =? 2 From 2 cos2 A = 1 + cos A, show that cos A = - a) 2 b also cos -= and cos =? Also derive the formule Air b(s)(s-a) a - s(s — ) ' tan B? 2 tan =? 2 76 PLANE TRIGONOMETRY. 63. Area of the triangle. In the figures of Art. 59 the area of the triangle ABC= A -1 pb But p = c sin A... a= bc sin A. (i) e sinBB Again, by law of sines, = sinB sin C C2 sin A sin B Substituting, A = c2sinA in c2sin A sin B (why?). (ii) 2 sin(A + B) A Finally, since sin A = 2 sin - cos, we have from (i) 2 2 AA be 2 sinA A ______(sa)(s- b)(s e 2 2 — 2 be - be or A=Vs (s - a)(s -b)(s-c) (iii) Find A; (1) Given a = 10, b = 12, C = 450* (2) Given a = 4, b =5, c = 6. (3) Given a = 2, B- 45, C = 60. SOLUTION OF TRIANGLES. 64. For the solution of triangles we have the following formulh, which should be carefully memorized: sin A sin B sin C a b c I tan(A - 1B) a-b tan 1 (A ~ B). i sin4- (S b (S), - or cos A <s(s - a) 2b 2 be or tanA= 2 sin(sA-sia) IV. A-1bc sinA c "SiAsin xs(s -a) (s -b) (s - c). 2a 2 sin (A~+ B) SOLUTION OF TRIANGLES. 77 Which of the above formulae shall be used in the solution of a given triangle must be determined by examining the parts known, as will appear in Art. 69. It is always possible to express each of the unknown parts in terms of three known parts. In solving triangles such as Case I, Art. 58, the law of sines applies; for, if the given side is not opposite either given angle, the third angle of the triangle is found from the relation A + B + C = 180~, and then three of the four sill A sin B quantities in sm - s being known, the solution gives a b the fourth. In Case II (6) the law of sines applies, but in II (a) two sin A sin B only of the four quantities in sin are known. a b Therefore, we resort to the formula tan (A-B) = a - tan (A + B), in which all the factors of the second member are known. In Case III, tan A (s ) (s) c) is clearly applicable, 2 s s(s - a) A A and is preferred to the formulae for sinA and cos-A; for, 2 2 first, it is more accurate since tangent varies in magnitude from 0 to oo, while sine and cosine lie between 0 and 1. (See Art. 27, 5.) Let the student satisfy himself on this point by finding, correct to seconds, the angle whose logarithmic sine is 9.99992, and whose logarithmic tangent is 1.71668. Does the first determine the angle? Does the second? And, second, it is more convenient, since in the complete solution of the triangle by sin A six logarithms must be taken A A from the table, by cos - seven, and by tan but four. 2 2 The right triangle may be solved as a special case by the law of sines, since sin C =1. PLANE TRIGONOMETRY. 65. Ambiguous case. In geometry it was proved that a triangle having two sides and an angle opposite one of them of given magnitude is not always determined. The marks of the undetermined or ambiguous triangle are: 1. The parts given are two sides and an angle opposite one. 2. The given angle is acute. 3. The side opposite this angle is less than the other given side. When these marks are all present, the number of solutions must be tested in one of two ways: (a) From the figure it is apparent that there will be no solution when the side opposite is less than the perpendicular p; one solution when side a equals p; and two solutions when a is greater than p. B B B A b A b C A b Ca A b CA b CA b C ' No Solution. One Solution. Two Solutions. FIG. 35. And since sin A = P, it follows that there will be no soluc tion, one solution, two solutions, according as sin A a. < c (b) A good test is found in solving by means of logarithms; and there will be no solutions, one solution, two solutions, according as log sin C proves to be impossible, zero, possible, i.e. as log sin C is positive, zero, or negative. This results from the fact that sine cannot be greater than unity, whence log sine must have a negative characteristic, or be zero. 66. In computations time and accuracy assume more than usual importance. Time will be saved by an orderly arrangement of the formulae for the complete solution, before opening the book of logarithms, thus: SOLUTION OF TRIANGLES. 79 Given A, B, a. Solve completely. C=180~-(A+B), b= a sinB c =asin A 1 ab sin. sin A sinA ' 180 log a = log a = A + B = log sin B = log sin C =.*. C = colog sin A = colog sin A = log b = log c =.'. b-.'. c= Check: log a log (s - b) = log b= log (s - c)= log sin C = colog s = colog 2 = colog (s - a)= log A = 2.'. A = * log tan A 2.~. A= 67. Accuracy must be secured by checks on the work at every step; e.g. in adding columns of logarithms, first add up, and then check by adding down. Too much care cannot be given to verification in the simple operations of addition, subtraction, multiplication, and division. A final check should be made by using other formulae involving the parts in a different way, as in the check above. As far as possible the parts originally given should be used throughout in the solution, so that an error in computing one part may not affect later computations. 68. The formulae should always be solved for the unknown part before using, and it should be noted whether the solution gives one value, or more than one, for each part; e.g. the same value of sin B belongs to two supplementary angles, one or both of which may be possible, as in the ambiguous case. 69. Write formulae for the complete solution of the following triangles, showing whether you find no solution, one solution, two or more solutions, in each case, with reasons for your conclusion: 80 80 ~~PLANE TRIGONOMETRY. a b c 1. 2. 3. 4. 0.75 5. 243 6..7. 0.058 8. 2986 9. 78.54 135.82 26.89 0.85 0.95 562 38.75 25.92 A B 810 2 6' 298" 440 1 1 630 18' 20" 530 28' 30" 780 la" 330 461 C 20"1 540 22' 12" 410 30' 18" 360 15' 40"1 630 50' 10" 300 1493 48 50 260 151 MODEL SOLUTIQNS. 1. Given a =0.78-5, b=-0.85, c=-0.633. Solve completely. A ( s-)s - c) B (_ -_a_________-_) s b tan. tan -t-)(-c - C= sa sh 2 - s (s a) 2.- s(s-b) 2n~- s(s- c) Check: A + B+ C =180'. A - v's(s -a)(s -b) (s -c). a = 0.735 b =0.85 C= 0.633 29)2.268 s = 1.134 s - a = 0.349 s - b =0.281 s - c = 0.501 Check: A =610 53' 38" B = 72' 46' 4" C= 450 20' 20" 1800 0' 2"1 log (s - h) = 9.4533~2 log (' - c) = 9.69984 colog s = 9.94539 colog (s - a) = 0.45717 2)19.555-72 log tan A= 9.77786 A 300 56' 49" A =61' 53' 38" log (s -a)=- 9.54283 log (s - b)=- 9.45332 colog s = 9.94539 colog (s -c)=- 0.30016 2)19.24170 log tan 1C - 9.62085 C 22' 40' 10" C=- 45' 20' 20" log (s -a) = 9.54283 log (s - c) = 9.69984 colog s = 9.94539 colog (s - b) =0.5466& 2)19.73474 log tan B = 9.86737 B - 360 23' T'" B = 720 46' 4" log s = 0.05461 log (s - a) = 9.54283, log (s - b) =9.45332 log (s - c) = 9.69-984 2)18.75060 log A~ = 9.37534 A= 0.2373 Solve: (1) Given a = 30, b = 40, c = 50. (2) Given a = 2159, b = 1431.6, c = 914.8. (3) Given a = 78.54, b = 32.56, c = 48.9. SOLUTION OF TRIANGLES. 81 2. Given A = 570 23' 12", C= 680 15' 30", c = 832.56. Solve completely. a c sin A bc sin B A.='bcsinA sin C sill C 2 B = 1800 -(A + - 540 91' 18'. loge = 2.92042 log sin A = 9.92548 colog sin C ' 0.03204 logoa = 2.87794 a= 754.98 Check: a 754.98 boo 728.38 c= 832.56 2)2315.92 s 1157.96 Check: tan s (s-a) 2 = s sS - a) log c = 2.92042 log sinl B = 9.90990 colog sin C = 0.03204 logb = 2.86236 logb = 2.86236 log c = 2.92042 log sin A = 9.92548 log 2 A = 5.70826 b = 728.38 A - 510811 = 255405.5 2 s-a= 402.98 s-bo= 429.58 s-c= 325.40 log(s - b) = 2.63304 log (s - c) = 2.51242 colog s = 6.93634 colog (s -a)= 7.39471 2)19.47651 log tan 1A = 9.73826 1A = 280 41' 38' A = 570 23' 16" Solve: (1) Given a = 215.73, B 920 15', C - 280 14'. (2) Given b = 0.827, A =780 14' 20", B 63' 42' 30". (3) Given b = 7.54, c = 6.93, B = 540 28' 40". 3. Given a = 25.384, c = 52.925, B - 28' 32' 20". Solve completely. (Why not use the same formulae as in Example 1, or 2?) tanC-A c a C + A - c sin B tan tan ~,b A - 1 ac sill B. 2 c + a 2 sin C ' 1800 - B = C + A = 1510 27' 40"..'. 1(C +A)= 750 43' 50"' Check: b = a sin B sin A c= 52.925 a= 25.384 c+a= 78.309 c-a= 27.541 log c = 1.72366 log sin B =9.67921 colog sin C = 0.11484 log b=1.51771 b= 32.939 log (c - a) = 1.43998 colog (c + a) = 8.10619 log tan 1(C+A) 0.59460 log tan 1(C -A)=0.14077 Check: log a -1.40456 log sin B= 9.67921 colog sin A = 0.43395 log b 1.51772 ' 1 (C -A)= 540 7'38" (C~A)= 75043/5011 adding, C=129051128/1 subtracting, A= 21036112// log a = 1.40456 log co =1.72366 log sin B = 9.67921 log 2 A = 2.70743 A 509.83 054965 2 82 PLANE TRIGONOMETRY. Solve: (1) Given a = 0.325, c = 0.426, B = 48~ 50' 10". (2) Given b = 4291, c = 3194, A =73~ 24' 50". (3) Given b = 5.38, c = 12.45, A = 62 14' 40". 4. Ambiguous cases. Since the required angle is found in terms of its sine, and since sin a = sin (180~ - a), it follows that there may be two values of a, one in the first, and the other in the second quadrant, their sum being 180~. In the following examples the student should note that all the marks of the ambiguous case are present. The solutions will show the treatment of the ambiguous triangle having no solution, one solution, two solutions. (a) Given b = 70, c = 40, C= 47~ 32' 10". Solve. Why ambiguous? sin B =b sin C log b = 1.84510 c log sin C = 9.86788 colog c = 8.39794 log sin B = 0.11092. B is impossible, and there is no solution. Why? Show the same by sin C > - (b) Given a = 1.5, c = 1.7, A = 61~ 55' 38". Solve. sin C c sin A log c = 0.23045 a log sin A = 9.94564 colog a = 9.82391 log sin C = 0.00000 C= 90~ and there is one solution. Why? Show the same by sinA =a. Solve for the remaining parts and check the work. SOLUTION OF TRIANGLES. 8~3 (c) Given a = 0.235, 6 = 0.189, B = 360 28' 20". Solve. i A a sin B b sill C6 sin B log a = 9.37107 log 6 = 9.27646 log sillB= 9.77411 log sin C = 9.99772 colog 6 = 0.72354 colog sin B = 0.22589 log sin A = 9.86872 log c = 9.50007 A = 470 39' 25" c = 0.31628 or 1320 20' 35"..'. C= 95' 52' 15" or 110 11' 5". 9.27646 or 9.28774 0.22589 or 8.79009 or 0.06167 6 Solve for A, and check. Show the same by Sin B < -. a Solve: (1) Given 6 = 216.4, c = 593.2, B = 98' 15'. (2) Given a = 22, 6 = 75, B = 320 20'. (3) Given a = 0.353, c = 0. 295, A = 46' 15' 20". (4) Given a = 293.445, 6 = 450, A = 40' 42'. (5) Given 6 = 531.03, c = 629.20, B = 340 28' 16". Solve completely, given: a b C A B 1. 2. 3. 4. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 50 60 10 4 5 C 780 27' 47" 930 351 40 352.25 0.573 107.087 51 513.27 0.394 11 6 10 1090 28' 16" 490 28' 32" 482.68 1120 41 56" 15' x/2 1170 340 27' 380 56' 54" 480 35' 450 730 15' 15" 420 18' 30" 490 8' 24" 197.63 246.35 4090 3850 3795 234.7 26.234 273 136 3811 185.4 22.6925 840 36' 720 25' 13"1 84 PLANE TRIGONOMETRY. APPLICATIONS. 70. Measurements of heights and distances often lead to the solution of oblique triangles. With this exception, the methods of Chapter V apply, as will be illustrated in the following problems. The bearing of a line is the angle it makes with a north and south line, as determined by the magnetic needle of the mariner's compass. If the bearing does not correspond to any of the points of the compass, it is usual to express it thus: N. 40~ W., meaning that the line bears from N. 40~ toward W. EXAMPLES. 1. When the altitude of the sun is 48~, a pole standing on a slope inclined to the horizon at an angle of 15~ casts a shadow directly down the slope 44.3 ft. How high is the pole? 2. A tree standing on a mountain side rising at an angle of 18~ 30' breaks 32 ft. from the foot. The top strikes down the slope of the mountain 28 ft. from the foot of the tree. Find the height of the tree. 3. From one corner of a triangular lot the other corners are found to be 120 ft. E. by N., and 150 ft. S. by W. Find the area of the lot, and the length of the fence required to enclose it. 4. A surveyor observed two inaccessible headlands, A and B. A was; W. by N. and B, N.E. He went 20 miles N., when they were S.W. and S. by E. How far was A from B? 5. The bearings of two objects from a ship were N. by W. and N.E. by N. After sailing E. 11 miles, they were in the same line W.N.W. Find the distance between them. 6. From the top and bottom of a vertical column the elevation angles of the summit of a tower 225 ft. high and standing on the same horizontal plane are 45~ and 55~. Find the height of the column. 7. An observer in a balloon 1 mile high observes the depression angle. of an object on the ground to be 35~ 20'. After ascending vertically and uniformly for 10 mins., he observes the depression angle of the same object. to be 55~ 40'. Find the rate of ascent of the balloon in niles per hour. 8. A statue 10 ft. high standing on a column subtends, at a point 100 ft. from the base of the column and in the same horizontal plane, thel same angle as that subtended by a man 6 ft. high, standing at the foot of the column. Find the height of the column. 9. From a balloon at an elevation of 4 miles the dip of the horizon is 2~ 33' 40". Required the earth's radius. TRIANGLES - APPLICATIONS. 85 10. Two ships sail from Boston, one S.E. 50 miles, the other N.E. by E. 60 miles. Find the bearing and distance of the second ship from the first. 11. The sides of a valley are two parallel ridges sloping at an angle of 30~. A man walks 200 yds. up one slope and observes the angle of elevation of the other ridge to be 15~. Show that the height of the observed ridge is 273.2 yds. 12. To determine the height of a mountain, a north and south base line 1000 yds. long is measured; from one end of the base line the summit bears E. 10~ N., and is at an altitude of 13~ 14'. From the other end it bears E. 46~ 30' N. Find the height of the mountain. 13. The shadow of a cloud at noon is cast on a spot 1600 ft. due west of an observer. At the same instant he finds that the cloud is at an elevation of 23~ in a direction \V. 14~ S. Find the height of the cloud and the altitude of the sun. 14. From the base of a mountain the elevation of its summit is 54~ 20'. From a point 3000 ft. toward the summit up a plane rising at an angle of 25~ 30' the elevation angle is 68~ 42'. Find the height of the mountain. 15. From two observations on the same meridian, and 92~ 14' apart, the zenith angles of the moon are observed to be 44~ 51'21" and 48~ 42' 57". Calling the earth's radius 3956.2 miles, find the dis- / Z=Zenith angle tance to the moon.nith 16. The distances from a point to three objects are 1130, 1850, 1456, and the angles subtended by the distances between the three objects are respectively 102~ 10', 142~, and 115~ 50'. Find the distances between the three objects. 17. From a ship A running N.E. 6 mi. an hour direct to a port distant 35 miles, another ship B is seen steering toward the same port, its bearing from A being E.S.E., and distance 12 miles. After keeping on their courses 1~ hrs., B is seen to bear from A due E. Find B's course and rate of sailing. 18. From the mast of a ship 64 ft. high the light of a lighthouse is just visible when 30 miles distant. Find the height of the lighthouse, the earth's radius being 3956.2 miles. 19. From a ship two lighthouses are observed due N.E. After sailing 20 miles E. by S., the lighthouses bear N.N.W. and N. by E. Find the distance between the lighthouses. 20. A lighthouse is seen N. 20~ E. from a vessel sailing S. 25~ E. A mile further on it appears due N. Determine its distance at the last observation. EXAMPLES FOR REVIEW. IN connection with each problem the student should review all principles involved. The following list of problems will then furnish a thorough review of the book. In solving equations, find all values of the unknown angle less than 360~ that satisfy the equation. 1. If tan a = 7, taln =, show that tan (/ -2 a ) = T 2. Prove tan a + cot a = 2 csc 2 a. A A A A 3. From the identities sin2- + cos2 = 1, and 2 sin - cos- = sin A, 2 2' 2 2 A prove 2 sin = ~ v1 + sill A ~ /1 - sin A, and 2 cos - = d +1 + sinll A T V1 - sin A. 4. Remove the ambiguous signs in Ex. 3 when A is in turn an angle of each quadrant. 5. A wall 20 feet high bears S. 59~ 5f E.; find the width of its shadow on a horizontal plane when the sun is due S. and at an altitude of 60~. 6. Solve sin x +sin2x+ sin 3 x = 1 + cosx+ cos 2 x. 1 + tan_1 71_ r. 7. Prove tan-1 + tan-' = 2 3 4 8. If A = 60~, B = 45~, C = 30~, evaluate tan A + tan B + tan C tan A tan B + tan B tan C + tan Ctan A 9. Prove cos ( + B) cos C 1 - tan A tan B cos(A + C) cos B 1 - tan A tan C 10. Solve completely the triangle whose known parts are b = 2.35, c = 1.96, C = 38~ 45'.4. 11. Find the functions of 18~, 36~, 54~, 72~. Let x =18~. Then 2 =36~, 3 x =54~, and 2x + 3x = 90~. 12. If cot a = P, find the value of sin a + cos a + tan a + cot a + sec a + csc a. 86 EXAMPLES FOR REVIEW. 87 13. Prove sin 3 a sin 2 a - sin 3 8 sin 2 a 1 + 4 cos cos. sin 2 a sin/3 - sin 2 / sin a 14. From a ship sailing due N., two lighthouses bear N.E. and N.N.E., respectively; after sailing 20 miles they are observed to bear due E. Find the distance between the lighthouses. 15. Solve 1 -2 sin x = sin 3 x. 16. Prove sin-'lia tan-'. a + b = b 17. If cos 0 - sin 0 = 2 sin 0, then cos 0 + sin 0 = /2 cos 0. 18. Solve completely the triangle ABC, given a = 0.256, b = 0.387r C = 102~ 20'.5. 19. Prove tan (30~ + a) tan (30~- a) =2 cos 2 1 2 cos 2 a + 1 20. Solve tan (45~ - 0) + tan (45~ + 0) = 4. 21. Prove sin2 a cos2 3 - cos2 a sin2 / = sin2 C - sin2 /. 22. Prove cos2 aE cos2 / - sin2 a sin2 / = cos2 a - sin2 /. 23. A man standing due S. of a water tower 150 feet high finds its elevation to be 72~ 30'; he walks due W. to A street, where the elevation is 44~ 50'; proceeding in the same direction one block to B street, he finds. the elevation to be 22~ 30'. What is the length of the block between A and B streets? 1 + tan_1 1 i 1 ~r 24. Prove tan-i1 + tan-l + tan-1 + tan-1 = 3 5 7 8 4 25. If P = 60~, Q = 45~, R = 30~, evaluate sin P cos Q + tan P cos Q sin P cos P + cot P cot R 26. If cos (90~ + a) - -, evaluate 3 cos 2 a + 4 sin 2 a. 27. If sin B + sin C =m, cos B + cos C = n, show that tanB = m. 2 n 28. Show that sin 2 /3 can never be greater than 2 sin /. 29. Prove sin-l + sin-1 5 = tan-1 56 30. Solve cot-lx + sin-l 1 = -7r 5 4 31. Solve sin-' x + sin- (1 - x) = cos- x. 32. A man standing between two towers, 200 feet from the base of the higher, which is 90 feet high, observes their altitudes to be the same,; 70 feet nearer the shorter tower he finds the altitude of one is twice that of the other. Find the height of the shorter tower, and his original distance from it. 88 PLANE TRIGONOMETRY. 33. Solve cos 3, + 8 cos3 8 = 0. 34. Solve cot m - tan (180~ + m) = sec m + sec (90~ - i). 35. Solve tant 2 cos 2t 1 + tan t 36. Prove cotA + cotB = sin (A + B). sin A sin B 37. Prove cotP-cotQ = _sin(P- Q). sin P sin Q 38. In the triangle ABC prove a = b sin C + c sin B, b = c sin A + a sin C, c = a sin B + b sin A. 39. Solve completely the triangle, given a = 927.56, b = 648.25, c = 738.42. 40. Prove cos2 a - sin (30~ + a) sin (30 ~- a)= |. 41. Prove tan 3 x tan x cos x co 4 x cos 2 x + cos 4 x 42. Simplify cos (270~ + a) + sin (180~ + a) + cos (900 + a). 43. Simplify tan (270~ - 0) - tan (90~ + 0) + tan (270~ + 0). 44. Solve cos 3 - cos 2 ) + cos q = 0. 45. Solve cosA + cos 3 A + cos5A + cos 7 A =0. 46. The topmast of a yacht from a point on the deck subtends the:same angle a, that the part below it does. Show that if the topmast be a feet high, the length of the part below it is a cos 2 a. 47. A horizontal line AB is measured 400 yards long. From a point in AB a balloon ascends vertically till its elevation angles at A and B.are 64~ 15' and 48~ 20', respectively. Find the height of the balloon. sin a pr o v o / no 48. If cos = n sin a, and cot = (, prove cos n tan/3 /l+ n2cos 2a 49. Find cos 3 a, when tan 2 a = -. 50. Solve completely the triangle, given a = 0.296, B = 28~47'.3, C = 84~ 25'. 51. Evaluate sin 300~ + cos 240~ + tan 2250. 52. Evaluate sec 2 _ cs 5 + tan 4 3 3 3 EXAMPLES FOR REVIEW. 89 53. If tan 0 = Sin a cos y - sin # sin y cos a cos y - cos / sill y and tan 0 = sin a sin y - sin / cos cos a sill y - cos / cos y show that tan( + 4) = tan(a + /3). 54. If tan 466' 15' 38" - 24, find the sine and cosine of 2330 7' 4911 55. Prove csc a - cot a sec at - tan a sec a + tan a csca(t + cot a 56. Prove cos(a - 3 /) - cos(3 a - a ) - 2 sin(a - /3). sin 2 a + sin 2/3 57. Prove sin 80' = sin 401 + sin 200. 58. Prove cos 20' cos 400 + cos 80". 59. Prove 4tan-'- tan-' 1 w 5 239 4 60. From the deck of a ship a rock bears N.N.W. After the ship has sailed 10 miles E.N.E., the rock bears due W. Find its distance from the ship at each observation. 61. Find the length of an arc of 800 in.a circle of 4 feet radius. 62. Given tan 9 =4, tan 4),',, evaluate sin(O + 4) + cos(9 -4)). 63. If tan 9 = 2 tan 4, show that sin(9 ~ 4)) = 3 sin(O - 4). 64. Prove cos(a+/3)cos (a-/3) ~sii(a+[3) sin (a-/)= 1 -tan2fl 1 +tan2/3 65. Solve 4 cos 2 ~ + 3 cosO0 = 1. 66. Solve 3 sin a 2 '~ sin (60' - a). 67. Prove (sin a - csc a)2 - (tan ae - cot a)2 + (COS a - sec a)2- 1. 68. Prove 2(sin6 a + cos6 a) + 1 = 3(sin4 a + cos4 a). 69. Prove csc 2/3 ~ cot 4/3 = cot/3 - csc 4/3. 5 _ _ _ _ _ 70. If tanp =-, cos 2 q 5,then csc _-q= 5V'T3. 12 625 2 71. Solve completely the triangle, given a 5 0.0654, b 0.092, B - 380 40'.4. 72. Solve completely the triangle, given b = 10, c = 26, B - 22 37'. 73. A railway train is travelling along a curve of I mile radius at the rate of 25 miles per hour. Through what angle (in circular measure) will it turn in half a minute? 90 PLANE TRIGONOMETRY. 74. Express the following angles in circular measure: 630, 40 30', 60 12' 36". 75. Express the following angles in sexagesimal measure: 7T 37r 177r 6 8 64 76. If A, B, C are angles of a triangle, prove cos A + cos B + COS C = 1 + 4 sinAin,B sin 2 2 2 77. Prove sin 2 x + sin 2 y + sin 2 z = 4 sin x sin y sin z, when x, y, z are the angles of a triangle. 78. Prove sec a = 1 + tan a tan2 -79. Prove sin2(a + /) - sin2(a - 8)= sin 2 a sin 2 3. 80. Prove cos2(a + 3) - sin2(a - )= cos 2 a cos 2/3. 81. Prove sin 19p + sin 17p = 2 cos 9p. sin 10p + sin 8p 82. Consider with reference to their ambiguity the triangles whose known parts are: (a) a = 2743, b = 6452, B = 43' 15'; (b) a = 0.3854, c = 0.2942, C - 380 20'; (c) b= 5, c = 53, B = 15' 22f' (d) a=20, b = 90, A = 63' 281.5. 83. From a ship at sea a lighthouse is observed to bear S.E. After the ship sailed N.E. 6 miles the bearing of the lighthouse is S. 270 30' E. Find the distance of the lighthouse at each time of observation. 84. Prove sin 6O + 3 4)+ sin (3 0 O 2 cos (9 +) sin 2 0 + sin 2 4 85. Prove cos 15' - sin 150 86. Show that cos (a + /3) cos (a - /) = cos2 a - sin2 / = cos2 / - sin2 a. 2 sin2 a - 1 87. Show that tan (a + 450) tan (a - 450) 2 2 cos2 a - 1 88. Sove sin x(X+V) si(X - y)=, os(x + y)cos (x-y)=O 1 +sin a cosa a 89. Prove - = tan 1 + sill a + COS a2 EXAMPLES FOR REVIEW. 91 90. Prove tan 2 6 + sec 2 cos 0 + sin 0 cos 0 - sin 9 91. If tan 4 = b, then a cos2 4< + bsin2 = a. a 92. Prove sin-l 1 + cot-l3 =r ao 4 93. Solve cos A + cos 7 A = cos 4 A. 94. Two sides of a triangle, including an acute angle, are 5 and 7, the area is 14; find the other side. 95. Show that 3 cos 3 - 2 cos 0 - cos5 0 tan 2 0. sin 5 0 - 3 sin 3 0 + 4 sin 0 96. A regular pyramid stands on a square base one side of which is 173.6 feet. This side makes an angle of 67~ with one edge. What is the height of the pyramid? 97. From points directly opposite on the banks of a river 500 yards wide the mast of a ship lying between them is observed to be at an elevation of 10~ 28'.4 and 12~ 14'.5, respectively. Find the height of the mast. 98. Show that (sin 60~ -sin 45~) (cos 30~ + cos 45~) = sin2 30~. 99. Find x if sin-1 x + sin-' x = 4r 2 4 100. Trace the changes in sign and value of sin a + cos a as a changes from 0~ to 360~. CHAPTER VIII. MISCELLANEOUS PROPOSITIONS. 71. The circle inscribed in a given triangle is often called the incircle of the triangle, its centre the incentre, and its radius is denoted by r. The incentre is the point of intersection of the three bisectors of the angles of the triangle (geometry). The circle circumscribed about a triangle is called the circumcircle, its centre the circumcentre, and its radius R. The circumcentre is the point of intersection of perpendiculars erected at the middle points of the three sides of the triangle (geometry). Incircle. Circumcircle. Escribed circle opposite A. FIG. 37. The circle which touches any side of a triangle and the other two sides produced is called the escribed circle; its radius is denoted by ra, rb, or r,, according as the escribed circle is opposite angle A, B, or C. Again, the altitudes from the vertices of a triangle meet in a point called the orthocentre of the triangle. Finally, the medians of a triangle meet in a point called the centroid, which is two-thirds of the length of the median from the vertex of the angle from which that median is drawn (geometry). Certain properties o'f the above will now be considered. 92 MISCELLANEOUS PROPOSITIONS. 93 72. To find the radius of the incircle. Let A, A', A," A"' represent C the areas of triangles ABC, COB, AOC, BOA, respectively. Then a A = A/ + /A + A/ A- 4 B FIG. 38. = l (a + b + c) r = sr. And since A = Vs(s - a)(s - b)(s - c), (Art. 63) A V(s - a)(s- b)(s -c) s s CoR. To express the angles in terms of r and the sides, divide each member of the above equation by s - a. Then r (s- b)(s-c) = tan 2 A. (Art. 62) s - a - s(s - a) r r In like manner tan 1 B = tan 1 C s s-b s - To find the radius of the circumcircle. a B a R A' FIGB c b3. FIG. 39. In the figure ABC is the given triangle, and A' C a diameter of the circumcircle. Then, angle A = A', or 180~-A'..-. sin A = sin A'. Since A'BC is a right angle, sinA' BC a A'C 2R R=n A sin A 94 PLANE TRIGONOMETRY. CoR. 1. As above, 2 R = a = = which is sin A sin B sin w another proof of the "law of sines." COR. 2. From R= a we have 2 sin A abe abe -R b- a = A' where A= area ABC. 2 be sin A 4 A 74. To find the radii of the escribed circles. r b FI. 40. FIG. 40. Represent areas ABC, BOA, AOC, BOC, by A,, Af, A, A'f, respectively. Then ra is the altitude of each of the triangles BOA, AOC, BOC. Now A = A' + A/f — A-'/ -- rc + ' rlb -- r a = -2i ra (c + b - a) = r(s - a). A ra -= s-a A A In like manner, rb = -; rc = S - b 8 - C 75. The orthocentre. Denote the perpendiculars on the sides a, b, c, by APa, BPb, CP,, and let it be required to find the distances from their intersection 0 to the sides of the triangle, and also to the vertices. OPb = APb tan CA O. FIG. 41. But APb=ccosA, and CAO=90~-C..0. OPb = c cos A cot C = Ccos A cos C. sin C = 2 R cos A cos C. (Art. 73, Cor. 1) MISCELLANEOUS PROPOSITIONS. 95 In like manner, OP. = 2 B cos B cos A, Opa = 21? cos C cos B. Again, the distances from the orthocentre to the vertices are, OA AP6 ccosA cos CA O sin C -2 R cos A. Also, OB 2 1 cos B, and OC= 2BR cos C. 76. Centroid and medians. The lengths of the medians may be computed as follows: In the figure the medians to the B sides a, 6, c, arc AMa, BAM,, CM,, C a meeting in the centroid 0. Now, by the law of cosines, from A b A b 2Mb the triangle BJi C, FIG. 42. BM2 =a2+HMC2- 2a. 1Mb4C cos C = ~-2 + h- ahcos C. 4 a2 ~ 62 -~f But, cos C-2 2 2 ab BJJ~T2-a2 +2 a2 + h2 - C2 2 a2 + 2 C2 b2 B b 4 2 4 whence, BMb= I V2 a2+ 2 C2 - b2 = 1 Va2 + C2 + 2 ac cos B since a2 + -6s2 cos B. 2 ae In like manner, CM, = 1 V2S + b2+2a2,c2 = I -\62 + a2+ 2 ba cos C, and A a = 1 V2 C2 + 2 b2 -a2 = lV c2+62+ 2 cb cos.A. 96 PLANE TRIGONOMETRY. EXAMPLES. 1. In the triangle, a = 25, b = 35, c = 45, find R, r, r,. 2. Given a = 0.354, b = 0.548, C = 28~ 34' 20", find the distances to C and B, from the circumcentre, the incentre, the centroid, and the orthocentre. 3. In the ambiguous triangle show that the circumcircles of the two triangles, when there are two solutions, are equal. 4. Prove that 1 + 1 + 1= ra rb rc r 5. In any triangle prove A = -r rrarbrc. 6. Prove that the product of the distances of the incentre from the vertices of the triangle is 4 r2R. 7. Prove that the area of all triangles of given perimeter that can be circumscribed about a given circle is constant. 8. Prove that the area of the triangle ABC is Rr(sin A + sin B + sin C). CHAPTER IX. SERIES DE MOIVRE'S THEOREM - HYPERBOLIC FUNCTIONS. 77. First consider some series by means of which logarithms of numbers and the natural functions of angles may be computed. For this purpose the following series is important: 1 I 1 e=l+ I+ + 1 + +~ It may be derived as follows: By the binomial theorem, 1 nx n ++ "' = 1+. 1 + nx(nx- 1) I + nx(nx-1) (nx-2) 1 =l+ + + +' "' 1\ + l+M +1\ 2+ WnI n n =1+x+ +-~ +++ This is called the exponential series, and is represented by ex, so that X2 X3 Xr e"= l+x+-+ -+.+.+.-.. It is shown in higher algebra that this equation holds for all values of x; whence, if x = 1, 11 1 e = I + I + l ++ + +.97 97 98 PLANE TRIGONOMETRY. This value of e is taken as the base of the natural or Naperian system of logarithms. This value e, however, is not the base of the system of logarithms computed by Napier, but its reciprocal instead. The natural logarithm is used in tl. theoretical treatment of logarithms, and, as will presently appear, it is customary to compute the common logarithm by first finding the natural, and then multiplying it by a constant multiplier called the modulus, Art. 82; i.e. in the Naperian system the modulus is taken as 1, and thei base is comnputed. In the common system the base 10 is chosen and the modulus computed. 78. From the exponential series the value of e may be computed to any required degree of accuracy. e=l + + 1 + 1 + 1+I 1I 1 1+1 + = 2.5 = 0.1666666666 0.0416666666 14 - 0.0083333333 15 1 0.0013888888 1 1= 0.0001984126 = 0.0000248015 = 0.0000027557 - 0.0000002755 Adding, e = 2.7182818, correct to 7 decimal places. SERIES. 99 79. To expand ax in ascending powers of x. Z2 Z3 zr e =l+z + +z + ++'". Let a' = eZ, then z = loge ax = x ~ loge a. (Arts. 35, 40) Substituting ax= 1 + x. loge a +x2 (loge a)2 + (oge a)3 + Now put 1 + a for a, and 1 + a)x = 1 + x log (1 + a) x2 [loge ( + a)]2 + X3 [1oge ( + a)]3 + + + +3. But by the binomial theorem, + a)= + xa +x(x- 1) a2 +x(x- 1)(x-2) +.. (1 +a^y = 1 + xa + a 2+ Equating coefficients of x in the second members of the above equations, a2 a3 a4 log (1 + a) = a - - + +**;,or writing x for a, x2 X3 X4 loe (1 + x) = - + - +-. In this form the series is of little practical use, since it converges very slowly, and only when x is between + 1 and - 1 (higher algebra). Put - x for x, and x2 x3 X4 loge (1 - x) = - x- - -- - -.'.log, log, (1 + )- ge (- +X) = loge 1 -(x + ) l-x 3 T~~`j 100 PLANE TRIGONOMETRY. Finally, put 1 for x, and n + 1+ log + = log, (n + 1) - log, n n t 2 n +1 32 nz + 1 5 1 1 1 3) 1og,(n+t)=1og~n+2~' +!- ( ___ (2n~1 3 2n~1}5' 2 +1/j a series which is rapidly convergent. 80. From this series a table of logarithms to the base e may be computed. To find log, 2 put n = 1. Then, since log, 1 = 0, the series, becomes lo,2 og +2 + + + + io&?=og~,~l ri 1 1 1 1 g g3 33~5 35~7 37~9.39 1 1 + 11.3" + + =0.693147. The computations may be arranged thus: 3 2.00000000 9.~G66666667.666~66667 9.07407407~ 3=.02469136 9.00823045~ 5=.00164609 9.000914~49 7 =.00013064 9.00010161 9=.00001129 9.00001129 ~ 11 =.00000103.00000125 1 3 =.00000009.69314717 whence loge 2 = 0.693147, correct to 6 decimal places. To find loge 3, put n = 2, and log, 3=log, e2+ 2(+ 353+5-55+7l57+59+.I. 3 ~ 53 5 i 9 59 SERIES. 101 5 2.00000000 25.40000000 =.40000000 25.01600000 3=.00533333 25.00640000 5 =.00012800 25.00025600 7=.00000366.00000102 9 =.00000011.40546510 loge,2=.69314717.'. loge 3 =1.098612, correct to 6 decimal places. loge 4 = 2 x loge 2, log, 6 = log, 3 + loge 2, etc. (Why?) The logarithms of prime numbers may be computed as above by giving proper values to n. 81. Having computed the logarithms of numbers to base e, the logarithms to any other base may be computed by means of the following relation: Let loga n = x; then ax = n. Also, log, n = y; then by = n,.-. ax = by. Hence, loga (a) = loga (b^), and.. x = y loga b. It follows that log n = logb n ~ loga b; logb = loga n. 1 log, b whence This factor 1 is called the modulus of the system of log, b logarithms to base b. Using it as a multiplier, logarithms of numbers to base b are computed at once from the logarithms of the same numbers to any other base a. 102 PLANE TRIGONOMETRY. 82. To compute the common logarithms. Common logarithms are computed from the Naperian by use of the modulus log 1; z.e. loge 10 By Art. 80, loge 10 can be found, and - =.434294, the modulus of the common system. log, 10 Ex. Compute the common logarithms of: 2, 3, 4, 6, 5, 10, 15, 216, 3375. COMPLEX NUMBERS. 83. In algebra it is shown that the general expression for complex numbers is a + bi, where a represents all the real terms of the expression, b the coefficients of all the imaginary terms, and i is so defined that i2 = - 1; whence i =-l, i2=, = —, = i4=1, etc. The laws of operation in algebra are found to apply to complex numbers. Moreover, it is further shown that if two complex numbers are equal, the real terms are equal, and the imaginary terms are equal; i.e. if a + bi = c + di, then a=b and c= d. Finally, the complex number may be graphically represented as roib follows: -x' - o a x The real number is measured along OX, a units; the imaginary parallel to OY, b units. The line Vr r is a graphic representation of FIG. 43. a + bi. DE MOIVRE'S THEOREM.lo 103 Since a = rcos 0 and b = rsin9,. a +bi = r(cos0~+i sill 9). The properties of complex numbers are best developed by using this trigonometric form. If r be taken as unity, then cos 6 + i sin 6 represents any complex number. 84. De Moivre's Theorem. To prove that, for any value of n, (cosO0 + i sin O)n cos nO ~ i sin nO. I. Wlhen n is a positive integer. By multiplication, (cos a + isin a) (cos /3 ~ i sin /3) = COS ae COS /3 - sin a sin /3 + i (sin a cos/3, + cos a sin /3) = COS (ae+/3)~+i sin (a +/3). In like manner, (cos a ~ i sinl a) (cos /3 ~ i sin /3) (cos y + i sinl y) = cos (a + /3 + y) ~ i sin (a ~ /3 ~ y); and finally, (cos a +isin a) (cos/3 + isin/3) (cosy + isin y).to n factors = COS(a~/38+'y+..)~+i sin(a~/+y8+7+...). Now let a=/3 = y=..., and the above becomes (cos ac + i sin a)n = cos fla + i sin na. II. When n is a negative integer. Let n=- m; -then (cos ae + i sin a)n = (coS a + i sin a>"'11 (cos a + i sin ae)"m cos ma + i sin ma cos ma - i sin ma (cos ma + i sin ma) (cos ma - i sin ma) cos ma - i sin ma cos2 Ma + sin2 ma =cos ma - i sin ma = Cos (- m) a + i Sinl (-m) a. 104 PLANE TRIGONOMETRY. Substituting n for - m, the equation becomes (cos a + i sin a)n = cos na + i sin na. III. When n is a fraction, positive or negative. Let n = -, p and q being any integers. Now aei ina a a COS = cOS +is1nq.-=cosa+isina (byl). / 1 Then Cos -~ + i sill =(cos a ~ i Sill a)q. Raising each member to the power p, P a. aP P P in1 (Cos a + i sin ae)=yeos ~ms1sin )= COs a+ sin-a. q q Yq COMIP:TATIONS,n~P OF NATURAL FUNCTIONS. 85. The radian measure of an acute angle is greater than its sine and less than its tangent, i.e. sin a < a < tan a. N A Let a be the circular, or rt FIG. 44. 4measure of any acute angle Then, in the figure, area of sector OAP < area of triangle OAT, i.e. 1OA arcAP< OA. AT. 2 2 are AP< AT. Now, since NP < arc AP, NP are AP AT K K OP OP OP are AP But a circular measure of AOP = a; OP -whence Sial a < tan a. idian, 4 OP. NATURAL FUNCTIONS. 105 86. Since sin a < a < tan a, a 1 1<. <. sill a Cos a Hence, however small a may be, lies between 1 and 1-a~ ~~~~~~~~s1 a. When a approaches 0, cos c approaches unity. cos a Therefore, by diminishing a sufficiently, we may make -- differ from unity by an amount less than any assignsin a able quantity. This we express by saying that when a approaches 0, sa approaches unity as a limit, i.e. = 1, sin a sin a approximately. Multiplying by cos a (= 1, nearly), we have a = 1, approximately. Whence, if a approaches 0, tan a tan a = sin a ==, approximately. 87. Sine and cosine series. cos na + i sin no = (cos a + i sin a)", (De Moivre's Theorem). Expanding the second member by the binomial formula, it becomes, osa cos" a i i + n cos"-, * i sin2 a + n(n- l)(n- 2) COsn-3 C. i3 Sin3 GC + n(n- 1)(nl - 2)(n- 3) Cos-4. i4 Sill a + Substituting the values of i2, i3, i4, etc., we have cos na + i sin nna = acos" a - 1cos "-2 a sin2 a 12 n (n - )(n - 2)(n -3) cos_"4 ( sin4 a.. + i(n cosn- a sin a - n (n ) 2) cos-3 a sin3 a + [3 106 PLANE TRIGONOMETRY. Equating the real and imaginary parts in the two members, cos nt = cos -n(n - 1) osn2 a sin2 a + n (n - 1)(n-2)(n -3) eosn-4 a sin4 a. + 4on — 1 - ccsin^-..., 1I4 and sin na = n cos1 n- )(n - n) os -3 a sin3... 13 Ex. 1. Find cos3a; sin3a. In the above put n = 3, and cos 3 a = os3 a - 3 cos a sin2 a = 4 cos3 - 3cos a; also sin3 a = 3 cos2 a sin ( - sin3 a = 3 sin a - 4 sin3 a. 2. Find sin 4ac; cos4ac; sin5a; cos 5a. It will be noticed that in the series for cos nc and sin nc the terms are alternately positive and negative, and that the series continues till there is a zero factor in the numerator. 88. If now in the above series we let na = 0, then W 1) cOS 0 = cosn, - COsn-2 a sin2 c u- - Cos- ( s-in 4a — - -coSn-4 a sin4 c-.. I4 = Cos" cc- (O ac)- COSn2 ( sia + (0-(t) (0-2 t)(O-3 ) CoSn4 csin c)a If now 0 remain constant, and a decrease without limit, then will n become indefinitely great, and sin c and every cc NATURAL FUNCTIONS. 107 power thereof, and cos a and every power of cos a will approach unity as a limit, so that 02 04 06 cos = 1 - + +.... 2L L6 Similarly, sin 0 = 0 - _ + - 0_ +.... By algebra it is shown that these series are convergent for all values of 0. By their use we can compute values of sin 0 and cos 0 to any required degree of accuracy. 03 2 05 Show from the above that tan 0 = 0 + - + - + *. 3 15 Ex. 1. Compute the value of sin 1~, correct to 5 places. 03 05 07 In sinl 0 = + --- +'.., make 0 the radian measure of 1~ = -7 = 0.01745 +. 180 Then, 6 = 0.01745 + 3= 0.0000008..'. sin = 0.01745 +. The terms of the series after the first do not affect the fifth place, so that the value is given by the first term, an illustration of the fact that, if a is small, sin a = a, approximately. Compare the value of tan 1~. 2. Show that sin 10~ = 0.17365; cos 10~ = 0.98481; sin 15~ = 0.25882; cos 60~ = 0.50000. 3. Find the sine and cosine of 18~ 30'; 22~ 15'; 67~ 45'. It is unnecessary to compute the functions beyond 30~, for since sin (30~ + ) + sin (30 - 0) = cos 0 (why?),.S. sin (30~ + 0) = cos 0 - sin (30~ - 0). So, also, cos (30~ + 0) = cos (30~ - ) - sin 0. Giving 0 proper values the functions of any angle from 30~ to 450 are determined at once from the functions of angles less than 30~. Thus, sin 31~ = cos 1~ - sin 29~; cos 31~ = cos 29~ - sin 1~. 4. Find sine and cosine of 40~; of 50~. 108 PLANE TRIGONOMETRY. 89. The following are sometimes useful in applied mathematics: Ex. 1. To find the sum of a series of sines of angles in A. P., such as sin a + sin (a + 3) + sin (a + 2 f) + *.. + sin (a + [n - 1]/3). 2sin asin cos= ) -coss (a + ) 2 2 2 2 sin (a + 8) sin: = cos (a + ) - cos (a + 3), 2 2 2 2 sin (a + 2 3) sing = cos(a + ) - cos (a + 5), 2 2 2 2 sin (a + n - 1]/,) sinf = cos (a + 2n-3)-cos ( +2n- 13). 2 2 \ 2i Adding 2 {sin sin ( + )sin (a2) ++ i ( + 2 + + sin (a + [n- 1] /)} sill = cos(a-) -cos (a +2n-1/) = 2 sin( + -2 /) sinn8..s. s sin ( + f) +sini (a + 2 f)+.sin (a + [n - 1]3) sin (r + n-l /) sin n sin 3 2 Similarly it can be shown that cos a + cos (a + /) + cos (a + 2 f/) + * + cos (a + [n - 1]) cos (a + - 1) sin n2 sill 2 HYPERBOLIC FUNCTIONS. 109 x2 x3 Xr 90. The series ex = 1 + x + ~- + + *. + - + *. is proved in higher algebra to be true for all values of x, real or imaginary. Then if x = i0, j2092 i393 irtr eie= 1 + iO+ + - +... +... 02 04 06 O3 05 07 = 1 — ~+ +* +[i ( + -+ )-.. e = cos + i sin 0 (Art. 87). In like manner, e-ie = cos - i sin 9. Adding, cos =e e-' 2 eio _ e-io subtracting, sin 0 = e-e 2i HYPERBOLIC FUNCTIONS. i _ e-ie e + e -i 91. Since sin = 2i and cos = 2 are 2i ' 2 true for all values of 0, let 0 = iO. en - e0.eiO e- e Then, sin (i)= = e 2 = i sinh 0, 2i 2 and cos (iO) = e + = cosh 0, 2 so that tan (iO) = sin (i{) =i sinh 0, cos (i9) cosh 0 where sinh 0, cosh 0, tanh 0, are called the hyperbolic sine, cosine, and tangent of 0. The hyperbolic cotangent, secant, and cosecant of 0 are obtained from the hyperbolic sine, cosine, and tangent, just as the corresponding circular functions, cotangent, secant, and cosecant, are obtained from tangent, cosine, and sine. The hyperbolic functions have the same geometric relations to the rectangular hyper 110 PLANE TRIGONOMETRY. bola that the circular f unctions have to the circle, hence the namne hyperbolic functions. sinh 0=..e sch 0= 2 2 e cosh0= e.. - sech 0 = 2 eo+ - tanh 0=ct 0= - eO +e0O' CtOeo -e-0 92. From the relations of Art. 91 it appears that to any relation between the circular functions there corresponds a, relation between the hyperbolic functions. Since cos2 (iO) ~ sin2 (iO) = 1, cosh2 0 + i2 sinh 2 0 = 1, or cosh 2 0 - sinh2 0 = 1. This may also be derived. thuis: cosh 2 9 - sinh 2 o -ej(0-6 e20 ~ 2 ~ e-20 - e20 ~ 2 e-2=1 4 Also since sin (ia + i13) = sill (ia) cos (ifi) + cos (ice) sin (ifi), i sinh (cc ~ 3) = i sinh cecoshfl + cosli a- i sinh 3, and sinh (a + /3) = sinh a cosh8 13 cosh a sinh 13. Let the student verify this relation from the exponential values of sinh and cosh. EXAM PLES. Prove 1. cosh (a~/3) = coshoaecosh [3 +sinh asinh /3. 2. cosh (a +/3)- cosh (a-3) = 2sinh asinh/[3. 3. cosh 29=1~+2 sin1i20 2 cosh260 — 1. 4. sinh 2 a=2sinh acoshac. EXAMPLES.,hp 1 + cosh 0 0 _/cosh 0 - 1 5. cosh0 j+ s; sinh h 6. sinh 3 = 3 sinh 0 + 4 sinh3 0. 7. sinh 0 + sinh = 2 sinh 0 +-t cosh 0. 2 2 8. sinh a +sinh (a + 3) + sinh (a + 2,3)+ *..+ sinh (a + [n-1]/3) silh a + n-1 - ) sinh n sinh / 2 9. tanh( )=tanh + tanh 4 9. tal h (O + () = + tanh ~ ta.h -; 1 + tanh 6 tallh ( 10. sinh-' x = cosh-l /1 + x2 = tanh-1 x \/1 + x2 11. cosh (at + /) cosh (a - /,) =cosh2 a + sinh2 3 = cosh2 / + sinh2 a. 12. 2 cosh nae cosh a = cosh (n + 1) a + cosh (n - 1) a. 13. cosh a = (ea + e-a) = 1 ) + 2 +.... |_2 L4 14. sinh a = (ea -e- a)= a + 5 + a+ b 15. tanh-1 a + tanh-1 b = tanh-1 a + b 1 + ab SPHERICAL TRIGONOMETRY. -- o, —o 0 95::9.-c - CHAPTER X. SPHERICAL TRIANGLES. 93. Spherical trigonometry is concerned chiefly with the solution of spherical triangles. Its applications are for the most part in geodesy and astronomy. The following definitions and theorems of geometry are for convenience of reference stated here. A great circle is a plane section of a sphere passing through the centre. Other plane sections are small circles. The shortest distance between two points on a sphere is measured on the arc of a great circle, less than 180~, which joins them. A spherical triangle is any portion of the surface of a sphere bounded by three arcs of great circles. We shall consider only triangles whose sides are arcs not greater than 180~ in length. The polar triangle of any spherical triangle is the triangle whose sides are drawn with the vertices of the first triangle as poles. If ABC is the polar of A'B'C', then A'B'C' is the polar of ABC. In any spherical triangle, The sum of two sides > the third side. The greatest side is opposite the greatest angle, and conversely. Each angle < 180~; the sum of the angles > 180~, and < 5400. Each side < 180~; the sum of the sides < 360~. 112 SPHERICAL TRIANGLES. 113 The sides of a spherical triangle are the supplements of the angles opposite in the polar triangle, and conversely. If two angles are equal the sides opposite are equal, and conversely. The sides of a spherical triangle subtend angles at the centre of the sphere which contain the same number of angle degrees as the arc does of arc degrees; i.e. an angle at the centre and its arc have the same measure numerically. The arc does not measure the angle for they have not the same unit of measuremeent, but we say they have the same numerical measure; i.e. the arc contains the unit arc as many times as the angle contains the unit angle. The angles of a spherical triangle are said to be measured by the plane angle included by tangents to the sides of the angle at their intersection. They have therefore the same numerical measure as the dihedral angle between the planes B of the arcs. In the figure the following have the same numerical meas- o \ ure ~ ~ arc a and angle a; A arc b and angle/3; \ arc c and angle y; FIG. 45. plane angle A'B C'; spherical angle B and dihedral angle A-BO-C; spherical angle C and dihedral angle B-CO-A; spherical angle A and dihedral angle C-A O-B. A'C'B and C'A'B have not the same measure as spherical angles C and A, for BA', A'C', C'B are not perpendicular to OA or OC. 94. In plane trigonometry the trigonometric functions were treated as functions of the angles. But since an angle and its subtending arc vary together and have the same 114 SPHERICAL TRIGONOMETRY. numerical measure, it is clear that the trigonometric ratios are functions of the arcs, and may be so considered. All the relations between the functions are the same whether we consider them with reference to the angle or the arc, so that all the identities of \ plane trigonometry are true for the func/ Y a tions of the arcs. ax- L Thus in the figure we may write, FIG. 46. sin a = - or sin a = -; r r sin2 a + cos2 a = 1, or sin2 a + os2 a = 1; cos 2 a = 2 cos2a- 1, or cos 2 a = 2 cos2a-1. GENERAL FORMULAE FOR SPHERICAL TRIANGLES. 95. The solutions of spherical triangles may be effected by formulae now to be developed: First it will be shown that in any spherical triangle cos a = cos b cos c + sin b sin c cos A, cos b = cos c cos a + sin c sin a cos B, cos c = cos a cos b + sin a sin b cos C. The following cases must be considered: I. Both b and c < 90~. III. Both b and c > 90~. II. b >90~, c< 90~. IV. Either b or c= 90~. V. b = = 900. The figure applies to Case I. Let ABC be a spherical tri- A angle, a, b, c its sides, and 0 the centre of the sphere. Draw AC' and AB' tangent ~ - --- \ to the sides b, c at A. (The C same result would be obtained by drawing AB', AC' perpen- dicular to OA at any point to FIG. 47. GENERAL FORMUL2E. 115 meet OB, 0 C.) Since these tangents lie in the planes of the circles to which they are drawn, they will meet OC and OB in C' and B', and the angle C'AB' will be the measure of the angle A of the spherical triangle ABC. Since OAB', OAC' are right angles, AOB', AOC' must be acute, and hence sides c, b are each < 90~. In the triangles C'AB' and C'OB', C'B'2 = AC'2 + AB'2- 2 AC'. AB' cos C'AB', and B'C'2 = OC12 + OB'2-2 2OC' OB' cos C'OB'. Subtracting and noting that cos C'AB' = cos A and cos C'OB' = cos a, we have 0 = OC'2 - A C2 + OB2 - AB'2 + 2 AC' AB' cos A- 2 OC'. OB' cos a. But C'2 - A C'2 = OA2 and OB'2- AB'2 = OA2. Hence, 0 = OA2 + A C'. AB' cos A - C'. OB' cos a; OA OA A AC' AB' or cos a + = ~ - ~ cos A. OC' OB' OC' OB'.'. cos a = cos b cos c + sin b sin e cos A. Similarly, cos b = cos a cos c + sin a sin c cos B, and cos c = cos a cos b + sin a sin b cos C. These formulae are important, and should be carefully memorized. c b II. b>90~; c< 900. A -:a.A In the triangle ABC, let b > 90~ --- and c< 90~. Complete the lune FIG. 48. BA CA'. Then in the triangle A'CB the sides a and A'C are both less than 90~, and by (I) cos A'B = cos A'C cos a + sin A'C sin a cos A'CB. 1166 SPHERICAL TRIGONOMETRY. But A'B = 1800-c, A'C=1800-6, and A'CB= l80'- C. cos (1800 - c)= cos (1800 - b) cos a + sin (1800 - 6) sin a cos (1800 - C); or - cos c =(- cos b) cos a ~ sin b sin a(- cos C) and cos c = cos a cos b + sin a sin b cos C. A similar proof will apply in case c > 900, 6 < 900. 9 6 III. Both b and c > 900. a A In the triangle ABC, let both b anid c >900. Complete the lune ABA'U. Then since A'C FIG. 49. and A'B are both < 900, cos a = cos A'C cos A'B + sin A'C sin A'B cos A'. But A' A, A'C= 1800 - 6, A'B = 180'- c. cos a - cos (1800 - 6) cos (1800 - c) ~ sin (180' - 6) sin (1800 - c) cos A; or cos a = cos b cos eC sin b sin c cos A. Cases IV and V are left to the student as exercises. 96. Since the angles of the polar triangle are the supplements of the sides opposite in the first triangle, we have A' a'=1800-A, 6'=1800 -B, e A c'=1800 - C, A'= 1800 - a. e b Substituting in C C/a cosat = COS bf Cose' e a' + sill 6' sin c' cos A', FIG. 50. we have - cos (1800 - A) = cos (1800 - B) cos (1800 - C) + sin (1800 - B) sin (1800 - C) cos (1800 - a); or - cos A = (- cos B)(- cos C) + sin B sin C( - cos a). GENERAL FORMULIE. 117 Changing signs, cos A = - cos B cos C + sin B sin C cos a. Similarly, cos B = - cos A cos C + sin A sin C cos b, and cos C =- cos A cos B + sin A sin B cos c. sinA sinB sill C 97. In any spherical triangle to prove si A = - si sina si b sin Since cos A = cos a - cos b cos c Since cos A = sin b sin c.S. sin2 A =1 (cos a - cos b cos C)2 sil b sin c sin2 b sin2 c - (cos a - cos b cos )2 sin2 b sin2 c (1 -cos2 b) (- cos2 ) - (cos a - os b cos c)2 sin2 b sina2 C 1- cos2 a - cos2 b - cos2 + 2 cos a cos b cos c sin2 b sin2 c Hence, n A -/1 - os2 a- cs2 b - cos2 c-2 cos a cos b cos c sin b sin c sin A /1 - cos2 a - cos2 b- cos2 c- 2 cos a cos b cos c and sin a sin a sin b sin c sin B sin C By a similar process, n B and si will be found equal sin b sin c to the same expression. sinA sin B sinC sin a sin b sin c 118 SPHERICAL TRIGONOMETRY. 98. Expressions for sine, cosine, and tangent of half an angle in terms of functions of the sides. We have 2 sin24= 1 - cos A 2 1 cos a - cos b cos c sin b sin c cos 6 cos c ~ sin 6 sin c - cos a sin b sin c - cos (6 - c) - cos a sin 6 sin c Then 2 sin2A 2 sin2(a~-c)sin:(a-6 ~c) (Art.51) 2 sin b sin c 2 sin (s - b) sin (s - e) sin b sin c when 2s=a6 + b + c. si__A sin (s - c) sin (s - c). 2 sint b sin c Similarly, sin sin(s - c) sin (s - a) 2 sin a sin c and sin C =sin (s - b) sin(s-a) sif 2 7\ sin a sin b Also from the relation A 2 cos2 A 1 + cos A 2 cos a - cos b cos c si= b sin c we have COS 4sin s sin(s - a) 2 sin b sin c Also, Cos jB sins sin(s - b) 2 sin c sin a and C sin s sin (s - c) co 2 = sin a sin b GENERAL FORMULX1 119 From the above, A tan A2 2 /lsin(s -b) sin(s -c) tan — CA sin s sin(s - a) 2 Also, B /sin (s - a) sin (S - C) n2 = sin s sin (s - b) and tan C = N Isin(s - a) sin (s - b) t sin s sin(s - c) Compare the formulae thus far derived with the corresponding formule for solving plane triangles. The similarity in forms will assist in memorizing the formube for solving spherical triangles. 99. From the formulhe of Art. 96, the student can easily prove the following relations: si a '- cos S cos(S - A) 1-1I., sin B sin C where 2S= A + B+ C. sin b =2 sin C =s n i 2 Cosa ICOs (S - B3) cos (S - C) c 2 Ntl sin B sint C 2 2 a =-NI - cos S Cos (S- A) tan n s 9 2 n os (S- B) os(S - C) tan b= - 2 tan = '=. 2 120 SPHERICAL TRIGONOMETRY. 100. Napier's Analogies. tan - A /sin (s- b)sin (s - c) Since ta2 sin s sin (s - a) Since --- = tanlB sin (s- c) sin (s- a) 2 sin s sin (s - 6) - sin12(s -) sin (s - ) sin2(s - a) sin (s - a) by composition and division, A B tan A + tan - 2 2 sin (s - b)+sin(s - a) ta A _ tan B sin (s - b)- sin (s - a)' tan - - tan - 2 2.A. B sin A sin B 2 2 A B COS -- Cos -- 2C 2 sin -I ( S - (2 - b) cos 1 (a -b) A _.B os 1 (2 - a-b) sinl (a - b) ln 2 (Art. 51) A B cos -- co 2 2 sin 2 (A + B) _ tan (2 s - a -) sin 2 (A - B) tan I (a-b) tan - 2 t= -n- -- since 2s-a-b=c. tan ~- (a - b) 2 l. b)8sin|(A-B)tc sin (A+B) To find an expression for tan -(A - B) we have only to consider the polar triangle, and by substituting 180~ - A for.a, etc., 180~ - a for A, etc., we have the following relations: (a - b)= (1800 - A- 1800 + B)=- A - B); 2 \" "/e2 2'~2 ' GENERAL FORMULzE. 121 also, -1 (A -B) =- - -( ); (A + B)= (1800 - a + 1800 - b) = 180~- 1(a + b); e 2 and =90 C. 2 2 The formula then becomes, applying Art. 29, sin 1 (a - b) tan - B)= 2 cotC sin (a+ b) Formulae for tan -- (a + b), tan I (A + B) are derived as follows: Since A B 4jsi (s-b) sin (s-c) - sin (s-c) sin (s-a) tan - tan -- 2 2 a i S i(-a sin s s - a) sn sin(s- b). A. B sin- si2 2 sin(s - e). A B sin s COS- COS - 2 2 By composition and division, A B.A.B cos-cos- + sin-sin2 2 + 2 2 sin s + sin (s - c) A B A sB sin s-sin (s-c)' cos -cos - -sin -sin - 2 2 2 2 cos 1 (A - B) tan I (a + b). 51 whence -2 -- 2 7 - - (Art. 51) cos (A + B) ta 2 \'" i tan 2 since 2 s-c = a + b, cos (A- B) or, tan(a+b)= s2 ( tan C. cos1 (A + B) 2 122 SPHERICAL TRIGONOMETRY. The value of tan I (A + B) is derived by substituting in terms of the corresponding elements of the polar triangle. cos (a-b) _-tan (A + B) -cos (a + b) ot 2 i COS e (a - b) 7. tan (A + B)= cot cos (a + b) Similar relations among the other elements of the triangle may be derived, or they may be written from the above by proper changes of A, B, C, a, b, c in the formulae. The student should write them out as exercises. 101. Delambre's Analogies. Since sin 1 (A + B) = sin - cos B + cos A sin B 2 ~"2 2 2 2 then sin (A + B) = sin (s - b) + sin (s - a) /sin s sin (s - c) ~~2 ~' ~-sin c sin a * sin b (Art. 98) H sill-A + B) sin(s - b) + sin (s - a) H ~2ence, 2 cc COs- sin c 2 2 sill cos - (a - ) 2 __=, (Art. 51) 2 sin - cos - 2 2 os 1 (a -) C and sin (A+B) = -- - os- C i2 22 COS In like manner derive sin (a-b) C sin (A- B)= cos; sin- 2 2 RIGHT SPHERICAL TRIANGLES. 123 cos 1 (a + b) cos (A + B)= 2 in -; 2 C cos 2 sin 1 (a + b) 2 cos (A - B)= 2 sin. These formulae are often called Gauss's Formulae, but they were first discovered by Delanbre in 1807. Afterwards Gauss, independently, discovered them, and published them in his Theoria Motus. 102. Formulae for solving right spherical triangles are derived from the foregoing by putting C= 90~, whence sin C= 1, cos C = 0. cos c = cos a cos b + sin a sin b cos C (Art. 95) becomes cos c = cos a cos b. (1) Substituting the value of cos a from (1), and simplifying, cosA cos a - cos b cosc (Art. 95) cos A = -^(Art. 95) sin b sin c becomes cos A =tan b (2) tan c ( Again, sinA si (Art. 97) sin a sin c in the right triangle is sin A =sin a. (3) sin a Dividing (3) by (2), tan A sin a cos b sin a cos a cos b sin a tan A =. =. = —. cos c sin b cos c cos a sin b cos a sin b since cos a cos b = cos c..'. tan A sin b = tan a. (4) 124 SPHERICAL TRIGONOMETRY. From (4) tan a = tan A sin b, also, tan b = tan B sin a. Multiplying, tan a tan b = tan A tan B sin a sin b, or, cot A cot B = cos a cos b = cos c. (5) From (2) and (3), by division, tan b cos A tan c cos c - --- = --- = cos a. sin B sin b cos b sin c.'. cos A = cos a sin B. (6) Let the student write formulae (2), (3), (4), (6) for B. It will be noticed that (1) and (5) give values for c only, while (2), (3), (4), (6) apply only to A and B. 103. Formulae (1)-(6) are sufficient for the solution of right spherical triangles if any two parts besides the right angle are given. They are easily remembered by comparison with corresponding formulae in plane trigonometry. Two rules, invented by Napier, and called Napier's Rules of Circular Parts, include all the formulae of Art. 102. Omitting C, and taking the comple90-B ments of A, c, and B, the parts of the triangle taken in order are a, b, 90~- A, /0 \/ a 90~ - c, 90~ - B. These are called the circular parts of the triangle. 0~-A b Any one of the five parts may be FIG. 51. selected as the middle part, the two parts next to it are called the adjacent parts, and the remaining two the opposite parts. Thus, if a be taken as the middle part, 90~- B and b are the adjacent parts, and 90~ - c, 90 - A the opposite parts. NAPIER'S RULES. 125 Napier's Two Rules are as follows: The sine of the middle part equals the product of the tangents of the adjacent parts. The sine of the middle part equals the product of the cosines of the opposite parts. It will aid the memory somewhat to notice that i occurs in sine and middle, a in tangent and adjacent, and o in cosine and opposite, these words being associated in the rules. The value of the above rules is frequently questioned, most computers preferring to associate the formulae with the corresponding formulae of plane trigonometry. These rules may be proved by taking each of the parts as the middle part, and showing that the formulae derived from the rules reduce to one of the six formulae of Art. 102. Then, if b is the middle part, by the rules, sin b = tan a tan (90~ - A) = tan a cot A, or tan A = tan sin b sill b = cos (90~ - c) cos (90~ - B) = sin c sin B, or sin B = sin sill C results which agree with (4) and (3), Art. 102. If any other part be taken as the middle part, the rules will be found to hold. 104. Area of the spherical triangle. If r = radius of the sphere, E = spherical excess of the triangle =A + B + - 1800, A = area of spherical triangle, then by geometry A=Er2x I. If the three angles are not known, E may be computed by one of the following methods, and A found as above. 126 SPHERICAL TRIGONOMETRY. Cagnoli's Method. sn sin = sin (A + C - 180~) 2 = sin 2 (A + B)sin - cos 1 (A + B)cos 2 2 2 sinC COS2 2 cos= [COS 2 - b) - coS 2 (a + b)] (Art. 101) a.b 2 2 sin - sin2 2 /sin s sin (s - a) sin (s -b) sin (s - c) c sin a sin b 2 (Arts. 51, 98) sE = Vsin s sin (s - a) sin (s - b) sin (s - c) sin 2 2 2 cos 2a cos b cos 2 2 2 Lhuilier's Method. taE sin l(A + B + C- 180~) 4 cos(A + B + C-1800) Now, multiply each term of the fraction by 2 cos (A + B - + 180~), and by Art. 51, (1) and (3), the equation becomes sin -(A + B)-costanE 2 cos j(A + B)+ sincos 2 (a- b)- cos cos [cosi (a + b - )+ (Art. s1) cos I2 (a + ) + cos 2 sn sin 2 (s-b) sin (s - a) sin s sin(s - c) CO s - C ) sin (s- a) sill( - b) cos 2 cos (Art. 51) AREA OF SPHERICAL TRIANGLES. 127 By Art. 52, introducing the coefficient under the radical, tan E a)tan (s-b)tan!(s-c). 4 2 2 ' If two sides and the included angle are given, BE may be determined as follows: cos-= CO-'s -IA ~ B~+ C - 1800) C C, CGos - (A + B)sin- ~ sin I (A + B) cos- 2e 2 2 -/c' 1 (+ C+CO COS +6) sC + cos (a - b) cosn2 (Art. 101) 222 2 a 6.a 6 cos cos -+ sin sin - cos C 2 2 2 2 CO Cos - 2 Si a, B n sin ~ 2 sin.cos. 2 2 2 2 But sin-= (Cagnoli's Method) 2 C 2 Dividing this equation by the above, a 6 sin - sin - siz C B 2 2 tan, 2 a 6 a.6 CoS-Cos + sin - sin - COS C 2 2 2 2 This formula is not suitable for logarithmic computations. Usually it is better to compute the angles by Napier's Analogies, and solve by A = Er2 x V 180 128 SPHERICAL TRIGONOMETRY. EXAM PLES. 1. Show that cos a = cos b cos c + sin b sin c cos A becomes sec A = 1 + sec a, when a = b = c. 2. If a+b+c=7r, prove B C (a) cos a = tan-tan-. 2 2 (h) cos24 cosa 2 sin b sin c (c) sin 24 cot b cot c. 2 (ci) cos A + cos B + cos C= 1..A 2' *2 (e) sin - + sin - + sin2 1. 2 2 2 sinEcos ( - E) sin sin(s - a) 2 2 3. Prove A a (Art. 104) sin - cos - 2 2 4. Show that cos a sin b = sin a cos b cos C -+ sin c cos A. CHAPTER XI. SOLUTION OF SPHERICAL TRIANGLES. 105. According to the principles of spherical geometry any three parts are sufficient to determine a spherical triangle; the other parts are computed, if any three are given, by the formulae of trigonometry. The known parts may be: I. Three sides, or three angles. II. Two sides and the included angle, or two angles and the included side. III. Two sides and an angle opposite one, or two angles and a side opposite one. It will appear that, as in plane geometry, III may be ambiguous. The signs of the functions in the formulae are important since the cosines and tangents of arcs and angles greater than 90~ are negative; whether the part sought is greater or less than 90~ is therefore determined by the sign of the function in terms of which it is found unless this function be sine. In this case the result is ambiguous, since sin a and sin (180~ - a) have the same sign and value. Thus if the solution gives log sin a = 9.56504, we may have either a = 21~ 33', or 158~ 27'. The conditions of the problem must determine which values apply to the triangle in question. The negative signs, when they occur, will be indicated thus: log cos 115~ 20' = 9.63135-, indicating, not that the logarithm is negative, but that in the final result account must be made of the fact that cos 115~ 20' is negative. 129 130 SPHERICAL TRIGONOMETRY. 106. FormulcE for the solution of triangles. j. sinA sinB sinC sin a sin b sin c II. ~~tanA A. 2 a)sinssin(s-a) III. tan a Cos -esCos (S -A) 2 cos (S - B) cos (S - C), sin I (A - B) IV. tana - b)= 2 tanc 2 sin 1I(A + B) 2" 2 Cos (A - B) V. tan 1 (a + b) 2 tancos1 (A + B) 2 2 sin (aC - b) VI. tan,(A - B)= cot 2 2~~~~ L2 ~~sin 2(a + - b) cos1 (a - b) C 2 coVII. tan(A + B)= cos (b 2ot 2~~~~o 1. (a +t b) 2 Vill. A =Er'? 180' where B is determined by tan -Vtan s tan! (s-a) tan (s-b)tan ) 4 2 2 2 2 Right triangles may be solved as special cases of oblique triangles, or by the following: (1) cos c = cos a cos b. (4) tanA sinb=tana. (2) cosA tanbc tanc (5) cotAcotBcosc. (3) sinA =sin _a (6. cosA =cos asinB. sin c The fornula to be used in any case may be determined by applying Napier's Rule of Circular Parts. 107. In solving a triangle the student should select formulas MODEL SOLUTIONS. 1_31 in which all parts save one are known, and solve for that one' (see page 77). Referring to Arts. 105 and 106, it will appear that solutions are effected as follows: Case I by formulke II, or III, check by I. Case II by formulae VI, VII, I, or IV, V, I, check by IV or VI. Case III by formulae I, IV, or I, VI, check by VI or IV. MODEL SOLUTIONS. 108. 1. Given a - 460 24' b = 670 14', c = 810 12'. Solve. tan A \in-tan 6SB i )sin (s - a) siXf (s - c) 2 sins sin (s - a) 2 sills sin (s - b) tan = lsin (s - a) sill (s - ) Check:si a sin 2 sil l sill (s - c) sin A sill B Arrange and solve as in Example 1, page 80. Ans. A = 46' 13'.5,B=, 0= Solve: (1),4 960451 B = 108- 30', C = 1160 15'. (Use formulae III in the same manner as in Example 1.) (2) a = 1080 14', b = 750 29', c = 56' 37'. (3) A = 570 50', B = 98' 20', C = 630 40'. 2. Given b = 1130 3', c = 820 39', A = 1380 50'. Solve. co b- A si A tan (B + C) cos=(2 -co) A tan(B - C) sin2 b-c) cot A cos (b + c) 2 sin I (b~+ c) 2' in=sill A sinl (B + C)~+I (B - C)= B, orC, sin sin b 2 ~~~~~~~~~~~sin Bj Check: t11 = tan (b - c) sinl -1(B + C) 2 sin 1 (B - C) b=1130 31 log cos 1 (b - c) = 9.98453 log sin (b - c) = 9.41861 c - 890 391 colog cos 1 (b + c) = 0.86461f colog sin - (b + c) = 0.00409 (b +c) = 970 511 A A log cot = 9.57466 log cot A 9.5746 4(b-c)= 150121 2 2 ~ A - 690 25' log tan I (B + C) = 0.42380 log tan 1 (B - C) = 8.99736 ~ (B-I C) =1100 391 1 (B - C) = 50 40'.6 (B - C) = 50 40'.6 B = 1160 19'.6 and 0=1040 58'.4 132 SPHERICAL TRIGONOMETRY. Check: log sin A = 9.81839 log tan I (b - c) = 9.43408 log sin b = 9.96387 log sin a (B + C) = 9.97116 cologs in B = 0.04756 colog sin (B - C) = 1.00474 log sin a = 9.82982 log tan a = 0.40998 a = 137~ 29' 2 a = 137~ 29' Notice that tan a (B + C) is -. Hence, A (B + C)is greater than 90~, i.e. 110~ 39'. Solve: (1) A = 68~40', B= 56~20', c= 84~30'. (Use formulae IV, V, I. Compare Example 2.) (2) a =102~ 22', b = 78~ 17', C = 125~ 28'. (3) A =130~ 5', B = 3226', c = 51~ 6'. 109. Ambiguous cases. By the principles of geometry the spherical triangle is not necessarily determined by two sides and an angle opposite, nor by two angles and a side opposite. The triangle may be ambiguous. By geometrical principles it is shown that the marks of the ambiguous spherical triangle are: 1. The parts given are two angles and the side opposite one, or two sides and the angle opposite one. 2. The side, or angle, opposite differs from 90~ more than the other given side, or angle. 3. Both sides, or angles, given are either greater than B 90~, or less than 90~. In the right triangle ABC2, i/sin a = sin a = sin A sin c. (formula (3)) /alp \ Therefore there will be no solution, one A X solution, or two solutions, according as sin a = sin A sin, i.e. according as a > FIG. 52. > > the perpendicular p. (See Art. 65.) But the most expeditious means of determining the ambiguity is found in the solution of the triangle. The use of formula I gives the solution in terms of sine, so that it is to be expected that two values of the part sought may be possible; and whether the triangle be ambiguous or not, there must be some means of determining which of the two AMBIGUOUS SPHERICAL TRIANGLES. 133 angles, a and 180~ - a, that have the same sine is to be used. If there are two solutions, both values are used. This is determined in the further solution of the triangle by formula VI, which may be written t b _ cos I (A + C) tan - (a + c) 2 cos - (A- C') Now - <90~, whence tan - is +. Then if for both values 2 2 of C, found by the sine formula, the second member is +, there are two solutions; if the second member is - for either value of C, there is but one solution; while if both values of C make the second member -, there is no solution. The various cases will be illustrated by problems. 3. Given a = 62 15'.4, b = 103~ 18'.8, A = 53~42'.6. Solve. sB sin sin sin A c os (A + B) tan (a + ) si B = tan - sin a 2 cos ~ (A - B) sin sinin A C ct tan i (A - B) sin I (a + b) sin C = -- Check: cot-= sin a 2 sin I (a - b) Solving the first formula gives log sin B = 9.94756, whence B1 = 62~ 24'.4, B2 = 117~ 35'.6. For each of the values B1 and B2, cos (A + B) tan ~ (a + b) cos a (A - B) is + and therefore equal to tan c- Hence there are two solutions. Find c = 153~ 9'.6, or 70~ 25'.4 and C = 155~ 43'.2, or 59~ 6'.2 4. Given a = 46~ 45.5, A = 73~11'.3, B = 61~ 18'.2. Solve. sin= sasin B cot = tan (A B) cos (a +b) sill A 2 cos ~ (a - b) si c sinsin in Check: tan tann (a - b) sin ~ (A + B). sin A 2 sin i ( - B) 134 SPHERICAL TRIGONOMETRY. Solving for b gives log sin b = 9.82446, whence b = 41~521.5, and b2 = 138~ 7'.5. For the value b1 the fraction tan 1 (A - B) cos ~ (a + b) cos I (a - b) is +, but for b2 cos ~ (a + b) is -, making the fraction -, and hence it can not equal cot C, which is +. There is then but one solution. Find C = 60~ 42'.7, c = 41~ 35'.1. 5. Given a = 162~ 30', A = 49 50', B = 57~ 52'. Solve. Solving gives log sin b = 9.52274, whence bi = 19~271.9, be = 160~ 32'.1. For both values, b1 and b2, cos ~ (a + b) is -. Therefore, tan ~ (A -B) cos a (a + b) cos 2 (a - b) is - and not equal to cot -. Hence the triangle is impossible. 2 Solve, testing for the number of solutions: (1) b = 106~ 24'.5, c= 40~20', C = 38~ 45'.6. (2) a = 8050, A = 131~40', B = 65~25'. (3) a = 60~ 31'.4, b = 147~ 32'.1, B = 143 50'. (4) a= 55~30', = 139~ 5', A= 43~25'. RIGHT TRIANGLES. 110. Right triangles are a special case of oblique triangles, but are usually solved by formulae (1) to (6), Art. 106. Students should have no difficulty in applying these. Computers generally question the utility of Napier's Rules of Circular Parts. For those who prefer the rules a problem will be solved by their use. SPECIES. 135 6. Given c = 86~ 51, B = 18 3'.5, C = 90~. The parts sought are a, b, A, and it is immaterial which is computed first. a and A are adjacent to c and B, while b is the middle part of c and B. Then by Napier's first rule sin (90~ - B) = tan (90 ~- c) tan a; cos B or tan a = -- = cos B tan c, \ cot c which is formula (2). By the same rule o0-A -- sin (90 ~- c) = tan (90~ - A) tan (90~ - B), FIG. 53. or cot A = cos c = cos c tan B, formula (5). cot B Finally by the second rule sin b = cos (90~ - c) cos (90~ - B) = sin c sin B, formula (3). The solutions give a = 86~ 41'.2, b = 18~ 1'.8, A = 88~ 58'.4. Verify. 111. Species. Two angles or sides of a spherical triangle are said to be of the same species if they are both less, or both greater, than 90~. They are of opposite species when one is greater and the other less than 90~. Since the sides and angles of a spherical triangle may, any or all, be less or greater than 90~, it is necessary in solutions to determine whether each part is more or less than 90~. The directions already given are sufficient in oblique triangles. In right triangles the sign of the function will determine if the solution gives the result in terms of cosine or tangent, but not if the result is found in terms of sine. Thus in Example 6, above, we have log sin b = 9.49068, whence b = 18~ 1'.8, or 161~ 58'.2. By formula (4) sin b = tan. Now sin b is taln A always +, therefore, tan a and tan A must be of the same sign, whence in any right spherical triangle an oblique angle and its opposite side must be of the same species. Again by formula (1) cos c = cos a cos b. Now cos c is + or - according as c is less or greater than 90~. If then c<90~, cos a and cos b are of the same sign, but if c>90~, cos a and cos b are of opposite sign. Therefore, if the 136 SPHERICAL TRIGONOMETRY. hypotenuse of a right spherical triangle is less than 90~, the other sides, and hence the angles opposite, are of the same species; but if the hypotenuse be greater than 90~, the other sides, and the angles opposite, are of opposite species. 112. Ambiguous right triangles. When the parts given are a side adjacent to the right angle, and the angle opposite this side, the triangle is ambiguous, for solving for the hypotAtc enuse by formula (3) gives C\ a/t b '. sin a sin c = sin A' B sin A FIG. 54. from which there result two values of c. By the last rule of species it follows that to the values of c, one <90~, the other >90~, there will correspond two values for b, one of the same species as a, the other of opposite species. Clearly sin c > 1, according as sin a sill A, and hence there will be no solution, one solution, or two solutions, according as sil a > sin A. Solve the spherical triangles, right angled at C, given: (1) b = 73~ 21'.4, c= 84~ 48'.7. (2) c = 54028', B = 128~ 12'.6. (3) b = 450 42', B = 135~ 42'. (4) a = 108~ 22'.3, b = 120~ 14'.5. (5) a = 70~ 50', A = 170~40'. (6) b = 328'.4, B =46~ 2'.8. (7) b = 34 28', c = 62~ 50'. (8) c = 102~35', B = 17~45'. (9) a = 92~16, c = 570 35'. EXAMPLES 137 EXAM PLES. Solve, given: a 1. 970 351 2. 3. 400'20 4. 6. 7. 1440 101 8. 9. 10. 620 421 11. 1200 301 132. 500 15' 13. 14. 840 141.5 15. 1000 16. 17. 630 50' 18. 19. 500 20. 1590 50' 21. 1240 121.5 22. 23. 760 36' 24. 25. 26. 9802911.7 27. 990 40'.8 b c 270 8'.4 1190 8'.4 670 331.4 940 51 700 401 820 391.5 1100 46'.4 A B C 990 57'.6 400 1160 20' 700 7 900 260 61.3 410 44'.92 490 441.3 1210 10'.4 1300 L320 16' 1390 441 1270 30' 115 50 51.3 I 1100 10' 500 121 700 20'.8 580 8' 700 20'.3 690 35I 750 301 1160 201 500 600 870 121 800 19I 880 121 500 301 340 151 630 15' 1590 43I 900 1040 591.1 1380 501.2 320 261.1 360 45'.4 900 420 15'.2 1210 36'.2 900 1230 40' 540 181 970 121.5 400 20' 280 45'.1 530 521 1150 13'.5 480 311.3 620 551.7 420 15'.2 440 22'.2 1220 25'.1 1250 181.9 440 53f 1090 50'.4 640 23'.2 900 950 38'.1 138 SPHERICAL TRIGONOMETRY. APPLICATIONS TO GEODESY AND ASTRONOMY. 113. Geodesy is concerned in measuring portions of the earth's surface, considering the earth as a sphere. To find the distance on the earth's surface between two points whose latitudes and longitudes are known. If A and B are two places on the jE e' earth, P the north pole, ECDE' the equator, and PEP' the principal meridian, e.g. the meridian of Greenwich, and if the latitude and longitude of A F. 5. and B are known, then AB can be FIG. 55. computed. For AP = 90~ - latitude A, BP = 90~- latitude B, angle APB = longitude A - longitude B..'. two sides and the included angle of the triangle APB are known, and AB can be computed. Ex. 1. Find the distance between Ann Arbor, 42~ 19' N., 83~ 43'.8 W., and San Juan, 18 29' N., 66~ 7' W. 2. Htow far is Manila, 14~36' N., 1200 58' E., from IHonolulu, 21~ 18' N., 157~ 55' W.? Honolulu from San Francisco, 37~47'.9 N., 122~ 24'.5 W.? San Francisco from Manila? 114. The celestial sphere. The heavenly bodies appear to be situated on a sphere of indefinitely great radius with the centre at the point of observation. This is called the celestial sphere. A tangent plane to the. earth at the point of observation cuts the celestial sphere in a great circle called the horizon. The points of the horizon directly south, west, north, east are called the south, west, north, east points. A vertical line through the point of observation cuts the celestial sphere above in the zenith, and below in the nadir, the zenith and nadir being poles of the horizon. APPLICATIONS. 139 The earth's axis produced is the axis of the celestial sphere, cutting it in the north and south poles of the equator. The altitude of a star is its distance from the horizon measured on an arc of a great circle drawn through the star;and the zenith. The azimuth, or bearing, of a star, is the arc of tie horizon measured from some fixed point to the foot of the great circle through the star and the zenith. The fixed point is usually the south point. The declination of a star is its distance from the celestial equator. The circle drawn through the pole and the star is the hour circle, and the angle at the pole between the prime meridian and the hour circle is the hour angle of the star. Let an observer be at 0 on the surface of the earth, and let P be the position of a star. Then Z is the zenith, Z' the nadir, H -:. —H' EQE' the celestial equator, N its north \E' pole, S its south pole, HRH' the horizon, NPS the meridian, or hour circle, of P, Z and ZNP the hour angle. The declination of the star is PQ, its altitude PR, and its azimuth, or bearing, NZP. The astronomical triangle NZP can be solved if any three of its parts are known. EXAMPLES. 1. What will be the altitude of the sun as 9 A.M. in Detroit, lat. 42~ 20' N., its declination being 27~ 30'.5? 2. At what time will the sun rise at San Francisco, lat. 37~47'.9, if its declination is 32~ 46'.2? 3. Find the azimuth and altitude of a star to an observer in lat. 42~ 20' N., when the hour angle of the star is 3 h. 42.3 m. E., and the declination is 42~ 31' N. 4. The latitude of Sayre Observatory is 40~ 36'.4 N.; the sun's altitude is 47~ 15'.3, its azimuth 80~23'.1. Find its declination and hour angle. 5. At Ann Arbor, March 13,1891, the altitude of Regulus is 32~ 10'.3, and the azimuth is 283~ 5'.1. Find the declination and hour angle. FIVE-PLACE LOGARITHMIC AND TRIGONOMETRIC TABLES ADAPTED FROM GAUSS'S TABLES BY ELMER A. LYMAN MICHIGAN STATE NORMAL COLLEGE AND EDWIN C. GODDARD UNIVERSITY OF MICHIGAN ALLYN AND BACON 3ostonu antiu bicago COPYRIGHT, -1899, BY ELMER A. LYMAN AND EDWIN C. GODDARD. Narb300b vre0s J. 8. Cushinog & Co. - Berwick & Smith Norwood Mass. U.S.A. TABLE I. THE COMMON LOGARITHMS OF NUMBERS FROM 1 TO 10009. a N.TL. O 1 2 3 4 5 6 7 8 9 P. P. I 100 IOI 102 103 104 105 io6 107 io8 0og 00000 043 087 130 I73 432 475 5i8 561 604 86o 903 94S 988 *030 01 284 326 368 410 452 703 745 787 828 870 02 II9 i6o 202 243 284 53I 572 612 653 694 938 979 *oig o6o *0Ioo 03 342 383 423 463 503 743 782 822 862 902 217 260 303 346 389 647 689 732 775 817 *072 *II *I57 *199 *242 494 536 578 620 662 9I2 953 995 *036 *078 325 366 407 449 490 735 776 8i6 857 898 I4I *i8i *222 *262 *302 543 583 623& 663 703 941 981 *02I *060 *ioo 110 04 I39 179 218 258 297 336 376 415 454 493 III 532 571 6io 65o 689 727 766 8o5 844. 883 112 922 961 999 *038 *077 *II *I54 *192 *23I *269 II3 05 308 346 385 423 461 500 538 576 614 652 114 690 729 767 805 843 88i 918 956 994 *032 II5 06070 io8 145 183 221 258 296 333 37I 408 ii6 446 483 521 558 595 633 670 707 744 781 117 819 856 893 930 967 *004 *04I *078 *II5 *I5I ii8 07 i88 225 262 298 335 372 408 445 482 518 iig 555 591 628 664 700 737 773 809 846 882 120 918 954 990 *027 *063 *099 *I35 *I7I *207 *243 I2I o8 279 314 350 386 422 458 493 529 565 6oo 122 636 672 707 743 778 814 849 884 920 955 123 99I *026 *06i *096 *132 *i67 *202 *237 *272 *307 124 09 342 377 4I2 447 482 5I7 552 587 621 656 125 691 726 760 795 830 864 899 934 968 *003 126 I0037 072 io6 140 175 209 243 278 312 346 127 380 415 449 483 517 55I 585 619 653 687 128 721 755 789 823 857 890 924 958 992 *025 129 II 059 093 126 i6o 193 227 261 294 327 361 130 394 428 46I 494 528 561 594 628 66i 694 131 727 760 793 826 86o 893 926 959 992 *024 132 12057 090 123 I56 I89 222 254 287 320 352 133 385 418 450 483 Pi6 548 58i 613 646 678 134 710 743 775 8o8 840 872 905 937> 969 *oo1 135 13 033 o66 098 I30 162 I94 226 258 290 322 136 354 386 418 450 481 513 545 577 6o9 640 137 672 704 735 767 799 830 862 893 925 956 138 988 *oI9 *05I *082 *II4 *145 *I76 *208 *239 *270 I39 14 301 333 364 395 426 457 489 520 55I 582 140 613 644 675 706 737 768 799 829 86o 891 141 922 953 983 *OI4 *045 *076 *io6 *I37 *i68 *198 142 15 229 259 290 320 351 381 412 442 473 503 143 534 564 594 625 655 685 715 746 776 8o6 144 836 866 897 927 957 987 *oI7 *047 *077 *107 145 i6 I37 167 I97 227 256 286 316 346 376 406 146 435 465 495 524 554 584 613 643 673 702 147 732 761 79i 820 85o 879 909 938 967 997 148 17 026 o56 085 II4 143 I73 202 231 260 289 I49 319 348 377 406 435 464 493 522 551 580 44 43 42 I 4,4 4,3 4,2 2 8,8 8,6 8,4 3 13,2 12,9 12,6 4 17,6 17,2 i6,8 5 22,0 21,5 21,0 6 26,4 25,8 25,2 7 30,8 30,I 29,4 8 35,2 34,4 33,6. 9 39,6 38,7 37,8 41 40 39 I 4,I 4,0 3,9 2 8,2 8,o 7,8 3 12,3 12,0 11,7 4 16,4 i6,o I5,6 5 20,5 20,0 19,5 6 24,6 24,0 23,4 7 28,7 28,0 27,3 8 32,8 32,0 3I,2 9 36,9 36,0 35,1 38 37 36 I 3,8 3,7 3,6 2 7,6 7,4 7,2 3 11,4 II, io,8 4 15,2 14,8 I4,4 5 19,0 I8,5 i8,o 6 22,8 22,2 21,6 7 26,6 25,9 25,2 8 30,4 29,6 28,8 9 34,2 33,3 32,4 35 I 3,5 2 7,0 3 10,5 4 14,0 5 1I7,5 6 21,0 7 24,5 8 28,0 9 31,5 32 I 3,2 2 64 3 9,6 4 12,8 5 16,o 6 19,2 7 22,4 8 25,6 9 28,8 34 33 3,4 3,3 6,8 6,6 10,2 9,9 13,6 13,2 17,0 i6,5 20,4 i9,8 23,8 23,I 27,2 26,4 30,6 29,7 31 30 3,I 3,0 6,2 6,o 9,3 9,0 12,4 12,0 15,5 15,0 i8,6 i8,o 21,7 21,0 24,8 24,0 27,9 27,0 150 609 638 667 696 725 754 782 8ii 840 869 N.jL. O 1 2 3 4 5 6 7 8 9 P. P. 0 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. 150 '5' 152 '53 '54 '55 156 '57 I58 '59 17609 638 667 898 926 95~ i8 184 2I3 241 469 498 526 752 780 8o8 696 725 984 *0I3 270 298 554 583 837 865 754 782 8ii 840 869 *04I *070 *099 *I27 *I56 327 35~ 384 412 441 6ii 639 667 696 724 893 921 949 977 *00a 19033 o6i 089 117 145 3I2 340 368 396 424 590 6i8 64~ 673 700 866 893 921 948 976 20 140 167 I94 222 249 173 201 229 257 285 451 479 507 535 562 728 756 783 8ii 838 *003 *030 *058 *o8~-*II2 276 303 330 358 385 160 412 439 466 493 520 548 575 602 629 656 i6i 683 7IO 737 763 790 817 844 871 898 925 162 952 978 *005 *032 *059 *o8~ *II2 *139 *i67 *I92 163 21 2I9 245 272 299 32S 352 378 405 43I 458 164 484 511 537 564 590 617 643 669 696 722 i65 748 775 8oI 827 854 88o 906 932 958 985 i66 22011 037 063 089 115 141 167 I94 220 246 167 272 298 324 350 376 401 427 453 479 505 i68 531 557 583 6o8 634 660 686 712 737 763 169 789 814 840 866 891 917 943 968 994 *oI9 170 23045 070 096 121 147 I72 198 223 249 274 I7I 300 325 350 376 401 426 452 477 502 528 172 553 578 603 629 654 679 704 729 754 779 173 8o5 830 855 88o go5 930 955 980 *005 *030 174 24055 o8o oS I30 I55 i8o 204 229 254 279 175 304 329 353 378 403 428 452 477 502 527 176 551 576 6oi 62' 65o 674 699 724 748 773 177 797 822 846 871 897 920 944 969 993 *oi8 178 25 042 o66 091 iis I39 164 i88 212 237 261 179 285 3IO 334 358 382 406 43I 455 479 503 180 527 55I 57~ 6oo 624 648 672, 696 720 744 i8i 768 792 8i6 840 864 888 912 935 959 983 182 26007 031 055 079 102 126 I~0 174 198 221 183 24~ 269 293 316 340 364 387 4II 435 458 184 482 507 529 553 576 600 623 647 670 694' i85 717 741 764 788 8ii 834 858 881 905 928 i86 951 975 998 *02I *045 *068 *091 *"II4 *38 *i6i 187 27 184 207 231 254 277 300 323 346 370 393 i88 416 439 462 485 5o8 531 554 577 600 623 189 646 669 692 715 738 761 784 807 830 852 190 87' 898 921 944 967 989 *oI2 *03~ *058 *081 191 28 I03 126 149 I7I I94 2I7 240 262 285 307 192 330 353 37S 398 421 443 466 488 5II 533 193 556 578 6oI 623 646 668 691 713 735 758 194 780 803 825 847 870 892 914 937 959 981 I95 29003 026 048 070 092 IIS 137 159 i8I 203 196 226 248 270 292 314 336 358 380 403 425 197 447 469 491 513 535 557 579 6oi 623 645 198 667 688 7IO 732 754 776 798 820 842 863 199 88S 907 929 951 973 994 *oi6 *038 *060 *o81 29 28 I 2,9 2,8 2 5,8 5,6 3 8,7 8,4 4 ii,6 11,2 5 14,5 I4,0 6 17,4 i6,8 7 20,3 19,6 8 23,2 22,4 9 26,1 25,2 27 26 I 2,7 2,6 2 5,4 5,2 3 8,i 7,8 4 io,8 IO,4 5 13,5 I3,0 6 16,2 I5,6 7 1i8,9 18,2 8 21,6 20,8 9 24,3 23,4 25 I 2,5 2 5,0 3 7,5 4 10,0 5 12,5 6 15,0 7 '7,5 8 20,0 9 22,5 24 I 2,4 2 4,8 3 7,2 4 9,6 5 12,0 6 14,4 7 i6,8 8 I9,2 9 21,6 22 I 2,2 2 4,4 3 6,6 4 8,8 5 11,0 6 13,2 7 '5,4 8 17,6 9 19,8 23 2,3 4,6 6,9 9,2 II,5 I3,8 i6,j 18,4 20,7 21 2,1 4,2 6,3 8,4 10,5 12,6 '4,7 i6,8 i8,9 200 I 30 103 125 146 i68 9go 211 233 255 276 298 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. I N. L.o 1, 2 3 4 5 6 7 8 9 P. 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P. 550 55' 552 553 554 74 036 044 05 2 o6o o68 II5 I23 I3I I39 I47 194 202 210 218 225 273 280 288 296 304 35I 359 367 374 382 o76 084 092 099 107 I55 162 170 178 i86 233 24I 249 257 265 3I2 320 327 33~ 343 390 398 406 414 421 555 429 437 445 453 461 468 476 484 492 500 556 507 5I~ 523 531 539 547 554 562 570 578 557 586 593 6oi 609 617 624 632 640 648 656 558 663 671 679 687 695 702 7IO 718 726 733 559 741 749 757 764 772 780 788 796 803 8ii 560 819 827 834 842 850 858 865 873 88i 889 56i 896 904 912 920 927 935 943 950 958 966 562 974 981 989 997 *005 *012 *020 *028 *035 *043 563 75 051 059 o66 074 082 089 097 105 113 120 564 128 136 143 151 I59 i66 174 182 189 I97 565 205 213 220 228 236 243 251 259 266 274 566 282 289 297 305 312 320 328 335 343 35I 567 358 366 374 381 389 397 404 412 420 427 568 435 442 450 458 46$ 473 481 488 496 504 569 5II 519 526 534 542 549 557 565 572 580 570 587 595 603 6io 6i8 626 633 641 648 656 571 664 671 679 686 694 702 709 717 724 732 572 740 747 755 762 770 778 785 793 8oo 8o8 573 8i~ 823 831 838 846 853 86i 868 876 884 574 891 899 906 914 921 929 937 944 952 959 575 967 974 982 989 997 *005 *oI2 *020 *027 *035 576 76042 050 057 065 072 o8o 087 095 103 II0 577 i18 I25 I33 140 I48 I5j i63 170 178 i8~ 578 I93 200 208 215 223 230 238 245 253 260 579 268 275 283 290 298 305 313 320 328 335 580 343 350 358 365 373 380 388 39~ 403 410 581 418 425 433 440 448 455 462 470 477 485 582 492 500 507 5I5 522 530 537 545 552 559 583 567 574 582 589 597 604 612 619 626 634 584 641 649 656 664 671 678 686 693 701 708 585 716 723 730 738 745 753 760 768 775 782 586 790 797 8o5 812 819 827 834 842 849 856 587 864 871 879 886 893 901. 908 916 923 930 588 938 945 953 960 967 975 982 989 997 *004 589 77 012 OI9 026 034 041 048 o56 063 070 078 590 o85 093 IOO I07 I15 122 129 137 144 151 591 159 i66 173 i8i i88 195 203 210 217 225 592 232 240 247 254 262 269 276 283 291 298 593 305 313 320 327 335 342 349 357 364 371 594 379 386 393 401 408 415 '422 4.30 437 444 595 452 459 466 474 481 488 495 503 510 517 596 525 532 539 546 554 56i 568 576 583 590 597 597 605 612 619 627 634 641 648 656 663 598 670 677 685 692 699 706 714 721 728 735 599 743 750 757 764 772 779 786 793 8oi 8o8 8 i o,8 2 i,6 3 2,4 4 3,2 5 4,0) 6 4,8 7 5,6 8 6,4 9 7,2 7 I 0,7 211,4 3 2,I 4 2,8 5 3,5 6 4,2 7 4,9 8 5,6 9 6,3 600 8i~ 822 830 837 844 85i 859 866 873 88o N. L. o 1 2 3 4 5 6 7 8 9 P. P. N.fL. 0 1 2 3 4 5 6 7 8 9 P. P. 600 6oi 602 603 604 I 77 8i~ 822 830 837 844 887 895 902 909 916 960 967 974 981 988 78 032 039 046 053 o6i 104 III II8 125 132 85i 859 866 873 88o 924 931 938 945 952 996 *oo3 *oIo *oI7 *025 o68 o75 082 089 097 140 147 154 i6i i68 605 I76 183 I90 197 204 211 219 226 233 240 606 247 254 262 269 276 283 290 297 305 3I2 607 3I9 326 333 340 347 355 362 369 376 383 608 390 398 405 412 419 426 433 440 447 455 609 462 469 476 483 490 497 504 512 519 526 610 533 540 547 554 56I 569 576 583 590 597 6ii 604 6ii 6i8 625 633 640 647 654 66i 668 612 67~ 682 689 696 704 711 -718 725 732 739 613 746 753 760 767 774 781 789 796 803 8io 614 817 824 831 838 84~ 852 859 866 873 88o 6I5 888 895 902 909 916 923 930 937 944 951 6i6 958 96,5 972 979 986 993 *000 *007 *oI4 *02I 617 79029 036 043 050 057 064 07I cq8 085 092 6i8 099 io6 113 120 127 134 141 148 I5~ 162 619 169 176 183 I90 197 204 211 218 225 232 620 239 246 253 260 267 274 281 288 295 302 621 309 316 323 330 337 344 351 358 36~ 372 622 379 386 393 400 407 414 421 428 435 442 623 449 456 463 470 477 484 491 498 505 511 624 5I8 52~ 532 539 546 553 56o 567 574 58I 625 588 595 602 609 616 623 630 637 644 65o 626 657 664 671 678 68~ 692 699 706 713 720 627 727 734 741 748 754 761 768 775 782 789 628 796 803 8io 817 824 831 837 844 85i 858 629 86~ 872 879 886 893 900 906 913 920 927 630 934 941 948 955 962 969 975 982 989 996 631 80003 oIo 017 024 030 037 044 051 o58 o65 632 072 079 o8~ 092 099 io6 113 120 127 134 633 140 147 154 i6i i68 175 182 i88 195 202 634 209 216 223 229 236 243 250 257 264 271 635 277 284 291 298 305 312 318 32~ 332 339 636 346 353 359 366 373 380 387 393 400 407 637 414 421 428 434 441 448 455 462 468 475 638 482 489 496 502 509 Pi6 523 530 536 543 639 550 557 564 570 577 584 591 598 604 6ii 640 6i8 625 632 638 64~ 652 659 66~ 672 679 641 686 693 699 706 713 720 726 733 740 747 642 754 760 767 774 781 787 794 8oi 8o8 814 643 82i 828 835 841 848 855 862 868 87~ 882 644 889 89~ 902 909 916 922 929 936 943 949 645 956 963 969 976 983 996 996 *003 *oIo *II7 646 8i 023 030 037 043 00 057 064 070 077 084 647 090 097 104 III 117 124 131 137 144 151 648 i58 164 171 178 184 191 198 204 211 218 649 224 231 238 245 251 258 265 271 278 285 8 i 0,8 2 i,6 3 2,4 4 3,2 5 4,0 6 4,8 7 5,6 8 6,4 9 7,2 7 I 0,7 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 I o,6 2 1,2 3 1,8 4 2,4 5 3,0 7 4,2 8 4,8 9 5A 650 I 291 298 305 311 318 325 33I 338 345 35I N. 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L. 0 1 2 3 4 |5 6 7 8 9 P.P. 650 651 652 653 654 8I 291 298 305 311 318 358 365 371 378 385 425 431 438 445 45I 491 498 505 511 5I8 558 564 571 578 584 325 33I 338 345 35I 391 398 405 411 418 458 465 471 478 485 525 531 538 544 551 591 598 604 611 617 655 624 631 637 644 65i 657 664 671 677 684 656 690 697 704 710 7I7 723 730 737 743 750 657 757 763 770 776 783 790 796 803 809 8i6 658 823 829 836 842 849 856 862 869 875 882 659 889 895 902 908 915 921 928 935 941 948 660 954 961 968 974 98I 987 994 *oo *007 *OI4 66 82 020 027 033 040 046 053 o60 o66 073 079 662 o86 092 099 Io5 112 119 I25 132 138 145 663 I5I 158 164 I71 178 184 I91 I97 204 210 664 217 223 230 236 243 249 256 263 269 276 665 282 289 295 302 308 3I5 32I 328 334 34I 666 347 354 360 367 373 380 387 393 400 406 667 413 4I9 426 432 439 445 452 458 465 471 668 478 484 491 497 504 5Io 5I7 523 530 536 669 543 549 556 562 569 575 582 588 595 601 670 607 614 620 627 633 640 646 653 659 666 671 672 679 685 692 698 705 711 718 724 730 672. 737 743 750 756 763 769 776 782 789 795 673 802 808 814 821 827 834, 840 847 853 860 674 866 872 879 885 892 898 905 911 918 924 675 930 937 943 950 956 963 969 975 982 988 676 995 oo oo008 *OI4 *020 *027 *033 *040 *046 *052 677 83 059 065 072 o78 o85 09I 097 I04 IIO 117 678 123 I29 I36 I42 I49 I55 I6I I68 I74 I8I 679 I87 I93 200 206 213 2I9 225 232 238 245 680 25I 257 264 270 276 283 289 296 302 308 68I 3IS 321 327 334 340 347 353 359 366 372 682 378 385 391 398 404 410 417 423 429 436 683 442 448 455 46I 467 474 480 487 493 499 684 5o6 512 5I8 525 53I 537 544 550 556 563 685 569 575 582 588 594 6oi 607 613 620 626 686 632 639 645 651 658 664 670 677 683 689 687 696 702 708 715 72I 727 734 740 746 753 688 759 765 77I 778 784 790 797 803 809 8i6 689 822 828 835 841 847 853 860 866 872 879 690 885 891 897 904 910 916 923 929 935 942 691 948 954 960 967 973 979 985 992 998 oo4 692 84 O 017 023 029 036 042 048 055 06i 067 693 073 o8o o86 092 o98 105 II II17 123 130 694 I36 142 148 I55 i6 I 67 I73 i8o i86 I92 695 I98 205 2II 2I7 223 230 236 242 248 255 696 261 267 273 280 286 292 298 305 311 3I7 697 323 330 336 342 348 354 36I 367 373 379 698 386 392 398 404 4IO 417 423 429 435 442 699 448 454 460 466 473 479 485 491 497 504 7 I 0,7 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 I o,6 2 1,2 3 I,8 4 2,4 5 3,0 6 3,6 7 4,2 8 4,8 9 5,4 I 700 510 5I6 522 528 535 54I 547 553 559 566 N. L. 0 1 2 3 4 5 6 7 8 9 P.P. i 0 N. L. O 1 2 3 4 5 6 7 8 9 P. P. 700 701 702 703 704 84510 516 522 528 535 572 578 584 590 597 634 640 646 652 658 696 702 708 714 720 757 763 770 776 782 54I 547 553 559 566 603 609 6i5 621 628 665 671 677 683 689 726 733 739 745 751 788 794 8oo 807 813 705 819 82~ 831 837 844 85o 856 862 868 874 706 88o 887 893 899 go9 911 917 924 930 936 707 942 948 954 960 967 973 979 985 991 997 708 85 003 009 oi6 022 028 034 040 046 052 058 709 065 071 077 083 089 095 II I07 114 120 710 126 132 138 144 i~o I56 163 169 I75 i8i 711 187 193 199 205 211 217 224 230 236 242 7I2 248 254 260 266 272 278 285 29I 297 303 7I3 309 315 321 327 333 339 345 352 358 364 714 370 376 382 388 394 400 406 412 4I8 425 715 431 437 443 449.455 461 467 473 479 485 716 491 497 503 509 5i6 522 528 534 540 546 717 552 558 564 570 576 582 588 594 6oo 6o6 718 612 6i3 625 631 637 641 649 655 66i 667 7I9 673 679 685 691 697 703 709 715 721 727 r720 733 739 745 751 757 763 769 775 781 788 721 794 8oo 8o6 812 8i8 824 830 836 842 848 722 854 86o 866 872 878 884 890 896 902 908 723 914 920 926 932 938 944 950 956 962 968 724 974 980 986 992 998 *004 *oIo *oi6 *022 *028 725 86034 040 046 052 058 064 070 o76 082 o88 726 094 oo0 io6 112 118 124 130 136 141 147 727 153 159 i6l I7I 177 i83 189 I9q 201 207 728 213 219 225 231 237 243 249 255 261 267 729 273 279 285 291 297 303 308 314 320 326 730 332 338 344 350 356 362 368 374 380 386 73I 392 398 404 4IO 415 421 427 433 439 445 732 451 457 463 469 475 481 487 493 499 504 733 510 5I6 522 528 534 540 546 552 558 564 734 570 576 58I 587 593 599 6o0 6ii 617 623 735 629 635 641 646 652 658 664 670 676 682 736 688 694 700 705 711 717 723 729 735 741 737 747 753 759 764 770 776 782 788 794 800 738 8o6 8I2 817 823 829 83~ 841 847 853 859 739 864 870 876 882 888 894 900 906 911 917 740 923 929 935 941 947 953 958 964 970 976 741 982 988 994 999 *005 *0II *oI7 *023 *029 *035 742 87040 046 052 058 064 070 075 o8i 087 093 743 099 105 Ini ii6 122 128 134 I40 146 I5I 744 157 i63 169 175 i8i i86 192 198 204 2IO 745 216 22I 227 233 239 245 251 256 262 268 746 274 280 286 291 297 303 309 315 320 326 747 332 338 344 349 355 361 367 373 379 384 748 390 396 402 408 413 419 425 431 437 442 749 448 454 460 466 471 477 483 489 495 500 7 I 0,7 2 I4 3 2,I 412,8 6 4,2 7 4,9 8 5,6 9 6,3 6 i 11o,6 ~1,8 1!1 412,4 5 13,0 6 3,6 7 4,2 8 4,8 9 5,4 5 I 0,5 2 I,O 3 1,5 412,0 5 I2,5 6. 3,0 7 3,5 8 4,0 9 4,5 I 750 506 5I2 5i8 523 529 53~ 541 547 552 558 N. 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L. 0 1 2 3 4 5 6 7 8 9 P.P. 750 75'1 752 753 754 87 506 512 518 523 529 564 570 576 58I 587 622 628 633 639 645 679 685 691 697 703 737 743 749 754 760 535 541 547 552 558 593 599 604 6io 616 65i 656 662 668 674 708 714 720 726 731 766 772 777 783 789 755 795 800 806 812 8i8 823 829 835 84I 846 756 852 858 864 869 875 88I 887 892 898 904 757 910 9I5 921 927 933 938 944 950 955 96I 758 967 973 978 984 990 996 *ooi *007 *013,oi8 759 88 024 030 036 041 047 053 058 o64 070 076 760 o8i 087 093 098 104 IIo ii6 121I 27 133 76I 138 144 I5 15 56 I6I 167 173 I78 184 190 762 I93 20I 207 213 218 224 230 235 241 247 763 252 258 264 270 275 281 287 292 298 304 764 309 315 32I 326 332 338 343 349 355 360 765 366 372 377 383 389 395 400 406 412 417 766 423 429 434 440 446 451 457 463 468 474 767 480 485 491 497 502 508 5I3 5I9 525 530 768 536 542 547 553 559 564 570 576 581 587 769 593 598 604 6Io 6ID 62I 627 632 638 643 770 649 655 660 666 672 677 683 689 694 700 77I 705 7II 717 722 728 734 739 745 750 756 772 762 767 773 779 784 790 795 8oi 807 812 773 8i8 824 829 835 840 846 852 857 863 868 774 874 88o 885 89I 897 902 908 913 919 925 775 930 936 941 947 953 958 964 969 975 98I 776 986 992 997 *003 *009 *oo 4 *020 *025 *03I *037 777 89 042 048 053 059 064 070 076 o8i 087 092 778 o98 104 109 I15 120 126 I3I I37 I43 148 779 154 I59 I65 170 176 182 I87 I93 I98 204 780 209 215 22I 226 232 237 243 248 254 260 78I 265 27I 276 282 287 293 298 304 310 3I5 782 32I 326 332 337 343 348 354 360 365 37I 783 376 382 387 393 398 404 409 415 421 426 784 432 437 443 448 454 459 465 470 476 481 785 487 492 498 504 509 5I5 520 526 531 537 786 542 548 553 559 564 570 575 58I 586 592 787 597 603 609 614 620 625 631 636 642 647 788 653 658 664 669 675 680 686 691 697 702 789 708 713 719 724 730 733 74I 746 752 757 790 763 768 774 779 785 790 796 8oI 807 812 791 8i8 823 829 834 840 845 851 856 862 867 792 873 878 883 889 894 900 905 911 916 922 793 927 933 938 944 949 955 960 966 971 977 794 982 988 993 998 o004 0oo09 015 020,026 031 795 90 037 042 048 053 059 o64 o69 075 o80 o86 796 09I 097 I02 Io8 113 119 124 129 135 140 797 I46 15I 157 I62 168 I73 I79 I84 I89 I95 798 200 206 211 217 222 227 233 238 244 249 799 255 260 266 271 276 282 287 293 298 304 6 0o,6 3 1 I,8 412,4 513,0 6 3,6 7 4,2 8 4,8 9 5,4 5 I o,5 2 1,0 3 x I,5 4 2,0 532,5 6 3,0 7 3,5 8 4,0 9 4,5 800 309 3I4 320 325 331 336 342 347 352 358 N. L. 0 1 2 3 4 5 6 7 8 9 P.P. I N. L. 0 1 2 3 4f5 6 7 8 9 P. P. 800 8oi 802 803 804 90309 314 320 325 331 363 369 374 380 385 417 423 428 434 439 472 477 482 488 493 526 53I 536 542 547 336 342 347 352 358 390 396 401 407 412 445 450 455 461 466 499 504 509 515 520 553 558 563 569 574 8o5 580 585 590 596 6oi 6o7 612 617 623 628 8o6 634 639 644 65o 65~ 66o 666 671 677 682 8o7 687 693 698 703 709 714 720 725 730 736 8o8' 741 747 752 757 763 768 773 779 784 789 809 795 8oo 8o6 8ii 8i6 822 827 832 838 843 810 849 854 859 865 870 87~ 88i 886 891 897 8ii 902 907 913 918 924 929 934 940 945 950 812 956 961 966 972 977 982 988 993 998 *oo4 813 91 009 oI4 020 025 030 036 04I 046 052 057 8I4 062 o68 073 o78 084 089 094 100 IO IIO 8I5 ii6 121 126 132 137 142 I48 I53 I58 164 8i6 169 174 i8o 185 190 196 201 206 2I2 217 8I7 222 228 233 238 243 249 254 259 265 270 8i8 275 281 286 291 297 302 307 3I2 318 323 819 328 334 339 344 350 355 360 365 371 376 820 381 387 392 397 403 408 413 418 424 429 821 434 440 445 4-50 455 461 466 471 477 482 822 487 492 498 503 508 514 519 524 529 535 823 540 545 55I 556 56I 566 572 577 582 587 824 593 598 603 609 614 619 624 630 635 640 825 64~ 65i 656 66i 666 672 677 682 687 693 826 698 703 709 714 719 724 730 735 740 745 827 751 756 761 766 772 777 782 787 793 798 828 803 8o8 814 819 824 829 834 840 845 8~o 829 85~ 86i 866 871 876 882 887 892 897 903 830 908 913 918 924 929 934 939 944 950 955 831 960 965 97I 976 981 986 991 997 *002 *007 832 92 012 oi8 023 028 033 038 044 049 054 059 833 065 070 075 o8o o8~ 091 096 ioi io6 iii 834 II7 122 127 132 137 I43 I48 I53 I58 163 835 169 174 179 184 189 195 200 20~ 2IO 2I1 836 221 226 231 236 241 247 252 257 262 267 837 273 278 283 288 293 298 304 309 314 319 838 324 330 335 340 345 350 355 361 366 371 839 376 381 387 392 397 402 407 412 418 423 840 428 433 438 443 449 454 459 464 469 474 841 480 485 490 495 500 50~ 5II 5i6 521 526 842 53I 536 542 547 552 557 562 567 572 578 843 583 588 593 598 603 609 614 619 624 629 844 634 639 645 65o 655 66o 66~ 670 67~ 68i 845 686 691 696 701 706 711 716 722 727 732 846 737 742 747 752 758 763 768 773 778 783 847 788 793 799 804 809 814 819 824 829 834 848 840 845 850 855 86o 86~ 870 87~ 88i 886 849 891 896 901 906 gII 916 921 927 932 937 6 i o,6 2 1,2 3 i,8 4 2,4 5 3,0 6 3,6 7 4,2 8 4,8 9 5,4 5 I 0,5 2 1I0 3 1,5 4 2,0 5 2,5 6 3,0 7 3,5 8 4,0 9 4,5 850 942 947 952 957 962 967 973 978 983 988 N.jL. O 1 2 3 4 5 6 7 8 9 P. P. 0 N. L. O 1 2 3 4 5 6 7 8 9 P. P. 850 85i 852 853 854 92 942 993 93 044 095 146 947 952 957 962 998 *003 *008 *OI3 049 054 059 064 Ioo 10 n1o I1 151 156 i6i i66 967 973 978 983 988 *oi8 *024 *029 *034 *639 069 075 o8o 085 090 120 125 131 136 141 171 176 i8i i86 I92 855 I97 202 207 212 217 222 227 232 237 242 856 247 252 258 263 268 273 278 283 288 293 857 298 303 308 3I3 318 323 328 334 339 344 858 349 354 359 364 369 374 379 384 389 394 859 399 404 409 414 420 425 430 435 440 445 860 450 455 460 465 470 475 480 485 490 49S 861 5OO 505 5Io 5IS 520 526 531 536 54I 546 862 55I 556 561 566 571 576 581 586 591 596 863 6oi 6o6 6ii 6i6 621 626 631 636 641 646 864 65i 656 66i 666 671 676 682 687 692 697 865 702 707 712 717 722 727 732 737 742 747 866 752 757 762 767 772 777 782 787 792 797 867 802 807- 812 817 822 827 832 837 842 847 868 852 857 862 867 872 877 882 887 892 897 869 902 907 912 917 922 927 932 937 942 947 870 952 957 962 967 972 977 982 987 992 997 87I 94 002 007 012 017 022 027 032 037 042 047 872 052 057 062 067 072 077 082 o86 ogi o96 873 ioi io6 iii ii6 121 126 131 136 141 146 874 151 156 i6i i66 171 176 i8i i86 igi 196 875 201 206 211 216 221 226 231 236 240 245 876 250 255 260 265 270 275 280 285 290 295 877 300 305 310 3I5 320 325 330 335 340 345 878 349 354 359 364 369 374 379 384 389 394 879 399 404 409 414 419 424 429 433 438 443 880 448 453 458 463 468 473 478 483 488 493 88i 498 503 507 512 517 522 527 532 537 542 882 547 552 557 562 567 571 576 581 586 591 883 596 6oi 6o6 6ii 6i6 621 626 630 63S 640 884 64S 65o 65S 66o 665 670 675 68o 685 689 885 694 699 704 709 714 719 724 729 734 738 886 743 748 753 758 763 768 773 778 783 787 887 792 797 802 807 8I2 8I7 822 827 832 836 888 841 846 851 856 86i 866 871 876 88o 885 889 890 89S goo 905 g9o 9I5 919 924 929 934 890 939 944 949 954 959 963 968 973 978 983 891 988 993 998 *002 *007 *OI2 *017 *022 *027 *032 892 95 036 041 046 051 056 o6i o66 071 07S o8o 893 08S 090 095 0oo 105 109 114 119 124 129 894 134 I39 I43 148 153 I58 163 i68 I73 177 6 i o,6 2 1,2 3 I,8 4 2A4 5 3,0 6 3,6 7 4,2 8) 4,8 9 5A4 5 I 0,5 2 I,O 3 1,5 4 2,0 5 2,5 6 3,0 7 3,5 8 4,0 9 4,5 4 I 0,4 2 o,8 3 1,2 4 i,6 5 2,0 6 2,4 7 2,8 8 3,2 9 3,6 895 896 897 898 899 900 182 187 192 I97 202 231 236 240 245 250 279 284 289 294 299 328 332 337 342 347 376 381 386 390 395 I 207 211 216 221 226 255 260 265 270 274 303 308 313 318 323 352 357 361 366 371 400 405 4IO 415 419 424 429 434 439 444 448 453 458 463 468 N. L. 0 1 2 3 4 5 6 7 8 9 P. P. N. L. O 1 2 3 4 5 6 7 8 9 P. P. 900 901 902 903 904 95 424 429 434 439 444 472 477 482 487 492 521- 525 530 535 540 569 574 578 583 588 6I7 622 626 631 636 448 453 458 463 468 497 501 5o6 511 516 545 550 554 559 564 593 598 602 607 612 641 646 65o 655 66o 905 665 670 674 679 684 689 694 698 703 708 906 7I3 718 722 727 732 737 742 746 75I 756 907 761 766 770 775 780 785 789 794 799 804 908 809 813 8i8 823 828 832 837 842 847 852 909 856 86i 866 871 875 88o 885 890 895 899 910 904 909 914 918 923 928 933 938 942 947 gII 952 957 961 966 97I 976 980 985 990 995 912 999 *004 *009 *oI4 *oI9 *.023 *028 *033 *038 *042 913 96047 052 057 o6i o66 071 076 080 o85 o09 914 095 099 104 109 114 ii8 123 128 133 137 9I5 142 I47 152 I56 i6i i66 I7I I75 i8o 185 916 190 194 199 204 209 213 218 223 227 232 9I7 237 242 246 251 256 261 265 270 275 280o 918 284 289 294 298 303 308 313 317 322 327 919 332 336 341 346 350 355 360 365 369 374 920 379 384 388 393 398 402 407 412 417 421 921 426 431 435 440 445 450 454 459 464 468 922 473 478 483 487 492 497 501 5o6 5II 515 923 520 525 530 534 539 544 548 553 558 562 924 567 572 577 58I 586 591 595 6oo 6o5 609 925 6I4 619 624 628 633 638 642 647 652 656 926 66i 666 670 675 68o 685 68q 694 699 703 927 708 7I3 7I7 722 727 731 736 74I 745 750 928 755 759 764 769 774 778 783 788 792 797 929 802 8o6 8ii 8i6 820 825 830 834 839 844 930 848 853 858 862 867 872 876 88i 886 890 931 895 900 904 909 914 918 923 928 932 937 932 942 946 951 956 960 965 970 974 979 984 933 988 993 997 *002 *007 *OIi i *oi6 *02I.025' *030 934 97 035 039 044 049 053 o58 063 067 072 077 935 o8i 086 9o o095 0oo 104 I09 114 1i8 123 936 128 132 137 142 I46 I5I I55 i6o 165 169 937 174 179 183 i88 192 197 202 206 211 216 938 220 225 230 234 239 243 248 253 257 262 939 267 27I 276 280 285 290 294 299 304 308 940 313 3I7 322 327 33I 336 340 345 350 354 941 359 364 368 373 377 382 387 391 396 400 942 405 410 414 419 424 428 433 437 442 447 943 451 456 460 465 470 474 479 483 488 493 944 497 502 5o6 5Ii 5I6 520 525 529 534 539 945 543 548 552 557 562 566 571 575 580 585 946 589 594 598 603 607 612 617 621 626 630 947 635 640 644 649 653 658 663 667 672 676 948 68i. 685 690 695 699 704 708 713 717 722 949 727 731 736 740 745 749 754 759 763 768 5 I 10,5 2 1,0 3 1,5 4 2,0 5 2-,5 6 3,0 7 3,5 8 4,0 9 4,5 4 I 0,4 2 o,8 3 1,2 4 i,6 5 2,0 6 2,4 7 2,8 8 3,2 9 3,6 950 1 772 777 782 786 791 795 8oo 804 809 813 N. L. 0 1 2 3 4[5 6 7 8 9 P. P. N. L. O 1 2 3 4f5 6 7 8 9 P. P. 950 95' 952 953 954 97 772 777 782 786 791 8i8 823 827 832 836 864 868 873 877 882 909 914 918 923 928 955 959 964 968 973 795 8oo 804 809 813 841 845 850 855 859 886 891 896 900 905 932 937 941 946 950 978 982 987 991 996 955 98 000 005 009 014 019 023 028 032 037 04I 956 046 050 055 059 064 o68 073 078 082 087 957 091 096 100 105 109 114 ii8 123 127 132 958 137 141 146 i5o 155 159 164 i68 173 177 959 182 i86 igi Ig9 200 204 209 2I4 218 223 960 227 232 236 24I 245 250 254 259 263 268 961 272 277 281 286 290 295 299 304 308 3I3 962 318 322 327 33I 336 340 345 349 354 358 963 363 367 372 376 381 385 390 394 399 403 964 408 412 417 421 426 430 435 439 444 448 965 453 457 462 466 471 475 480 484 489 493 966 498 502 507 5II 516 520 525 529 534 538 967 543 547 552 556 561 565 570 574 579 583 968 588 592 597 6oi 605 610 614 619 623 628 969 632 637 641 646 65o 655 659 664 668 673 970 677 682 686 691 695 700 704 709 713 7I7 97I 722 726 73I 735 740 744 749 753 758 762 972 767 771 776 780 784 789 793 798 802 8o7 973 8ii 8i6 820 825 829 834 838 843 847 85I 974 856 86o 865 869 874 878 883 887 892 896 975 900 905 909 '9I4 918 923-, 927 932 936 94I 976 945 949 954 958 963 967 972 976 981 985 977, 989 994 998 *003 *007 *0OI2 *01 6 *02I *025 *029 978 99 034 038 043 047 052 056 o6i o65 069 074 979 078 083 087 092 096 0oo 105 1og 114 ii8 980 123 127 131 136 140 145, 149 154 I58 162 981 167 171 176 i8o 185 I9 193 198 202 207 982 211 216 220 224 229 233 231 242 247 251 983 255 260 264 269 273 277 282 286 291 295 984 300 304 308 3I3 3I7 322 326 330 335 339 985 344 348 352 357 361 366 370 374 379 383 986 388 392 396 401 405 410 414 419 423 427 987 432 436 441 445 449 454 458 463 467 471 988 476 480 484 489 493 498 502 5o6 5II 515 989 520 524 528 533 537 542 546 550 555 559 990 564 568 572 577 58I 585 590 594 599 603 991 607 612 6i6 621 625 629 634 638 642 647 992 65i 656 66o 664 669 673 677 682 686 691 993 695 699 704 708 712 717 721 726 730 734 994 739 743 747 752 756 760 765 769 774 778 995 782 787 79I 795 800 804 8o8 813 817 822 996 826 830 835 839 843 848 852 856 86i 865 997 870 874 878 883 887 891 896 900 904 gog 998 913 9I7 922 926 930 935 939 944 948 952 999 957 961 965 970 974 978 983 987 991 996 5 I 0,5 2 I,O 210 3 1,5 4 2,0 5 2,5 7 3,5 8 4,0 9 4,5 4 I 0,4 2 o,8 3 1,2 4 i,6 5 2,0 6 2,4 7 2,8 8 3,2 9 3,6 1000I 00000 004 009 03 017 022 026 030 035 039 N.JL. 0 1 2 3 4 5 6 7 8 9 P. P. NOTES ON TABLES I AND II. The logarithms of numbers are in general incommensurable. In these tables they are given correct to five places of decimals. If the sixth place is 5 or more, the next larger number is used in the fifth place. Thus log 8102 = 3.908549+; in five-place tables this is written 3.90855, the dash above the 5 showing that the logarithm is less than given. So log 8133 = 3.910251-; in five-place tables this is written 3.91025, the dot above the 5 showing that the logarithm is more than given. In the natural functions of the angles (Table II) all numbers are decimals for sine and cosine (why?), and for tangent and cotangent, except where the decimal point is used to indicate that part of the number is integral. When no decimal point is printed in the tables it is to be understood. When the natural function is a pure decimal the characteristic of the logarithm is negative. Accordingly, in the tables 10 is added, and in the result this must be allowed, for. Thus nat. sin 44~ 20' = 0.69883, log sin 44~ 20' = 1.84437, or, as printed in the tables, 9.84437, which means 9.84437 -10. TABLE II. THE LOGARITHMIC AND NATURAL SINES, COSINES, TANGENTS, AND COTANGENTS OF ANGLES FROM 0~ TO 90~. 0 I f iNat. Sin Log. d. jNat.COSLog.lNat.TaflLog. c.d. llop,.COtNat-l 0 0 I 2.3 4 6 7 8 9 I' 12 '3 '4 15 '7 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 144 45 46 47 48 49 50 5' 52 53 54.56 57 58 59 60 00000 -02,9 6.46373 059 6.76476 087 6.94085 ii6 7.06579 00145 7.16270 175 7.24i88 204 7.30882 233 7.36682 262 7.41797 00291 7.46373 320 7.505I2 349 7.5429I 378 7 —57767 407 7-60985 00436 7.63932 465 7.66784 495 7.694I7 524 7.7I900 553 7.74248 00582 7.76475 6iI 7.78594 640 7.8o0i$ 669 7.82545 698 7.84393 00727 7.86166 756 7.87870 785 7.89509 814 7.9I088 844 7.926I2 00873 7.94084 902 7.95508 931 7.96887 960 7.98223 989 7.99520 oioi8 8.00779 047 8.02002 076 8.03I92 I05 8.04350 I34 8.0,5478 011i64 8.065-78 I93 8.07650 222 8.08696 251I 8.097i8 280 8.I0717 01309 8.ii693~ 338 8.i2647 367 8.I3581 396 8.14495 425 8.I53911 01454 8.16268 483 8.117128 513 8.I7971 542 8.i8798 571 8.i96i0 oi6oo 8.20407 629 8.2ii89 658 8.21958 687 8.22713 716 8.23456 745 8.24186 30I03 17609 12494 9691 7918 6694 5800 SI'S 4576 4139 3779 3476 3218 2997 2802 2633 2483 2348 2227 2I19 2021 1930 1848 10000 0.00000 000 0.00000 000 0.00000 000 0.00000 000 0.00000 00000 - 029 6.46373 o58 6.76476 087 6.9408S ii6 7.06579 10000 0.00000 00145 7.16270 000 0.00000 I75 7.24188 000 0.00000 204 7.30882 000 0.00000 233 7.36682 000 0.00000 262 7.41797 10000 0.00000 99999 0.00000 999 0.00000 999 0.00000 999 0.00000 99999 0.00000 999 0.00000 999 9.99999 999 9.9-9999 998 9.99999 99998 9.99999 998 9.99999 998 9.99999 998 9.99999 998 9.99c999 00291I 7.46373 320 7.50512 349 7-5429I 378 7.57767 407 7.60986 00436 7.63982 465 7.66785 495 7.69418 524 7.71900 553 7.74248 00582 7.76476 6iI 7.7859$ 640 7.80615 669 7.82546 698 7.84394 I. 'I I I I'. I17 I! I,. I: I,. I: 'I., 704 539 579 524 99997 9.99999 997 9.99999 997 9.99999 997 9.99999 99c6 o.cooo8 00727 7.8616Y7 756 7.8787I 785 7.895i0 8I5 7.9I089 844 7.026i1A 172 ----- 124 99996 9.99998 00873 7.94086 39 996 9.99998 902 7.95510 336 996 9.99998 93I 7.96889 336 959999 960 7.98225 259 99 9.99998 989_7___522 2399995-9.99998 oioi8 8.0078i [90 995 9.99998 047 8.02004 158 994 9.99997 076 8.03I94 128 994 9.99997 105 8.04353 100 4 9-99 135 8.05481 372 99993 9.99997 oi i64 8.o658i 2)46 993 9-99997 I93 8.07653 222 99 9.99997 222 8.08700 99 992 9.99997 251 8.09722 96 992 9.99996 280 8.10720 5499991 9.99_996 01309 8.ii696 934 991 9.99996 338 8.I2651 94 99,.99996 367 8.13585 86 990 9.99996 396 8.14500 877 990 9.99996 425 8.15395 86o 99989 9.99995 01455 8.16273 83 989 9.99995 484 8.I7133 843 989 9.99995 5I3 8.I7976 812 988 9.99995 542 8.i8804 707 988 9.99995 571 8.i96i6 - 30103 3-53627 3437.7 176og 3.23524 171i8.9 I443-05915 I1145.9 12494 2-93421 859.44 79 i8 2.83730 687.55 6694 2.758I2 572.96 5800 2.69ii8 49I.1I 5II 2.633i8 429.72 4576 2.58203 381i.97 4139 2.53627 343.77 3792.49488 312.52 3476 2.45709 286.48 3219 2.42233 264.44 2996 2-390I4 245.55 2803' 2.36018 229.18 2-33215 214.86 2633 2.30582 202.22 234821 2 28100 190.98 234 2.25752 180.93 2119 2.23524 171.89 2020 2245137 1931 2.i19385 156.26 1 848 2.174% I49.47 2.15606 143.24 1704 2.13833 I37-5I 1639 2.I2I29 132.22 i692.I0490 I27.32 I579 2.08911 I122.77 1524 2.07387 11I8.54 1424 2.059114 I114.59. I442.04490 110o.89 1336 2.03III I07.43 1297 2.01775 I04.17 I259 2.00478 101.11 1223 I.99219 98.2i8 I I-0 i.97996 95.489 1I59 i.96806 92.908 1128 I1.95647 90.463 1100 I.945i9 88.i44 1072 i.934i9 85.940 I04 11.92347 83.844 I022 1.91300 8i.847 998 11.90278 79.943 96 i.89280 78.126 1.88304 76.390 955 i.87349 74.729 9I5 8415 73.I39 89 185500 7I.615.878~ i.846o0570.1I53 86o 1.83727 68.750 84 1.82867 67.402 828 1.82024 66.105 812 i.81196 64.858 79 1.80384 63.657 782 1.79587 62.499 769 1.78805 6I.383 756 1.78036 60.306 742 1.77280 59.266 73.I76538 58.26i 73 175808 57.290 I 60 59 58 57 52 5' 49 48 47 46 -45 44 43 42 4' 40 39 38 37 36 -35 3l4 33 32 3' 30 29 28 27 26 25 24 2-3.22 21 20 '9 i8 '7 1 5 '4 '3 12 II 10 9 8 7 6 6 4 3 2 I 0 782 769 755 743 730 99987 9.99994 987 9.99994 986 9.99994 986 9.99994 985 9.99994 983 9.99993 oi6oo 8.20413 629 8.21195~ 658 8.21964 687 8.22720 716 8.23462 746 8.24192 Nat.COSI~og. d. INat. SiflLog. Nat.COtLog.~ c. d. ILog.TaflNat] 1 890 1 0 f Nat. Sin Log. d. Nat. COSLog.jNat.TanlLog.1 c~d.ILog. 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Cot Nat-j 0 I1 2 3 4 5 6 7 8 9 II 12 '3 '4 i6 '7 20 21 22 23 24 25~ 26 27 28 29 05234 8.7i88o 263 8.72I20 292 8.72359 321 8.72597 350 8.72834 05379 8.73069 408 8.73303 437 8.7353~ 466 8.73767 495 8.73997 05524 8.74226 553 8.74454 582 8.74680 6ii 8.74906 640 8.75I30 05669 8.75353 698 8.75575 727 8.7579$ 756 8.76015 785 8.76234 058I4 8.7645I 844 8.76667 873 8.76883 902 8.77097 931 8.773I0 05960 8.77522 989 8.77733 o6oi8 8.77943 047 8.78I52 076 8.78360 240 239 238 237 235 234 232 232 230 229 228 226 226 224 223 222 220 220 2I9 2I7 216 216 2I4 213 212 211 210 209 208 99863 9.99940 86i 9.99940 86o 9.99939 858 9.99938 857 9.99938 99855 9.99937 854 9.99936 852 9.9993 851 9.99935 849 9.99934 99847 9.99934 846 9.99933 844 9.99932 842 9.99932 841 9.9993I 05341 8.7I940 27o 8.72i8i 299 8.72420 328 8.72659 357 8.72896 05387 8.73I32 416 8.73366 445 8.73600 474 8.73832 503 8.74063 05533 8.74292 562 8.7452I 591 8.74748 620 8.74974 649 8.75I99 241 239 239 237 2-36 234 234 232 231I 229 229 227 226 225 224 222 1222 220 2I9 2I9 2I7 216 215 214 I.28060 i9.081I 1.278i9 i8.976 I.27580.871 I.27341.768 I.27I04.666 I.26868 18.564 1.26634.464 1.26400.366 I.26i68.268 I.25937.17I I.25708 i8.075 1.25479 I7.980 I.25252.886 I.25026.793 1.24801.702 I.24577 I7.6iI I.24355.52I 1.24133.43I I.239113.343 I.23694.256 1.23475 I7.1i69 1.23258.084 I.23042 i6.999 1.22827.9I5 1.22613.832 99839 9.99930 838 9.99929 836 9.99929 834 9.99928 833 9.99927 99831 9.99926 829 9.99926 827 9.99925 826 9.99924 824 9.99923 99822 9.99923 82I 9.99922 819 9.99921 817 9.99920 8I5 9.99920 05678 8.7542~ 708 8.75645 737 8.75867. 766 8.76087 795 8.76306 05824 8.76525 854 8.76742 883 8.76958 912 8.77173 941 8.77387 05970 8.77600 999 8.77811 06029 8.78022 o58 8.78232 087 8.7844i 30 o6Io5 8.78568 206 9981I3 9.99919 o6ii6 8.78649 31 134 8.78774 25 8I2 9.99918 I45 8.7885~ 32 163 8.78979 205 810 9.999I7 I75 8.79061 33 192 8.79183 204 8o8 9.99917 204 8.79266 34 221 8.79386 23 8o6 9.99916 23 8.97 35 36 37 38 39 46 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 57 58 59 60 06250 8.79588 279 8.79789 308 8.79990 337 8.8oi89 366 8.80388 06395 8.8058~ 424 8.80782 453 8.80978 482 8.81173 5Ii 8.8I367 06540 8.8I56o 569 8.81752 598 8.8i944 627 8.82134 656 8.82324 o6685 8.82513 7I4 8.82701 743 8.82888 773 8.83079 802 8.83261 06831 8.83446 86o 8.83630 889 8.83813 918 8.83996 947 8.84177 976 8.84358 26I 20I '99 '99 '97 '97 196 '95 '94 '93 I92 192 189 i88 187 187 i86 185 184 183 183 99804 9.99915~ 803 9-999I4 8oi 9.99913 799 9.99913 797 9.99912 99795 9.999II 793 9.999I0 792 9.99909 790 9.99909 788 9.99908 99786 9.99907 784 9.99906 782 9.99905 780 9.99904 778 9.99904 99776 9.99903 774 9.99902 772 9.99901 770 9.99900 768 9.99899 99766 9.99898 764 9.99898 762 9.99897 760 9.99896 758 9.99895 756 9.99894 06262 8.79673 291 8.7987~ 321 8.80076 350 8.80277 379 8.80476 06408 8.80674 438 8.80872 467 8.8Io68 496 8.81264 525 8.81459 06554 8.81653 584 8.81846 613 8.82038 642 8.82230 67I 8.82420 211 211 210 209 208 206 206 205 204 203 202 201 201 '99. 198 198 196 196 '195 '94 '93 192 192 190 190 189 i88 i88 i86 i86 185 184 184 182 182 I.22400 i6.750 I.22I89.668 I.2i978.587 1.2I768.507 1.21559.428 1.21351 i6.350 1.211I45.272 1.20939.195 1.20734 II19 1.20530.043 I.20327 I5.969 1.20I25.895 1.19924.82I 1.19723.748 1.19524.676 I.I9326 I5.605 I.I9128.534 1.8932.464 1.8736.394 1.8541.325 i.i8347 15.257 1.18154 i189 1.17962 I122 I.17770.o56 1.17580 I4.990 1.17390 I4.924 1.17201.86o 1.17013.795 i.i682~.732 1.1~6639.669 i.i6453 I4.606 i.i6268.544 i.i6084.482 I.15900.42I I.157i8.361 I.15536.30I 60 59 57 55 54 53,52 5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 '9 '7 i6 '4 '3 12 II 9 8 7 6 -5 4 3 2 I I I I I 06700 8.826i0 730 8.82799 759 8.82987 788 8.83I79 817 8.83361 06847 8.83547 876 8.83732 905 8.839i6 934 8.84100 963 8.84282 993 8.84464 m INat. Co0SLog.7Y.Nat. S in Log.INat. COt Logj1 c.d. lLog. Tanl Nat.I' I 860 __ _40 f Nat. sin Log. d. Nat. COS Log.jNat.TanLog.~ c.d. Log. Cot Nat.I I 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 15 '7 '9 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 36 37 38 39 40 4' 42 43 44 45 -46 47 48 49 50 5' 52 53 54 56 57 58 06976 8.84358 07005 8.84539 034 8.847i8 063 8.84897 092 8.85075 07121 8.85252 150 8.85429 I79 8.856o5 208 8.85780 237 8.85955 07266 8.86128 295 8.8630I 324 8.86474 353 8.86645 382 8.868i6 07411 8.86987 44o 8.87156 469 8.87325 498 8.87494 527 8.8766i 07556 8.87829 585 8.87995 614 8.88i6i 643 8.88326 672 8.88490 07701 8.88654 730 8.888I7 759 8.8898o 788 8.89142 8I7 8.89304 07846 8.89464 875 8.89625 904 8.89784 933 8.89943 962 8.90102 07991 8.90260 08020 8.90417 049 8.90574 078 8.90730 107 8.90885 08136 8.91040 165 8.9i1195 194 8.91349 223 8.91502 252 8.91655 08281 8.91807 310 8.9I959 339 8.92IIo 368 8.92261 397 8.9241I 08426 8.92561 455 8.92710 484 8.92859 513 8.93007 542 8.93154 08571 8.93301 6oo 8.93448 629 8.93594 658 8.93740 687 8.93885 716 8.94030 181 '79 '79 I78 '77 '77 176 '75 '75 '73 '73 '73 '7' '7' '7' 169 169 169 167 i68 i66 i66 164 164 163 163 162 162 i6o '59 '59 '59 '57 '57 I56 '55 '55 '55 '54 '53 '53 152 I52 'I5 'I5 I50 I50 '49 '49 I48 '47 '47 '47 146 146 '45 '45 99756 9.99894 754 9.99893 752 9.99892 750 9.9989I 748 9.9989I 99746 9.99890 744 9.99889 742 9.99888 740 9.99887 738 9.99886 99736 9.99885 734 9~.99884 73I 9.99883 729 9.99882 727 9.9988i 99725 9.99880 723 9.99879 721 9.99879 7I9 9.99878 716 9.99877 06993 8.84464 07022 8.84646 051 8.84826 o8o 8.850o6 i'o 8.85i8g 07139 8.85363 i68 8.85540 I97 8.85717 227 8.85893 256 8.86069 07285 8.86243 314 8.864I7 344 8.86591 373 8.8676~ 402__8.86935 0743' 8.87Io6 461 8.87277 490 8.87447 519 8.876i6 548 8.87785 99714 9.99876 7I2 9.99875 710 9.99874 708 9.99873 705 9.99872 99703 9.9.987I 701 9.99870 699 9.99869 696 9.99868 694 9.99867 99692 9.99866 689 9.99865 687 9.99864 685 9.99863 683 9.99862 99680 9.9986i 678 9.99860 676 9.99859 673 9.99858 67I 9.99857 99668 9.99856 666 9.99855 664 9.99854 66i 9.99853 6-59 9.008512 07578 8.87953 607 8.88I20 636 8.88287 665 8.88453 695 8.886i8 07724 8.88783 753 8.88948 782 8.89111 8I2 8.89274 841 8.8943q7 07870 8.89598 899 8.89760 929 8.89920 958 8.90080 987 8.90240 080I7 8.90399 046 8.90557 075 8.907IS 104 8.90872 I34 8.9I029 08163 8.91188 192 8.9I340 221 8.91495 251 8.9i6g0 280 8.9i803 08309 8.9I957 339 8.92110 368 8.92262 397 8.92414 427 8.02.56A 182 i8o i8o '79 I78 '77 '77 176 I76 '74 '74 '74 I72 172 '7' '17' I70 169 169 168 167 167 I66 165 163 163 163 162 i6o i6o i6o '59 '57 '57 156 '55 '55 '55 '53 '54 '53 152 I52 Is' 'I5 ISO I50 '49 148 '49 '47 '47 '47 146 146 1.I5536 I4.30I 1.15354.24I 1.15174 i182 I.14994.124 1.14815.o65 I.14637 I4.008 1.4460 I3.95I I.14283.894 I.I4I07.838 I.I393I.782 I-I3757 I3.727 I.13583.672 I.13409.6I7 1.13237.563 1.13065.510O 31.12894 I3.457 1.I2723.404 1.12553.352 I.12384.300 I.122I5.248 J.12047 I3.1197 mi~i8o.I46 I.II7I3.096 I.11547.046 I.1I382 I2.996 1. 11217 I2.947 1.1I052.898 1.1889.850 I.I0726.8oi I.10563.754 1.10402 I2.7o6 I.10240.659 m~oo8o.6I2 1.09920.566 1.09760.520 1.09601 12.474 I.0944~.429 1.09285.384 1.091128.339 1.08971.295 i.o88i5 12.25I i.0866.207 i.o8509 i163 i.08350.120 1.08197.077 1.08043 I2.035 1.07890 11i.992 1.07738.950 1.07586.909 1.07435.867 1.07284 11.826 1.07I34.785 1.06984.745 i.06835.705 1.06687.664 i.o6538 ii.625 i.o639i.585 i.o6244.546 i.06097.507 I.0595I.468 1.058og.430 60 59 58 57 55 54 53 52 SI 50 49 48 47 46 45 44 43 42 4' 40 39 38 3)7 3 6 35 34 33 3 2 3'I 3 0 29 28 27 26 2 5 24 23 22 2I 20 ' 9 '7 15 '4 ' 3 12 II 109 8 7 6 5 4 3 2 I 0 99657 9.9985I 654 9.99850 652 9.99848 649 9.99847 647 o.Q0846 99644 9.99845 08456 8.92716 642 9.99844 485 8.92866 639 9.99843 1514 8.93016 637 9.99842 I 54 8.93i65 635 9.9984Ij 573 8.933I3 99632 630 627 625 622 619 9.99840 9.99839 9.99838 9.99837 9.99836 9.99834 08602 632 66i 690 720 749 8.93462 8.93609 8.93756 8.93903 8.94049 8.94i95 0 - I - I TNat. COS Log: d. Nat. Sin Log. Nat. Cot Log. c~d. Log. Tan Nat fY 5 0 f Nat. sin Log. d. INat. COS Log.lNat.Tafl Log.1 c.d.ILog. Cot Nat.f 0 I 2 3 4 5 6 7 8 9 'I 12 '3 '4 15 '7 '9 20 21 22 23 24 25 26 27 28 29 3' 32 33 34 36 37 38 39 40 4' 42 43 44 46 -47 48 49 50 5' 52 53 54 55 57 58 59 60 08716 8.94030 745 8.94I74 774 8.943I7 803 8.9446i 831 8.94603 o886o 8.94746 889 8.94887 918 8.95029 947 8.95170 976 8.953110 09005 8.95450 034 8.95589 0)63 8.95728 092 8.95867 121 8.96005 09150 8.96I43 I79 8.96280 208 8.96417 237 8.96553 266 8.96689 09295 8.96825 324 8.96_960 353 8.97095 382 8.97229 411 8.97363 09440 8.97496 469 8.97629) 498 8.97762 527 8.978_94 -55 8.98026 09585 8-98I57 6I4 8.98288 642 8.98419 671 8.98549 700 8.98679 09729 8.98808 758 8.98937 707 8.99066 8i6 8.9_9194 845 8.99322 09874 8.99450 903 8.99577 932 8.99704 961 8.99830 990 8.99956 10019 9.00082 048 9.00207 077 9.00332 io6 9.00456 I35 9.00581 10164 9.00704 192 9.00828 221 9.00951 250 9.01074 279 9.01196 10308 9.01318 337 9.0I440 366 9.0156i 395 9.01682 424 9.01803 453 9.01923 '44 '43 '44 I42 '43 '4' 142 '4' 140 140 '39 '39 '39 138 138 I37 '37 136 136 I36 '35 '35 134 '34 '33 '33 '33 132 I32 '3' '3' '3' I30 130 129 129 129 128 128 128 I27 127 126 I26 I25 125 124 I25 123 124 123 123 122 I22 122 121 121 121 I20 99619 9.998,34 6I7 9.99833 614 9.998,32 612 9.998,3i 609 9.99830 99607 9.99829 604 9.99828 602 9.99827 599 9.99825 596 9.99824 99594 9.99823 591 9.99822 588 9.9,9821 586 9.99820 583 9.998I9 99580 9.998I7 578 _9.9_98i6 575 9.99815 572 9.99814 570 9.998I3 99567 9.998I2 564 9.998i0 562 9.99809 559 9.99808 5, 9.99807 99553 9.99806 55I 9.99804 548 9.99803 545 9.99802 542 9.99801 99540 9.99800 537 9.99798 534 9.99797 53I 9.99796 528 9-99795 08749 8.9419~ 778 8.94340 8o7 8.9448~ 837 8.94630 866 8.94773 08895 8.949I7 9.25 8.95060 954 8.95202 983 8.95,344 09013 8.95486 09042 8.95627 o7i 8.95767 ioi 8.95908 130 8.96047 I59 8.96187 09189 8.9632~ 218 8.96464 247 8.96602 2177 8.96739 306 8.96877 09335 8.97013 365 8.915 394 8 0128c~ 423 8.97421 453 8.97556 09482 8.9769i 5Ii 8.97825 541 8.97959 570 8.98092 6oo 8.9822~ 09629 8.98358 658 8.98490 688 8.98622 7I7 8.98753 746 8.98884 '45 '45 '45 '43 '44 '43 142 142 I42 141 140 '39 140 138 '39 138 '37 138 136 '37 '35 I36 '35 '35 '34 '34 '33 '33 '33 132 132 '3' '3' '3' 130 130 I30 129 128 129 128 127 128 I27 126 126 126 125 125 124 124 124 123 123 123 122 I22 122 1.05805g II.430 I.0566o.392 I.055I5.354 1.05370.316 1.05227.279 I.05o83 II. 242 1.04940.205 I.04798.x68 I.04656 J132 I.045I4.095 I.04373 II.059 I.04233.024 I.040_92 i0.988 1.03953.953 I.03813.9i8 1.03675 i0.883 1.035,36.848 I.03398.814 1.03261.780 1.03123.746 I.02987 IO.7I2 1.02850.678 1.02715.645 I.02579.612 1.02444.579 I.02309 IO.546 I.02175.514 I.02041.481i i.01908.449 I.0I775.4I7 i.01642 I0.385 I.015I0.354 1.01378.322 1.01247.29I i.oiii6.260 I.00985 10.229 I.00855.199 I.0072$ ji68 I.00595 I138 I.00466.io8 I.00338 1O0.078 I.00209.48 i.ooofi.019 0.99954 9.9893 0.99826 6oi 0.99699 9.9310O 0.99573 021 0.99447 9.8734 0.9932I 448 0.99195 164 0.99070 9.7882 0.98945 6oi 0.9882I 322 0.98697 044 0.98573 9.6768 0.98450 9.6493 0.98327 220 0.98204 9.5949 0.98082 679 0.97960 411 0.97838 144 60 59 58 57 54 53 52 -5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' -30 29 28 27 26 25 24 23 22 21 20 '9 '7 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 99526 9.99793 523 9.99792 520 9.99791 51I7 9.99790 -514 9.99788 995II 9.99787 5o8 9.99786 5o6 9.99785 503 9.99783 500 9.99782 99497 9.99781 494 9.99780 491 9.99778 488 9.99777 485 9.99776 99482 9.99775 479 9.99773 476 9.99772 473 9.9977I 470_9.99769 99467 9.99768 464 9.99767 461 9.99765 458 9.99764 455 9.99763 452 9.9976i 09776 8.99oig 805 8.99I45 834 8.99275 864 8.99405 893 8.99534 09923 8.99662 952 8.9979I 981 8.999I19 10011 9.00046 040__9.00174 10069 9.00301 099 9.00427 128 9.00553 I58 9.00679 187 9.00805 10216 9.00930 246 9.01055 275 9.0II79 305 9.011303 334 9.01427 10363 9.01550 393 9.01T673 422 9.0L1796 452 9.019I8 481 9.02040 510 9.02162. Nat. COS Log. d. INat. Sifl Log. lNat. CotLog. c~d. Log.TanlNat~lI 84~ 0 f Nat. Sin Log. d. INat.COS Log.jNat.TanlLog.l c.d. lLog. Cot Nat.I 0 0 I 2 3 4 5 6 7 8 9 10 'I 12 '3 '4 15 i6 '7 20 21 22 23 24 26 27 28 29 30 32 33 34 36 37 38 39 40 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 56 57 58 59 60 10453 9-0I923 48,2 9.02043 511 9.02I63 540 9.02283 569 9.02402 I0597 9.012520 62.6 9.02639 655 9.02757 684 9.02874 713 9.02992 I0742 9.03I09 771I 9.03226 8oo 9.03342 829 9.03458 858 9.03574 10887 9.03690 916 9.03805 945 9.03920 973 9.04034 11002 9.04I49 11031 9.04262 o6o 9.04376 089 9.04490 ii8 9.0460q I47 9.047I5 11176 9.04828 205 9.04940 234 9.05052 263 9.05164 291 9.05275 I1320 9.05386 349 9.05497 378 9.05607 407 9.05717 436 9.05827 II465 9.05937 494 9.06046 523 9.06i55 552 9.06264 -58o 9.06372 11609 9.0648i 638 9.06589 667 9.06696 696 9.06804 725 9.069i1 11754 9.070i8 783 9.07124 812 9.07231 840 9.07337 869 9.07442 11898 9.07548 927 9.07653 956 9.07758 985 9.07863 120)I4 9.07968 12043 9.08072 071 9.08I76 100 9.08280 129 9.08383 158 9.08486 187 9.08589 12 12 12 I I 11 I I II I I I I II I I I I I I I I I I I I I I I I I I II I I I I I I I I I I I I I I I] I I I I I I I]I I I I] I I IC IC IC IC IC IC IC IC IC IC IC IC IC IC IC I( I( I( I( D:0 0 9 99452 449 446 443 440 9.99761 9.99760 9.99759 9.99757 9.99756 10510 9.02162 540 9.02283 569 9.02404 599 9.0252-5 628 9.02645 99437 9.99755 io657 9.027 66 89.95 687 9.02885 7 43I 9.99752 716 9.03005 8 428 9.991751 746 9.03I24 7 424 9.99749 775 9.03242 79942I 9.99748 10805 9.0336I 6 418 9.99747 834 9.03479 6 4I5 9.99745 863 9.03597 6 412- 9.99744 893 9.037I4 6 409 9.99742 922 9.03832 ~599406 9.99741 10952 9.03948.5 402 9.99740 981 9.04065 4399 9.99738 11011I 9.04i8i 396 9.99737 040 9.04297 '5 393 9.99736 070 9.044I3 ~499390' 9.99734 11I099 9.04528.4 386 9.99733 12-8 9.04643 383 9.9973I I58 9.04758 3 380 9.99730 187 9.04873 377 9.99728 217 9.04987:2997 9.99727 II246 9.05101:2 370 9.99726 276 9.052I4 367 9.99724 305 9.05328.2 364 9.99723 335 9.0544I:1 360 9.9972I 364 9.05553:199357 9.99720 11394 9.05666.0 354 9.997i8 423 9.05778:0 351 9.997I7 4529.50:0905o:0 347 9.99716 482 9.0600.2:0 344 9.997I4 511 9.06113 )g9934I 9.997I3 1154I 9.06224 )g 337 9.997II 570 9,.06335 )q 334 9.997I0 600 9.06445 )8 331 9.99708 629 9.06556 327 9.99707 659 9.06666 )8 99324 9.99705 Ii688 9.06775 Y7 320 9.99704 718 9.06885 18 3I7 9.99702 747 9.06994 )7 3I4 9.99701 777 9.07I03 17 310 9.99699 8o6 9.0721I )6 99307 9.99698 11836 9.07320 )7 303 9.99696 865 9.07428 )6 300 9.99695 895 9.07536 )5 297 9.99693 924 9.07643 )6 239-99 954 9.07751 )599290 9.99690 11983 9.07858 )5 286 9.99689 120I3 9.07964 '5 283 9.99687 042 9.08071 )5 279 9.99686 072 9.08I77 '5, 276 9.99684 101 9.08283 121 121 121 120 121 I 19 120 I 19 "I9 "I7 II 8 ii6 I 17 ix6 "15 115 "I4 "I4 "I3 "I4 "I3 I12 "13 112 I12 112 III III III hO0 "II hO0 109 hlO 109 109 io8 109 io8 io8 I07 io8 I07 io6 I07 io6 io6 io6 io6 105 I05 105 I04 0.97838 9.5I44 0.97717. 9.4878 0.97596 614 0.97475 352 0.97355 090 0.97234 9.383I 0.97II$ 572 0.96995 3I5 0.96876 o6o 0.96758 9.2806 0.96639 9.2553 0.9652I 302 0.96403 052 0.96286 9.i803 o.96168 555 0.96052 9.1I309 0.95935 o65 0.958i9 9.082I 0.95703 579 0.95587 338 0.95472 9.oo98 0.95357 8.9860 0.95242 623 0.95I27 387 0.950I3 152 0.94899 8.8919 0.94786 686 0.94672 455 0.94559 225 0.94447 8.7996 0.94334 8.7769 0.94222 542 0.94II0 3I7 0.93998 093 0.93887 8.6870 0.93776 8.6648 0.93665 427 0.93555 208 0.93444 8.5989 0.93334 772 0.93225 8.5555 0.93II5 340 0.93006 126 0.92897 8.49I3 0.92789 70I 0.92680 8.4490 0.92572 280 0.92464 07I 0.92357 8.3863 0.92249 656 0.92142 8.3450 0.92036 245 0.91929 041 0.91823 8.2838 0.91717 636 0.91611 8.2434 0.9I505 -234 0.91400 035 0.9I295 8.i837 0.91190 640 0.91086 443 60 59 57 55 52.50 49 48 47 46 44 43 42 39 38 37 36 34 33 32 3' 29 28 27 26 25 24 23 22 21 290 '7 15 '4 '3 12 'II 1 0 9 8 7 6 5 4 3 2 I 0 '4 '3 )3 99272 269 265 262 258 255 9.99683 9.9968i 9.99680 9.99678 9.99677 9.99675 I2I31 i6o 190 2I9 249 278 9.08389 9.08495 9.08600 9.08705 9.08810 9.o89I4 INat. COS Loa. d. Nat. Sin Log. Nat. Cot Log.~ c.d. Log. Tan Nat.J1 830 70 0 ' Nat. Sin Log. d. Nat. COS Log. Nat.Tan Log. c.d. Log. Cot Nat.r,,,~~~ 1 0 I 2 3 4 5 6 7 8 9 10 II 12 I3 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 12187 9.08589 216 9.08692 245 9-08795 274 9.08897 302 9.08999 1233I 9.09101 360 9.09202 389 9.09304 418 9.09405 447 9.09506 12476 9.09606 504 9.09707 533 9.09807 562 9.09907 591 9.iooo6 12620 9.10106 649 9.10205 678 9.10304 706 9.10402 735 9.10501 12764 9.10599 793 9.I0697 822 9.10795 851 9.10893 880 9.10990 12908 9.11087 937 9.11184 966 9.11281 995 9.11377 13024 9.II474 13053 9.11570 081 9.11666 IIo 9.11761 139 9.11857 i68 9.11952 13197 9.12047 226 9.12142 254 9.12236 283 9.12331 312 9.12425 13341 9.12519 370 9.12612 399 9.12706 427 9.12799 456 9.12892 13485 9.12985 514 9.13078 543 9.13171 572 9.13263 600 9.I3355 13629 9.13447 658 9.I3539 687 9.13630 716 9.13722 744 9.13813 13773 9I.3904 802 9.13994 831 9.1408$ 860 9.14175 889 9.14266 917 9.14356 103 103 102 102 102 IOI 102 IOI IOI I00 IOI I00 I00 99 I00 99 99 98 99 98 98 98 98 97 97 97 97 96 97 96 96 95 96 95 95 95 94 95 94 94 93 94 93 93 93 93 93 92 92 99255 999675 251 9-99674 248 9.99672 244 9.99670 240 9.99669 99237 9.99667 233 9.99666 230 9.99664 226 9.99663 222 9.99661 99219 9.99659 215 9.99658 211 9.99656 208 9.99655 204 9.99653 12278 9.089I4 308 9.09019 338 9.09123 367 9.09227 397 9.09330 12426 9.09434 456 9.09537 485 9.09640 515 9.09742 544 9.09845 12574 9.09947 603 9.I0049 633 9.10150 662 9.10252 692 9.I0353 I05 I04 104 I03 104 I03 103 102 I03 102 102 IOI 102 IOI IOI 10I IOI I00 I00 100 I00 99200 9.9965I 12722 9.10454 197 9.99650 751 9.I0555 193 9.99648 781 9.1o656 189 9.99647 8Io 9.I0756 I86 9.99645 840 9.I0856 _~ 99182 9.99643 178 9.99642 175 9.99640 171 9-99638 167 9.99637 99163 9.99635 i6o 9.99633 156 9.99632 152 9.99630 148 9.99629 99144 9.99627 141 9.99625 137 9-99624 133 9.99622 129 9.99620 99125 9.996I8 122 9.99617 118 9.996I5 II4 9.996I3 1io 9.99612 99106 9.99610 I02 9.99608 098 9.99607 094 9.99605 091 9.99603 99087 9.9960I 083 9.99600 079 9.99598 075 9-99596 071 0.05905 12869 9.10956 899 9.11o56 929 9.1155 958 9.11254 988 9.11353 13017 9.11452 047 9.11551 076 9.11649 io6 9.11747 136 9.11845 13165 9.I943 I95 9.12040 224 9.12138 254 9.I2235 284 9.12332 13313 9.12428 343 9.12525 372 9.12621 402 9.12717 432 9.12813 13461 9.12909 491 9.I3004 52I 9.I3099 550 9.13194 580 9.13289 13609 9.I3384 639 9.13478 669 9.I3573 698 9.I3667 728 9. I3761 99 99 99 99 99 98 98 98 98 97 98 97 97 96 97 96 96 96 96 95 95 95 95 95 94 95 94 94 93 94 93 93 93 93 92 92 93 9I 92 0.91086 8.I443 0.90981 248 0.90877 054 0.90773 8.o860.90670 - 667 0.90566 8.0476 0.90463 285 0.90360 095 0.90258 7.9906 0.90155 718 0.90053 7-953~ 0.8995I 344 0.898o50 58 0.89748 7.8973 0.89647 789 0.89546 7.8606 o.89445 424 0.89344 243 0.89244 062 0.89144 7.7882 0.89044 7-7704 0.88944 525 o.88845 348 0.88746 171 0.88647 7.6996 o.88548 7.6821 0.88449 647 0.88351 473 0.88253 301 0.88I55 129 0.88057 7.5958 0.87960 787 0.87862 6i8 0.87765 449 0.87668 281 0.87572 7.5II3 0.87475 7.4947 0.87379 781 0.87283 615 0.87187 451 0.87091 7.4287 0.86996 124 o.86901 7.3962 o.868o6 800 0.86711 639 o.866i6 7.3479 0.86522 319 0.86427 i6o 0.86333 002 0.86239 7.2844 0.86146 7.2687 0.86052 531 0.85959 375 0.85866 220 0.85773 o66 0.85680 7.I9I2 0.85588 759 0.85496 607 0.85403 455 0.85312 304 0.85220 154 6 5' 5 5 -5' 51 5 5, 5: S5 5' 4' 4: 44 4, 4 4: 4: 4 41 3' 3 3' 3 3 3 3. 3 3 21 2 2' 24 2. 2, 2 2 2 I, I, I' I I, I I I 0 9 8 7 6 5 4 3 2 I 0 9 8 7 5; 5 4 3 2 I 0 9 8 7 6 5 4 3 2 I 0 9 8 7 6 5 4 3 2 I 0 9 8 7 6 5 4 3 2 I 0 9 8 7 6 5 4 3 2 I 0 92 9 99067 9-99593 13758 9-13854 9 063 9-9959I 787 9.13948 9 059 9.99589 817 9.14041 055 9.99588 846 9.I4I34 I 051 9.99586 876 9.14227 -1 76 9-I4227~~~~~~~~~~~~..... 90 9I 90 91 90 99047 043 039 035 03I 027 9.99584 9.99582 9.9958I 9.99579 9.99577 9.99575 13906 935 965 995 14024 054 9.14320 9.14412 9.I4504 9.I4597 9.14688 9.14780 I _. _ I~~ INat. COSLog. d. |Nat. Sin Log. Nat.COt Log. c.d. ILog.TanNat. ' 1 82~ I F Nat. Sin Log. d. INat. COS Log.INat.Tan Log.j c.d. lLog. Cot Nat. 0 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 15 i6 '7 i8 '9 20 21 22 23 24 295 26 27 28 29 30 3' 32 33 -34_ 35 36 37 38 39 40 42 43 44 45 46 47 48 49 50j 5' 52 53 54 55 57 58 59 60 13917 9.i435 946 9.114445 975 9.14535 14004 9.14624 033 9.114714 14061 9.114803 090 9.14891 119 9.14980 148 9.15069 177 9.15157 I4205 9.15245 234 9.15333 263 9.15421 292 9.15508 320 9.15596 I4349 9.15683 378 9.15770 407 9.15857 436 9.115944 464 9.16030 I4493 9.1i611 522 9.16203 551 9.i6289 58o 9.16374 6o8 9.16460 14637 9.16545 666 9.16631 695 9.1676 723 9.16801 752 9.i6886 14781 9.i6970 8io 9.17055 838 9.17139 867 9.17223 896 9.17307 14925 9.17391 954 9.17474 982 9.17558 I50II 9.17641 040 9.117724 15069 9.17807 097 9.1789o 126 9.179731 155 9.18055 184 9.18137 15212 9.i8220 241 9.18302 270 9.i8383 299 9.18465 327 9.18547 15356 9.i8628 385 9.i8709 4I4 9.1I8790 442 9.i887I 471 9.18952 15500 9.19033 529 9.19I113 557 9.1191193 586 _9.19273 6I5 9.19353 643 9.119433 89 90 89 90 89 88 89 89 88 88 88 88 87 88 87 87 87 87 86 86 87 86 86 86 84 84 84 84 84 83 84 83 83 83 83 83 82 82 83 82 82 82 82 83 82 8o 8o 99027 9.99575 023 9.99574 019 9.99572 015 9.99570 OIl 9.99568 99006 9.99566 002 9.99565 98998 9.99563 994 9.9956i 990 9.99559 I4054 9.14780 084 9.14872 113 9.114963 I43 9.15054 I73 9.115145 14202 9.15236 232 9.15327 262 9.15417 29I1 9.15508 32I 9.15598 98986 9.99557 I435I 9.15688 982 9.99556 381 9.115777 978 9.99554 410 9.15867 973 9.99552 440 9.I5956 969 9.99550 470 9.16046 98965 9.99548 14499 9.16135 961 9.99546 529 9.16224 957 9.99545 559 9.163I2 953 9.99543 588 9.164011 948 9.9954I 6I8 9.16489 98944 9.99539 I4648 9.i6577 940 9.99537 678 9.16665 936 9.99535 707 9.16753 93I 9.99533 737 9.1684-1 927 9.99532 767 9.16928 98923 9.99530 14796 9.I70I6 9I9 9.99528 826 9.17103 914 9.99526 856 9.17190 910 9.99524 886 9.I7277 906 9.99522 915 9.17363 98902 9.99520 I4945 9.17450 897 -9.995i8 975 9.17536 893 9.99517 15005 9.17622 889 9.99515 '034 9.17708 -884 9.99513 064 9-I7794 98880 9.99511 15094 9.117880 876 9.99509 124 9.17965 871 9.99507 I53 9.i805I 867 9.99505 183 9.i8I36 863 9.99503 213 9.18221 98858 9.9950I 15243 9.i8306 854 9.99499 272 9.1839I 849 9.99497 302 9.i8475 845 9.99495 332 9.18560 84I 9.99494 362 9.i8644 98836 9.99492 I539I 9.18728 832 9.99490 42I 9.188I2 827 9.99488 45I 9.18896 823 9.99486 48I 9.18979 8i8 9.99484 511 9.i9063 988I4 9.99482 I5540 9.19146 809 9.99480 570 9.19229 8o5 9.99478 6oo 9.19312 8oo 9.99476 630 9.19395 796 9.99474 66o 9.I9478 92 9' 9' 9' 9' 9' 9' 89 90 89 90 89 89 88 89 88 88 88 88 88 87 88 87 87 87 86 87 86 86 86 86 86 86 85 85 85 84 84 84 84 84.83 84 83 83 83 83 83 83 82 82 82 82 82 0.85220 7.11I54 0.85I28 004 0.85037 7.0855 0.84946 706 o.8485g 5 0.84764 7.0410 0.84673 264 0.8458,3 117 0.84492 6.9972 0.84402 827 0.84312 6.9682 0.84223 538 0.84I33 395 0.84044 252 0.83954 110 o.8386~ 6.8969 0.83776 828 0.83688 687 0.83599 548 0.83511 408 0.83.423 6.8269 0.83339 131 0.83247 6.7994 0.83159 856 0.83072 720 0.82984 6.7584 0.82897 448 0.82810o 3I3 0.82723 I79 0.82637 045 0.825~o 6.6912 0.82464 779 0.82378 646 0.82292 5I4 0.82206 383 0.82120 6.6252 o.8203S 122 0.81949 6.5992 0.8i864 863 0.8I779 734 0.8i694 6.56o6 0.8i609 478 0.8I525 350 0.8i440 223 0.81356 097 0.81272 6.4971I o.8ii88 846 0.81104 721 0.81021 596 0.809 37 472. o.8o854 6.4348 0.8077I 225 0.80688 103 o.80ogo 6.3980 0.80522 859 0.80439 6.3737 0.80357 6I7 0.80275 496 0.80193 376 o.8oiii 257 0.80029 138 60 59 57 55 54 53 52 -5' 50 49 48 47 46 44 43 42 4' 40i 39 38 37 36 34 33 32 3' 30 29 28 27 26 24 23 22 21 20 '9 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 98791 9.99472 787 9.99470 782 9.99468 778 _9.99466 773 9.99464 769 9.99462 15689 9.1956i 7I9 9-I9643 749 9.I9725 779 9.I9807 809 9.i9889 838 9.I997I 0 lNat. COSLog. d. INat. S in Log.]INat. C ot Log.Ic.d. I og. T an Nat.Il I 810 9 0 f 0 fNat. Sin Log. d. Nat. CosLog-jNat.'TanlLog.1 c.d. Log. Cot Nat.T 0.0 I 2 3 4 S 6 7 8 9 10 II 12 '3 '4 15 i6 '7 22 23 24 26 27 28 29 30 3' 32 33 34 35 36 37 38 39. 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 56 57 59 60 I5643 9.I9433 672 9.195I3 701 9.I9592 730 9.19672 758 9.1975i 15787 9.19830 8i6 9.i19909 845 9.i9988 873 9.20067 902 9.20145 I593I 9.20223 959 9.20302 988 9.20380 16017 9.20458 046 9.20535 16074 9.20613 103 9.20691 132 9.20768 i6o 9.20845 189 9.20922 16218 9.20999 246 9.21076 275 9.2II53 304 9.2I229 333 9.2I306 16361 9.2I382 390 9.2I458 419 9.2I534 447 9.2i6i0 476 9.21685 16505 9.2I761 533 9.2I836 562 9.2I912 591 9.21987 620 9.22062 76648 9.22I37 677 9.22211 706 9.22286 734 9.22361 763 9.22435 16792 9.22509 820 9. 22583 849 9. 2265,7 878 9. 22731 906 9. 22805 16935 9.22878 964 9.22952 992 9.23025 I702I 9.23098 050 9.23171 17078 9.2,3244 I07 9.23317 I36 9.23390 164 9.23462 193 9.23535 17222 9.23607 250 9.23679 279 9.23752 308 9.23823 336 9.23895 365 9.23967 8o 79 80 79 79 79 79 79 78 78 79 78 78 77 78 78 77 77 77 77 77 77 76 77 76 76 76 76 75 76 75 76 7 5 75 75 74 75 75 74 74 74 74 74 74 73 74 73 73 73 73 73 73 72 73 72 72 73 7' 72 72 98769 9.99462 764 9.99460 760 9.99458 755 9.99456 75I 9.99454 98746 9.99452 74I 9.9-9450 737 9.99448 732 9.99446 728__9.99444 98723 9.99442 71I8 9.99440 714 9.99438 709 9.99436 704 0.00434 I 15838 9.19971 868 9.20053 898 9.20134 928 9.20216 958 9.20297 15988 9.20378 16017 9.20459 047 9.20540 077 9.2062I 107 9.20701 16137 9.20782 167 9.20862 196 9.20942 226 9.21022 256 0.2II02 98700 9.99432 695 9.99429 690 9.99427 686 9.99425 68i 9.99423 98676 9.9942I 671 9.99419 667.9.99417 662 9.994I5 657 9.99413 98652 9.994II 648 9.99409 643 9.99407 638 9.99404 633 9.99402 98629 9.99400 624 9.99398 619 9.99396 6I4 9.99394 609 9.99392 98604 9.99390 6oo 9.99388 595 9.99385 590 9.99383 585 9.9938I 98580 9.99379 575 9.99377 570 9.99375 56 9.99372 56i 9.99370 16286 9.2Ii82 316 9.21261 346 9.2I34I 376 9.2I420 405 9.2I499 16435 9.2I578 465 9.21657 495 9.21736 525 9.2i8I4 555 9.2i893 16585 9.2197I 6I5 9.22049 645 9.22I2 7 674 9.22205 704 9.22283 16734 9.22361 764 9.22438 794 9.225i6 824 9.22593 854 9.22670 16884 9.22747 914 9.22824 944 9.22901 974 9.22977 I7004 9.23054 I7033 9.23130 063 9.2,3206 093 9.23283 123 9.23359 I53 9.23435 82 82 82 8i 8o 8o 8o 8o 8o 8o 8o 79 79 79 79 78 79 78 78 78 78 78 78 77 78 77 77 77 77 77 76 77 76 77 76 76 75 76 75 76 75 75 75 75 75 74 75 74 75 74 74 74 0.80029 6.3138 0.79947 019 0.79866 6.290I 0.79784 783 0.79703 666 0.79622 6.2549 0.79541 432 0.79460 316 0.79379 200 0.79299 085 0.792i8 6.I970 0.79I38 856 0.79058 742 0.78978 628 0.78898 5I5 0.788i8 6.I402 0.78739 290 0.78659 178 0.78580 o66 0.78501 6.095 5 0.78422 6.o844 0.78343 734 0.78264 624 0.78186 514 0.78I07 405 0.78029 6.0296 0.7795I i88 0.77873 o8o 0.77795 5.9972 0-777I7 865 0.77639 5.9758 0.77562 65I 0.77484 545 0.77407 439 0.77330 333 0.77253 5.9228 0.77176 124 0.77099 019 0.77023 5.89I5 0.76946 8ii 0.76870 5-.8 70 8 0.76794 605 o.767I7 502 0.7664I 400 0.76565 298 0.76490 5.81I97 0.764I4 095 0.76339 5.7994 0.76263 894 0.76i88 794 0.76I13 5.7694 0.76038 594 0.75963 495 0.75888 396 o.758I4 297 0.75739 5.7I99 0.75665 101 0.75590 004 0.755i6 5.6906 0.75442 809 0.75368 71 60 59 58 57 55 -54 53 52 SI 50 49 48 47 46 45 44 43 42 4' 4o6 39 38 37 36 35 34 33 32 3' 29 28 27 26 295 24 23 22 21 20 '9 '7 i6 15 14 '3 12 10 9 8 7 6 4 3 2 I 0 98556 9.99368 1783 9.235I0 551 9.99366 213 9.23586 546 9.99364 243 9.2366i 541 9.99362 273 9.23737 536 9.99359 303 9.23812 98531 9.99357 17333 9.23887 526 9.99355 363 9.23962 521 9.99353 393 9.24037 5i6 9.9935I 423 9.24II2 511 9.99348 453 9.24I86 98506 501 496 49' 486 481 9.99346 9.99344 9.99342 9.99340 9.99337 9.99335 I7483 513 543 573 603 633 9.2426i 9.24335 9.244I0 9.24484 9.24558 9.24632 INat. COS Log. d. INat. Sin Log~lNat. Cot Log. Ic.dJ Log. Taflnat I 800 10 0 __ _ INat. sin Log. d. INat. COS Log. d. INat.TanlLog. ~c.d.ILg o a. 0 I 2 3 4 -5 6 7 8 9 10 II 12 '3 '4 15 '7 i8 '9 21 22 23 24 26 27 28 29 30 3' 32 33 34 36 37 38 39 -40 4' 42 43 44 46 -47 48 49 5' 52 53 54 56 57 59 60 17365 9.23967 393 9.24039 422 9.24II0 451 9.24I8I 479 9.24253 I75o8 9.24324 537 9.24395 565 9.24466 594 9.24536 623 9.24607 I7651 9.24677 68o 9.24748 708 9.248i8 737 9.24888 766 9.24958 I7794 9.25028 823 9.25098 852 9.25i68 88o 9.25237 909 9.25307 I7937 9.25376 966 9.25445 995 9.25514 18023 9.25583 052 9.25652 i8o8i 9.2572I I09 9.25790 138 9.25858 i66 9.25927 I95_9.25995 182 24 9.26063 252 9.26I31 281 9.26i99 309 9.26267 338 9.26335 18367 9.26403 395 9.26470 424 9.26538 452 9. 26605 481 9.26672 18509 9.26739 538 9.26806 567 9.216873 595 9.26940 624 9.27007 18652 9.27073 68i 9.27140 710 9.27206 738 9.27273 767 9.27339 18795 9.27405 824 9.27471 852 9.27537 88i 9.27602 910 9.27668 1c8938 9.27734 967 9.27799 995 9.27864 I9024 9.27930 052 9.27995 o8i 9.28060 72 7' 7' 72 7' 7' 7' 70 7' 70 7' 70 70 70 70 70 70 69 70 69 69 69 69 69 69 69 68 69 68 68 68 68 68 68 68 67 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 65 65 66 65 65 98481I 9.99335 476 9.99333 47I 9.99331 466 9.99328 461 9.99326 98455 9.99324 450 9.99322 445 9.993I9 440 9.99317 435_9.99315 98430 9.99313 425 9-9913I0 420 9.99308 414 9.99306 409_9.99304 98404 9.9930I 399 9.99299 394 9.99297 389 9.99294 383 9.99292 98378 9.99290 373 9.99288 368 9.99285 362 9.99283 -357 9.99281 98352 9.99278 347 9.99276 341 9.99274 336 9.99271 -33I 9.99269 98325 9.99267 320 9.99264 315 9.99262 310 9.99260 -304 9.99257 98299 9.99255 294 9.99252 288 9.99250 283 9.99248 __277 9.99245 98272 9.99243 267 9.9924I 261 9.99238 256 9.99236 250 9.99233 98245 9.99231 240 9.99229 234 9.99226 229 9.99224 223 9.9922I 98218 9.992i9 212 9.99217 207 9.99214 201 9.992I2 196 9.99209 98190 9.99207 I85 9.99204 I79 9.99202 I74 9.99200 168 9.99197 163 9.99195 2 2 3.2 2 2 3 2 2 2 3 2 2 2 3 2 2 3 2:2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 3 2 2 3 2 23 2 3 2 3 2 3 2 2 3 2 I7633 9.24632 663 9.24706 693 9.24779 723 9.24853 753 9.24926 I7783 9.25000 813 9.25073 843 9.25146 873 9.25219 903 9.25292 I7933 9.25365 963 9.25437 993 9.255I0 18023 9.25582 0,53 9.25,655 18083 9.25727 113 9.25799 143 9.2587I:173 _9.25943 203 9.26015 18233 9.26086 263 9.26158 293 9.26229 323 9.2630i 353 9.26372 18384 9.26443 414 9.26514 444 9-2658$ 474 9.26655~ __504 9.26726 18534 9.26797 564 9.26867 594 9.26937 624 9,27008 654 9.27078 18684 9.27148 714 9.272i8 745 9.27288 775 9.27357 805 9.27427 18835 9.27496 865 9.27566 895 9.27635 925 9.27704 955 9.27773 18986 9.27842 19016 9.279T1I 046 9.27980 076 9.28049 io6 9.28iI7 19136 9.28i86 i66 9.28254 197 9.28323 227 9.2839I 257 9.28459 I9287 9.28527 317 9.28595 347 9.28662 378 9.28730 408 9.2,879)8 438 9.28865 74 73 74 73 74 73 73 73 73 73 72 73 72 73 72 72 72 72 72 7' 72 7' 72 7'1 7' 7' 70 7' 7' 70 70 7' 70 70 70 70 69 70. 69 70 69 69 69i 69 69 69 68 69 68 69 68 68 68 68 67 68 68 67 0.75368 5.6713 0.75294 6I7 0.752-21 521 0.75147 425 0.75074 329 0.75000 5-.6234 0.74927 I40 0.74854 045 0.74781 5.5951I 0.74708 857 0.74635 5.5764 0.74563 671 0.74490 578 0.744i8 485 0.74345 393 10.74273 5.5301 0.74201 209 0.74129 II8 0.74057 026 0.73985 5.4936 0.739I4 5.4845 0.7,3842 755 0.73771 605 0.73699 575 0.73628 486 0.73557 5.4397 0.73486 3o8 0.734y5 219 0.73345 13I 0.73274 043 0.73203 5.3955 0.73I33 868 0.73063 781 0.72992 694 0.72922 6o7 0.72852 5.3521 0.72782 435 0.727I2 349 0.72643 263 0.72573 I78 0.72504 5.3093 0.72434 oo8 0.72365 5.2924 0.72296 839 0.72227 755 0.72158 5.2672 0.72089 588 0.72020 505 0.7I95' 422 0.71883 339 0.718I4 5.2257 0.7i746 I74 0.7I677 092 0.71609 OIl 0.7I541 5.I929 9.7I47~ 5.1848 0.7I405 767 0.7I338 686 0.7I270 6o6 0.7I202 526 0.7II35 446 60 59 57 -56 54 53 52 5' 50 49 48 47 46 45 44 43 42 4' -46 39 38 37 36 34 33 32 3' 29 28 27 26 25 24 23 22 21 20 '9 '7 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 lNat.COS Log. d. Nat. Sifl Log. d. Nat. COtLog. c.d.lLog.TaflNat.1 I 79 0 _ _ 1 1 I ' Nat. Sin Log. d-. Nat. COS Log. dJ.Nat.TanlLog. 1c.d. Log. C ot Nat.I I 0 I 2 3 4 5 -6 7 8 9 10 II 12 '3 '4 15 '7 20 21 22 23 24 25 26 27 28 30 32 33 34 -57 36 37 38 39 4' 42 43 44 45 46 47 48 49 50 5I 52 53 54 5-5 56 57 58 19081I 9.28060 109 9.28I25 138 9.28i90 167 9.28254 I95 9.283I9 19224 9.28384 25~2 9.28448 28I 9.285I2 309 9.28577 -338 9.28641 19366 9.28705 395 9.28769 423 9.28833i 452 9.28896 481 9.28960 19509 9.29024 538 9.29087 566 9.29150 595 9.29214 623 9.29277 19652 9.29340 68o 9.29403 709 9.29466 737 9.29529 766 9-2959I I9794 9.29654 823 9.29716 85I 9.29779 88o 9.2984I 908 9.29903 I9937 9.29966 965 9.30028 994 9.30090 20022 9.30151 051 9.302I3 20079 9.30275 io8 9.30336 136 9.30398 i65 9.30459 193 9.3052I 20222 9.305q82 250 9.30643 279 9.30704 307 9.30765 336 9.30826 20364 9.30887 393 9.30947 42I 9.31008 450 9-3I068 478 9.3II29 20507 9.3I189 535 9.3I250 563 9.3I3I0 592 9.3I370 620 9.3I430 20649 9.31490 677 9.3I549 706 9.31609 734 9.31669 763 9.31728 79I 9-3I788 I 64 65 64 64 64 64 64 64 63 64 64 63 63 64 63 63 63 63 63 62 63 62 63 62 62 63 62 62 62 62 62 6i 62 62 6o 6o 6o 6o 6o 6o 6o 59 6o 6o 59 6o 981i63 9.99195 157 9.99I92 152 9.99190 I46 9.99i87 140 9.99i85 98135 9.99182 129 9.99180 124 9.99177 ii8 9.99I75 112 9.99172 98I07 9.99I70 i01 9.99167 096 9.99i65 090 9.99162 084 9.99i60 98079 9.99I57 073 9.99I55 067 9.99I52 061 9.99I50 o56 9.99147 98050 9-99I45 044 9.99I42 039 9.99I40 033 9.99137 027 9.99I35 98021 9.99I32 oi6 9.99I30 010 9.99127 004 9.99124 97998 9.99122 97992 9.99II9 987 9.9911I7 981 9.99114 975 9.99II2 969 9.99i09 97963 9.99i06 958 9-99I04 952 9.99I10 946 9.99099 940 9.99096 97934 9.99093 928 9.99091 922 9.99088 916 9.99086 910 9.99083 97905 9.99080 899 9.99078 893 9.99075 887 9.99072, 88i 9.99070 97875 9.99067 869 9.99064 863 9.99-062, 857 9.99059 85I 9.99056 97845 9.99054 839 9.9905I 833 9.99048 827 9.99046 821 9.99043 8I5 9.99040 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 3 2 3 2 3 2 3 3 2 3 2 3 3 2 19438 9.28865 468 9.28933 498 9.29000 529 9.29067 559 9.29I34 19589 9.2920I 619 9.29268 649 9.29335 68o 9.29402 710 9.29468 I9740 9.29535 770 9.29601 8oi 9.29668 831 9.29734 86i 9.29800 19891 9.29866 92I 9.29932 952 9.29998 982 9.30064 20012 9.30I30 20042 9.30195. 073 9.3026i 103 9.30326 I33 9.30391 164 9.30457 20194 9.30522 -224 9.30587 254 9.30652 285 9.30717 3I5 9.30782 20345 9.30846 376 9.3091 I 406 9.30975 436 9.31040 466_9.31104 20497 9..31168 527 9.31233 557 9.31297 588 9.3136i 6i8 9.31425 20648 9.31489 679 9.31552 709 9.3i6i6 739 9.31679 770 9.3I743 20800 9.3i806 830 9.31870 86I 9.3I933 891 9.31996 921 9.32059 20952 9.32122 982 9.32185~ 21013 9.32248 043 9.323II 073 9.32373 21104 9.32436 134 9.32498 164 9.3256i 195 9.32623 225 9.32685 256 9.32747 68 67 67 67 67 67 67 67 66 67 66 67 66 66 66 66 66 66 66 665 66 65 64 65 64 64 64 64 64 64 64 63 64 63 64 63 64 63 63 63 63 63 63 63 62 63 62 63 62 62 62 I 0.13 i.446 0.7I067 366 0.7I000 286 0.70933 207 0.70866 128 0.70799 5.1I049 0.70732 5.0970 0.7066 892 0.705983 814 0.70532 736 0.70465 5.0658 0.70399 58I 0.70332 504 0.70266 427 0.70200 350 0.70134 5.0273 0.70068 I97 0.70002 121 0.69936 045 0.69870 4.9969 0.69805 4.9894 0.69739 819 0.69674 744 0.69609 669 o.69543 594 0.69478 4.9520 0.69413 446 0.69348 372 0.69283 298 0.69218 225 0.69154 4.9152 0.69089 078 0.69025 oo6 0.68960 4.8933 0.68896 86o 0.68832 4.8788 0.68767 716 0.68703 644 0.68639 573 o.68575 501 0.6851 I4.8430 o.68448 359 0.68384 288 0.6832I 218 0.68257 147 0.68194 4.8077 o.68130 007 0.68067 4.7937 0.68004 867 0.6794I 798 0.67878 4.7729 0.67815i 659 0.67752 59I 0.67689 522 0.67627 453 0.67564 4.7385 0.67502 317 0.67439 249 0.67377 i8i 0.673I5 114 0.67253 046 60 59 58 57 55 -54 53 52 -5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 -36 35 34 33 32 3' 30 29 28 27 265 25 241 23 22 21 20 '9 '7 1i5 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 I I I 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 i I I: lNat.CosLog. d. INat. Sin Log. d. iNat. COtLog. c.d. Log.TaflNat.1 ' 0 780 m 120 f Nat. Sin Log. d. INat. COS Log. d. INat.TanlLog.1c.d.lLog. COt Nat, 0 I 2 3 4 -w 6 7 8 9 10 II 12 '3 '4 15 i6 '7 Ii8 '9 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 36 37 38 39 4 I 42 43 44 45 47 48 49 -5' 52 53 54 55 -56 57 59 60 2079I 9.3I788 820 9.31847 848 9.31907 877 9.3i966 905 9.32025 20933 9.32084 962 9.32143 990 9.32202 2I019 9.3226i 047 9.323I9 2i076 9.32378 104 9.32437 132 9.32495 i6i 9.32553 189 9.326I2 21218 9.32670 246 9.32728 275 9.32786 303 9.32844 33I 9.32902 2I360 9.32960 388 9.33018 417 9.33075 445 9.33133 474 9.33I90 21502 9.33248 530 9.33305 559 9.33362 587 9.33420 6i6 9.33477 21644 9.33534 672 9.33591 70I 9.33647 729 9.33704 758 9.3376i 21786 9.338i8 814 9.33874 843 9.33931 87I 9.33987 899 9.34043 2I928 9.34100 956 9.34156 985 9.3421I2 22013 9.34268 041 9.34324 22070 9.34380 098 9.34436 126 9.3449i 155 9.34547 183 9.34602 22212 9.34658 240 9.34713 268 9.34769 297 9.34824 325 9.34879 22353 9.34934 382 9.34989 410 9.35044 438 9.35099 467 9.35154 495 9.35209 59 6o 59 59 59 59 59 59 59 59 59 58 57 57 57 57 58 57 57 57 57 57 57 57 57 55 55 55 55 55 55 55 55 55 55 55 97815 9.99040 809 9.99038 803 9.99035 797 9.99032 79I 9.99030 97784 9.99027 778 9.99024 772 9.99022 766 9.990I9 760 9.99016 97754 9-990I3 748 9.99011 742 9.99008 735 9.99005 729 9.99002 97723 9.99000 7I7 9.98997 711 9.98994 705 9.9899i 698 9.98989 97692 9.98986 686 9.98983 68o 9.98980 673 9.98978 667 9.98975 97661 9.98972 655 9.98969 648 9.98967 642 9.98964 636 9.9896i 97630 9.98958 623 9.98955 617 9.98953 6ii 9.98950 604 9.98947 97598 9.98944 592 9.98941 585 9.98938 579 9.98936 573 9.98933 97566 9.98930 560 9.98927 553 9.98924 547 9.9892I 54I 9.98919 97534 9.989i6 528 9.98913 521 9.9)8910 S's 9.98907 5o8 9.98904 97502 9.98901 496 9.98898 489 9.98896 483 9.98893 476 9.98890 97470 9.98887 463 9.98884 457 9.9888i 450 9.,98878 444 9.98875 437 9.98872 2 3 3 2 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 2 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 2 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 21256 9.32747 286 9.328I0 316 9.32872 347 9.32933 377 9.32995 2I408 9.33057 438 9.33119 469 9,33i80 499 9.33242 529 9.33303 -2I560 9.33365 590 9.33426 62I 9.33487 65I 9.33548 682 9.33609 21712 9.33670 743 9-3373I 773 9.33792 804 9.33853 834 9.339I3 21864 9.33974 895 9.34034 925 9.34095 956 9.34155 986 9.342I5 22017 9.34276 047 9.34336 078 9.34396 io8 9.34456 I39 9.345i6 22169 9.34576 -200 9.34635 231 9.34695 261 9.34755 292 9.348I4 22322 9.34874 353 9.34933 383 9.34992 414 9.3505I 444 9.35"'I 22475 9.35170 505 9.35229 536 9.35288 567 9.35347 597 9.35405 22628 9.35464 658 9.35523 689 9.3558i 719 9.35640 750 9.35698 22781 9.35757 8ii 9.358i5 842 9.35873 872 9.3593I 903 -9.35989 22934 9.36047 964 9.36i05 995 9.36i63 23026 9.3622I o56 9.36279 087 9.36336 63 62 62 62 62 62 62 62 6i 6o 6o 6o 6o 6o 6o 6o 6o 6o 6o 6o 59 6o 59 59 59 6o 59 59 59 59 58 59 59 59 58 59 58 58 57 0.67253 4.7046 o.67I90 4.6979 0.67128 912 0.67067 845 o.67oo9 779 0.66943 4.67I2 o.6688i 646 0.66820 580 o.66758 514 0.66697 448 o.6663~ 4.6382 0.66574 3I7 o.665I3 252 0.66452 187 0.6639I 122 0.66330 4.6057 0.66269 4.5993 0.66208 928 0.66147 864 0.66087 800 o.66026 4.5736 0.65966 673 o.659o~ 609 o.65849 546 o.65789 483 0.65724 4.5420 0.65664 357 0.65604 294 0.65544 232 0.65484 169 0.65424 4.51I07 0'6536$ 045 0. 65305 4.4983 0.65245 922 o.65i86 86o 0.65126 4.4799 0.65067 737 0.65008 676 0.64949 6I5 0.64889 555 0.64830 4.4494 0.6477I 434 0.64712 373 0.64653 313 o.64599 253 0.64536 4.4I94 0.64477 I34 0.644i9 075 0.64360 015 0.64302 4.3956 0.64243 4.3897 o.64I8~ 838 0.64127 779 0.64069 72I 0.64011 662 0.63953 4.3604 0.63895 546 0.63837 488 0.63779 430 o.63721 372 0.63664 315 60 59 57 -56 54 53 52. SI 50 49 48 47 44 43 42 4' 40f 39 38 37 36 34 33 32 3'0 29 28 27 26 25 24 23 22 21 20 '9 '7 -15 '4 '3 12 II 10 9 8 7 6 4 3 2 I 0 INat. COS Log. d. Nat. Sin Log. d. NatOCOtLog. c~d.Log.TaflNat.1 770 m 13~ ' Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog. c.d. Log. Cot Nat.l 0[22495 9.35209 97437 9.98872 23087 9.36336 0.63664 4.33I5 60 I 523 9.35263 5 430 9-98869 2 117 9.36394 58 o.63606 257 59 2 552 9.353I8 424 9-98867 I48 9.36452 o.63548 200 58 3 580 9.35373 55 4I7 9.98864 3 I79 9.36509 57 0.63491 143 57 4 608 9.35427 54 4I 9.9886 3 209 9.36566 57 0.63434 o86 56 54 3 5326379.5835488 5 22637 93548I 5 97404 998858 23240 9.36624 0.63376 4.3029 55 6 665 9-35536 55 398 9.98855 3 27I 9.36681 57 o.63319 4.2972 54 7 693 9.35590 54 39I 998852 3 30I 936738 0.63262 9I6 53 8 722 9.35644 54 384 9.98849 3 332 9.36795 57 0.63205 859 52 9 750 9.35698 378 9.98846 3 363 936852 5 0.63148 803 51 - 2378 54 3 57 1022778 9-35752 5 9737I 9.98843 23393 9.369095 o.6309I 4.2747 50 I 807 9.35806 54 365 9.98840 3 424 9.36966 57 0.63034 691 49 I2 835 9.35860 54 358 9.98837 3 455 9.37023 570.62977 635 48 I3 863 9-359I4 54 351 9.98834 - 485 9.37080 57 0.62920 580 47 I4 892 9.35968 54 345 9.98831 3 56 9.37I37 5 0.62863 524 46 15 254 3 56 - 15 22920 9.36022 97338 9.98828 23547 9.37193 0.62807 4.2468 45 i6 948 9.36075 53 33 9.98825 3 578 9-37250 5 0.6270 4I3 44 I7 977 9-36129 54 325 9.98822 608 9.37306 0.62694 358 43 18 23005 9.36182 53 318 9.988I9 3 639 9.37363 0.62637 303 42 I9 033 9.36236 1 3 31 9.98816 3 670 9.37419 0.6258I 248 4I 20 23062 9.36289 5397304 9.98813 3 23700 9.37476 5 0.62524 4.2I93 40 2I 090 9.36342 298 9.98810 3 731 9.37532 6 0.62468 I39 39 22 i8 9.36395 291 9.98807 3 762 9.37588 56 0.62412 o84 38 23 146 9-36449 54 284 9.98804 3 793 937644 56 o.62356 ~3~ 37 24 I75 9.36502 53 278 9.98801 3 823 9.37700 56.62300 4976 3 2523203 9-36555 97271 9.98798 23854 9.-37756 0.62244 4.1922 35 26 23I 9.36608 53 264 9.98795 885 9.37812 0.62188 868 34 27 260 9.36660 52 257 9.98792 916 9.37868 0.62132 8I4 33 28 288 9.3673 53 251 9.98789 946 9.37924 6 0.62076 760 32 29 316 9.36766 53 244 9.98786 3 977 9.37980 5 0.62020 706 3 o.619 4. 653 30 30 23345 9-368i9 2 97237 9-98783 24008 9.38035 56 31 373 9-36871 5 230 9.98780 3 0399.3809.6i909 600 29 32 401 9.36924 53 223 9-9877 3 o69 9.3847 5 0.61853 547 28 33 429 9.36976 5217 9.98774 3 I00 9.38202 55 o0.6I798 493 27 34 458 9.37028 52 2I0 9.9877I 3 I3I 938257 5 o.6I743 44I 26 3523486 9.3708i 5 97203 9.98768 24162 9.38313 5.6i687 4-I38825 36 5I4 9-37I3. 52 i96 9-98765 3 I93 9.38368 55 0.61632 335 24 37 542 9.37185 52 I89 9.98762 3 223 9.38423 56 0.61577 28223 38 57I 9.37237 I82 9.98759 3 254 9.38479 56 0.61521 230 22 39 599 9.37289 52 176 9.98756 3 285 9.38534 55 0.61466 I78 2I 40 23627 9.37341 297169 9-98753 24316 938589 55.614I 4.1126 20 41 656 937393 52 162 9.98750 3 347 9.38644 55 0.61356 07419 42 684 937445 52 I55 998746 377 9.38699 55 0.6130 022 i8 43 712 9.37497 52 I48 998743 3 48 9.38754 54 0.61246 4.0970 17 44 740 9 37549 I41 9.98740 3 439 9.38808 o.6iI92 9i8 i6 45 23769 9.37600 52 97I34 9.98737 24470 9.38863 55 o.6II37 4.0867 15 46 797 9-37652 15 I27 9.98734 3 5oi 9-3898 54 0.61082 8i5 14 47 825 9.37703 I20 9.9873I 3 532 9.38972 55 0.61028 764 13 48 853 9-37755 II3 9.98728 3 562 9.39027 55 o.60973 7I3 I2 49 882 9.37806 I52 6 9.98725 3 593 9.39082 | 0.60918 662 II 50 239 9.37858 5 9700 9.987223 24624 9.39136.60864 4.06II 10 51 938 9.37909 5 093 9.9879 3 655 939905 0.6o8io 560 9 52 966 9.37960 5I o86 9.98715 4 686 9.39245 54 o.60755 5o9 8 53 995 9-380I11 5I 79 9.98712 3 7I7 9.39299 54 0.60701 459 7 54 24023 9.38062 0 o72 9.98709 747 9.39353 54.60647 408 6 55 24051 9.38113 5I 97065 9.98706 3 24778 9.39407 540.60593 4.0358 5 56 079 9.38164 I o58 9.98703 3 8o0 9-39461I 5 0.60539 308 4 57 io8 9.3826I5 I 05I 9.98700 3 840 9395I 54 o0.60485 257 3 58 I36 9.38266 0 044 9.98697 3 87I 9.39569 54 0.6043 207 2 59 I64 9.38317 5 I 037 9.98694 902 9.39623 o0.60377 I58 60 192 9.38368 5 030 9.98690 4 933 9.39677 54 0.60323 io8 0 I _ I __. I -- I I __ I |Nat. COS Log. d. INat. Sin Log. d. |Nat. COtLog.lc.d.lLog.TanNat.l ' i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ 76~ 140 0 fNat. Sin Log. d. INat. COS Log. d. jNat.TanLog. c~d. Log. Cot Nat. I. 1 0 I 2 3 4 - 6 I 7 8 9 10 II 12 I3 I4 15 I6 '7 i8 '9 20 21 22 23 24 25 26 27 28 29 3' 3I 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 SI 52 53 54 55 56 57 58 59 60 24192 9.38368 220 9.384i8 249 9.38469 277 9.38519 305 9.38570 24333 9.38620 362 9.38670 390 9.38721 418 9.38771 446 9.3882I 24474 9.38871 503 9.3892I 53I 9-3897I 559 9.39021 587 9.3907I 24615 9.39I2I 644 9.39170 672 9.39220 700 9.39270 728 9.393I9 24756 9.39369 784 9.394i8 813 9.39467 84I 9.395I7 869 9.39566 24897 9.396i5 925 9.39664 954 9.39713 982 9.39762 25010 9.398II 25038 9.39860 ob6 9.39909 094 9.39958 122 9.40006 151 9.40055 25179 9.40103 207 9.40152 235 9.40200 263 9.40249 291 9.40297 25320 9.40340 348 9.40394 376 9.40442 404 9.40490 432 9.40538 25460 9.40586 488 9.40634 516 9.40682 545 9.40730 573 9.40778 25601 9.40325 629 9.40873 657 9.40921 685 9.40968 713 9.41016 25741 9.4I063 769 9.41111 798 9.4II58 826 9.4I205 854 9.41252 882 9.41300 I 50 SI 50 SI 50 50 SI 50 50 50 50 50 50 50 50 49 50 50 49 50 49 49 50 49 49 49 49 49 49 49 49 49 48 49 48 49 48 49 48 49 48 48 48 48 48 48 48 48 48 47 48 48 47 48 47 '48 47 47 47 48 97030 9.98690 023 9.98687 015 9.98684 oo8 9.98681 oo0 9.98678 96994 9.98675 987 9.98671 980 9.98668 973 9.98665 966 9.98662 96959 9.98659 952 9.98656 945 9.98652 937 9.98649 930 9.98646 96923 9.98643 916 9.98640 909 9.98636 902 9.98633 894 9.98630 96887 9.98627 88o 9.98623 873 9.98620 866 9.98617 858 9.98614 96851. 9.986io 844 9.98607 837 9.98604 829 9.98601 822 9.98597 96815 9.98594 807 9.98591 800 9.98588 793 9.98584 786 9.98581 96778 9.98578 771 9.98574 764 9.98571 756 9.98568 749 9.98565 96742 9.98561 734 9.98558 727 9.98555 719 9.9855I 7I2 9.98548 96705 9.98545 697 9.9854I 690 9.98538 682 9.98535 675 9.98531 96667 9.98528 66o 9.98525 653 9.98521 645 9.98518 638 9.98515 96630 9.985II 623 9.98508 6I5 9.98505 6o8 9.98501 6oo 9.98498 593 9.98494 3 3 3 3 3 4 3 3 3 3 3 4 3 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 3 4 3 3 4 3 3 3 4 3 24933 9.39677 964 9.3973I 995 9.39785 25026 9.39838 056 9.39892 25087 9.39945 ii8 9.39999 149 9.40052 i80 9.40106 211 9.40159 25242 9.40212 273 9.40266 304 9.40319 335 9.40372 366 9.40425 25397 9.40478 428 9.40531 459 9.40584 490 9.40636 521 9.40689 25552 9.40742 583 9.40795 614 9.40847 645 9.40900 676 9.40952 25707 9.41005 738 9.4I057 769 9.4II09 800 9.41161 831 9.41214 25862 9.4I266 893 9.4138 924 9.41370 955 9.41422 986 9.41474 26017 9.41526 048 9.4I578 079 9.41629 II0 9.41681 14I 9.41733 26172 9.4I784 203 9.41836 235 9.41887 266 9.41939 297 9.41990 26328 9.42041 359 9.42093 390 9.42144 421 9.42195 452 9.42246 26483 9.42297 515 9.42348 546 9.42399 577 9.42450 6o8 9.42501 26639 9.42552 670 9.42603 701 9.42653 733 9.42704 764 9.4275$ 795 9.42805 54 54 53 54 53 54 53 54 53 53 54 53 53 53 53 53 53 52 53 53 53 52 53 52 53 52 52 52 53 52 52 52 52 52 52 52 SI 52 52 SI 52 51 52 SI SI 52 5' 51 5I 5' 5' 5' SI S~I 5I SI 50 SI SI 50 0.60323 4.0108 0.60269 o58 o.602I 009 0.60162 3.9959 o.6oio8 gio o.60059 3.9861 o.6oooi 812 0.59948 763 0.59894 714 0.5984I 665 0.59788 3.9617 0.59734 568 0.59681 520 0.59628 471 0.59575 423 0.59522 3.9375 0.59469 327 0.59416 279 0.59364 232 0.59311 184 0.59258 3.9I36 0.59205 089 0.59153 042 0.59100 3.8995 0.59048 947 0.58995 3.8900 0.58943 854 0.58891 807 0.58839 760 0.58786 714 0.58734 3.8667 0.58682 621 0.58630 575 0.58578 528 0.58526 482 0.58474 3.8436 0.58422 391 0.58371 345 0.58319 299 0.58267 254 0.58216 3.8208 0.58164 163 0.58113 II8 o.5806f 073 0.58010 026 0.57959 3.7983 0.57907 938 0.57856 893 0.57805 848 0.57754 804 0.57703 3.7760 0.57652 7I5 0.57601 671 0.57550 627 0.57499 583 0.57448 3.7539 0.57397 495 0.57347 45I 0.57296 408 0.57245 364 0.57195 321 60 59 58 57 56 54 53 52,I 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 II 16 9 8 7 6 5 4 3 2 I 0 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 3 4 3 4 - INat. COS Lo. d. Nat. Sin Log. dL Nat. COtLog. c.d. Log.Taf Nat] II 7b0 15 INat. Sin Log. d. Nat. COS Log. d. Nat.Tan Log. c.d. Log. Cot Nat. I 0I 25882 9.41300 I,' 9IO 9.4I347 47 2 938 9.4I394 47 3 966 9-4I44I 47 4 994 9.41488 47 47 5 26022 9.4I535 6 050 9.4I582 46 7 079 9.4i628 8 107 9.4i67 47 47 9 135 9.41722 46 10 26163 9.41768 II 19 9.4i8I5 46 12 2I9 9.4186 4 13 247 9.41908 46 14 275 9.4I954 47 15 26303 9.4200I 46 i6 331 9.42047 46 17 359 9.42093 x8 387 9.42140 46 Ig 415 9.42186 46 20 26443 9.42232 46 21 47I 9.42278 46 22 500 9.42324 46 23 528 9.42370 46 24 556 9.42446 25 2 n6584 9.4246I 46 26 612 9.42507 46 27 640 9.42553 46 28 668 9.42599 46 29 696 9.42644 45 — 46 30 26724 9.42690 46 31 752 9.42735 46 32 780 9.42781 33 8o8 9.42826 45 34 836 9.42872 46 45 -95 26864 9-42917 4 36 892 9.42962 46 37 920 9.43008 38 948 9.43053 45 39 976 9.43098 45 40 27004 9.43143 4I 032 9.43i88 45 42 o6o 9.43233 4 43 o88 9.43278 45 44 ii6 9.43323 45 44 45 27144 9.43367 46 172 9.434I2 45 47 200 9.43457 48 228 9.43502 45 49 256 9.43546 44 -- ~~45 50 27284 9.43591 51 312 9.43635 44 52 340 9.43680 45 53 368 9.43724 44 54 396 9.43769 45 55 27424 9.43813 56 452 9.43857 44 57 480 9.43901 44 58 5o8 9.43946 45 59 536 9.43990 44 60 564 9.44034 44 96593 9.98494 585 9.9849I 578 9.98488 570 9.98484 562 9.9848i 96555 9.98477 547 9.98474 540 9.9847I 532 9.98467 524 9.98464 96517 9.98460 509 9.98457 502 9.98453 494 9.98450 486 9.98447 96479 9.98443 471 9.98440 463 9.98436 456 9.98433 448 9.98429 96440 9.98426 433 9.98422 425 9.98419 417 9.984I5 410 9.98412 96402 9.98409 394 9.98405 386 9.98402 379 9.98398 371 9.98395 96363 9.98391 355 9.98388 347 9.98384 340 9.9838I 332 9.98377 96324 9.98373 316 9.98370 308 9.98366 301 9.98363 293 9.98359 96285 9.98356 277 9.98352 269 9.98349 261 9.98345 253 9.98342 96246 9.98338 238 9.98334 230 9.98331 222 9.98327 214 9.98324 96206 9.98320 198 9.983I7 190 9.983I3 182 9.98309 174 9.98306 96166 9.98302 I58 9.98299 I50 9.98295 142 9.98291 134 9.98288 I26 9.98284 3 3 4 3 4 3 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 3 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 3 4 3 4 4 3 4 3 4 3 4 4 3 4 3 4 4 3 4 26795 9.42805 826 9.42856 857 9.42906 888 9.42957 920 9.43007 26951 9.43057 982 9.43io8 27013 9.43I58 044 9.43208 076 9.43258 27IO7 9.43308 138 9.43358 169 9.43408 201 9.43458 232 9.43508 27263 9.43558 294 9.43607 326 9.43657 357 9.43707 388 9.43756 27419 9.43806 451 9.43855 482 9.43905 5I3 9.43954 545 9.44004 27576 9.44053 607 9.44102 638 9-4415I 670 9.44201 701 9.44250 27732 9.44299 764 9.44348 795 9.44397 826 9.44446 858 9.44495 27889 9.44544 921 9.44592 952 9.4464i 983 9.44690 28015 9.44738 28046 9.44787 077 9.44836 109 9.44884 140 9.44933 172 9.4498i 28203 9.45029 234 9.45078 266 9.45126 297 9.45I74 329 9.45222.28360 9.45271 39I 9.453I9 423 9.45367 454 9.454I5 486 9.45463 28517 9.455II 549 9.45559 580 9.45606 6I2 9.45654 643 9.45702 675 9.45750 0.57I95 3.7321 0.57I44 277 50 0.57094 234 0.57043 191 50 0.56993 148 50 0.56943 37105 50 0.56892 062 50 0.56842 oI9 50 0.56792 3.6976 50 0.56742 933 50 0.56692 3.6891 50 0.56642 848 50 0.56592 8o6 50 0.56542 764 50 0.56492 722 50 0.56442 3.6680 50 0.56393 638 50 0.56343 596 50 0.56293 554 50 0.56244 512 50 4 0.56194 3.6470 50 0:56I4$ 429 49 0.56095 3877 50 0.56046 346 50 0.55996 305 0.55947 3.6264 49 0.55898 222 50 0.55849 i8i 50 0.55799 140 49 0.55750 100 905570I 3.6059 49 0.55652 018 49 0.55603 3.5978 0.55554 937 49 0.55505 897 49 48 0.55456 3.5856 0.55408 8i6 49 0.55359 776 480.55310 736 48 0.55262 696 0.55213 3.5656 48 0.55164 6I6 48 4 0.55II6 576 48 0.55067. 536 48 0.55019 497 905497I 3.5457 48 0.54922 418 48 0.54874 379 48 0.54826 339 0.54778 300 48 0.54729 3.5261 48 0.54681 222 48 0.54633 183 48 0.54585 144 0.54537 105 48 0.54489 3.5067 4 05444 028 48 0.54394 3.4989 48 0.54346 95I 48 0.54298 912 0.542~0 874 60 59 58 57 56 55 54 53 52 5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 INat. COS Log. d. INat. Sin Log. d. INat. COt Log. c.d. Log.Tan Nat. 740 160 I Nat. Sin Log. d. INat. Cos Log. d. iNat.TanLog. cd. Log.CotNatI I 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 15 i6 '7 I8 '9 21 22 23 24 f25 26 27 28 29 30 3I 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5I 52 53 54 55 56 57 58 59 60 27564 9.44034 592 9.44078 620 9.44122 648 9.44i66 676 9.44210 27704 9.44253 731 9.44297 759 9.44341 787 9.44385 815 9.44428 27843 9.44472 871 9.44516 899 9.44559 927 9.44602 955 9.44646 27983 9.44689 280II 9.44733 039 9.44776 067 9.448i9 095 9.44862 28123 9.44905 150 9.44948 178 9.44992 206 9.45035 234 9.45077 28262 9.45120 290 9.45163 318 9.45206 346 9.45249 374 9.45292 28402 9.45334 429 9.45377 457 9.45419 485 9.45462 513 9.45504 28541 9.45547 569 9.45589 597 9.45632 625 9.45674 652 9.45716 28680 9.45758 708 9.45801 736 9.45843 764 9.45885 792 9.45927 28820 9.45969 847 9.4601i 875 9.46053 903 9.46095 931 9.46136 28959 9.46I78 987 9.46220 29015 9.46262 042 9.46303 070 9.46345 29098 9.46386 126 9.46428 154 9.46469 182 9.465II 209 9.46552 237 9.46594 44 44 44 44 43 44 44 44 43 44 44 43 43 44 43 44 43 43 43 43 43 44 43 42 43 43 43 43 43 42 43 42 43 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 4' 42 42 42 4' 42 4' 42 4' 42 4I 42 96126 9.98284 ii8 9.98281 iio 9.98277 102 9.98273 094 9.98270 96086 9.98266 o78 9.98262 070 9.98259 062 9.98255 054 9.9825I 96046 9.98248 037 9.98244 029 9.98240 021 9.98237 013 9.98233 96005 9.98229 95997 9.98226 989 9.98222 981 9.982i8 972 9.982I5 95964 9.98211 956 9.98207 948 9.98204 940 9.98200 931 9.98196 95923 9.98192 9I5 9.98i89 907 9.98i85 898 9.98181 890 9.98i77 95882 9.98174 874 9-98I70 865 9.98i66 857 9.98162 849 9.98I59 95841 9.98I55 832 9.98I5I 824 9.98i47 8i6 9.98i44 8o7 9.98I40 95799 9.98I36 791 9.98I32 782 9.98I29 774 9.98125 766 9.98121 95757 9.98iI7 749 9.98I13 740 9.98II0 732 9.98106 724 9.98102 95715 9.98098 707 9.98094 698 9.98090 690 9.98087 68i 9.98083 95673 9.98079 664 9.98075 656 9.9807I 647 9.98067 639 '9.98063 630 9.98060 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 3 4 4 4 4 3 4 4 4 4 4 3 4 4 4 4 4 4 3 28675 9.45750 706 9.45797 738 9.45845 769 9.45892 8oi 9.45940 28832 9.45987 864 9.46035 895 9.46082 927 9.46I30 958 9.46177 28990 9.46224 29021 9.46271 053 9.46319 084 9.46366 ii6 9.46413 29I47 9.46460 179 9.46507 210 9.46554 242 9.46601 274 9.46648 29305 9.46694 337 9-4674I 368 9.46788 400 9.46835 432 9.46881 29463 9.46928 495 9.46975 526 9.47021 558 9.47068 590 9.47I14 29621 9.47i60 653 9.47207 685 9.47253 716 9.47299 748 9.47346 29780 9.47392 8ii 9.47438 843 9.47484 875 9.47530 906 9.47576 29938 9.47622 970 9.47668 3000I 9.47714 033 9.47760 065 9.47806 30097 9.47852 128 9.47897 i6o 9.47943 I92 9.47989 224 9.48035 30255 9.48080 287 9.48I26 319 9.48I7I 351 9.48217 382 9.48262 30414 9.48307 446 9.48353 478 9.48398 509 9.48443 541 9.48489 573 9.48534 47 48 47 48 47 48 47 48 47 47 47 48 47 47 47 47 47 47 47 46 47 47 47 46 47 47 46 47 46 46 47 46 46 47 46 46 46 46 46 46 46 46 46 46 4.6 45 46 46 46 45 46 45 46 45 45 46 45 45 46 45 0.54250 3.4874 0.54203 836 0.54I55 798 0.54108 760 0.54060 722 0.540I3 3.4684 0.53965 646 0.539i8 6o8 0.53870 570 0.53823 533 0.53776 3.4495 0.53729 458 0.53681 420 0.53634 383 0.53587 346 0.53540 3.4308 0.53493 271 0.53446 234 0.53399 197 0.53352 I6o 0.53306 3.4124 0.53259 087 0.53212 050 0.53165 oI4 0.53II9 3.3977 0.53072 3.394I 0.53025 904 0.52979 868 0.52932 832 0.52886 796 0.52840 3.3759 0.52793 723 0.52747 687 0.52701 652 0.52654 6i6 0.52608 3.3580 0.52562 544 0.52516 509 0.52470 473 0.52424 438 0.52378 3.3402 0.52332 367 0.52286 332 0.52240 297 0.52194 261 0.52148 3.3226 0.52103 191 0.52057 I56 0.52011 122 0.51965 087 0.51920 3.3052 0.51874 017 0.51829 3.2983 0.51783 948 0.51738 914 0.5i693 3.2879 0.51647 845 0.51602 8ii 0.51557 777 051I5II 743 0.51466 709 60 59 58 57 -56 56 -54 53 52 51 50 49 48 47 46 45 44, 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 Nat. Co 0Log. d. Nat. Sin Log. d. Nat. COtLog.Ic.d. Log.Tan Nat. I.1 730 170 Nat. SinLog. d. INat. COS Log. d. INat.Tan Log. Ic.d. Log. Cot Nat. 0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 i6 '7 i8 19 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 5' 52 53 54 55 56 57 58 59 60 29237 9.46594 265 9.46635 293 9.46676 321 9.467I7 348 9.46758 29376 9.46800 404 9.4684i 432 9.46882 460 9.46923 487 9.46964 29515 9.47005 543 9.47045 571 9.47086 599 9.47127 626 9.47i68 29654 9.47209 682 9.47249 710 9.47290 737 9.47330 765 9.47371 29793 9.47411 821 9.47452 849 9.47492 876 9.47533 904 9.47573 29932 9.47613 960 9.47654 987 9.47694 30015 9.47734 043 9.47774 30071 9.47814 098 9.47854 126 9.47894 154 9.47934 182 9.47974 30209 9.48014 237 9.48054 265 9.48094 292 9.48133 320 9.48i73 30348 9.482I3 376 9.48252 403 9.48292 431 9.48332 459 9.48371 30486 9.484ii 514 9.48450 542 9.48490 570 9.48529 597 9.48568 30625 9.48607 653 9.48647 68o 9.48686 708 9.48725 736 9.48764 30763 9.48803 791 9.48842 819 9.4888i 846 9.48920 874 9.48959 902 9.48998 41 4' 41 4' 42 4I 4' 4I *4I 4I 40 41 4' 4I 14' 40 4' 40 4I 40 41 40 41 40 40 4' 40 40 40 40 40 40 40 40 40 95630 9.98060 622 9.98056 613 9.98052 605 9.98048 596 9.98044 95588 9.98040 579 9.98036 571 9.98032 562 9.98029 554 9.98025 95545 9.98021 536 9.980I7 528 9.980I3 519 9.98009 511 9.98005 95502 9.98001 493 9.97997 485 9.97993 476 9.97989 467 9.97986 95459 9.97982 450 9.97978 441 9.97974 433 9.97970 424 9.97966 95415 9.97962 407 9.97958 398 9.97954 389 9.97950 380 9.97946 95372 9.97942 363 9.97938 354 9.97934 345 9.97930 337 9.97926 95328 9.97922 3I9 9.979i8 310 9.979I4 301 9-979IO 293 9.97906 95284 9.97902 275 9.97898 266 9.97894 257 9.97890 248 9.97886 95240 9.97882 231 9.97878 222 9.97874 2I3 9.97870 204 9.97866 95195 9.97861 i86 9.97857 177 9.97853 i68 9.97849 159 9.97845 95150 9.97841 142 9.97837 133 9.97833 124 9.97829 II5 9.97825 io6 9.97821 4 4 4 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 3 4 4 4 4 4 i4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 4 4 4 4 4 4 4 4 4 4 30573 9.48534 605 9.48579 637 9.48624 669 9.48669 700 9.487I4 30732 9.48759 764 9.48804 796 9.48849 828 9.48894 86o 9.48939 30891 9.48984 923 9.49029 955 9.49073 987 9.49II8 3I019 9.49i63 31051 9.49207 083 9.49252 115 9.49296 147 9.49341 178 9.49385 3I2IO 9.49430 242 9.49474 274 9.49519 306 9.49563 338 9.49607 31370 9.49652 402 9.49696 434 9.49740 466 9.49784 498 9.49828 31530 9.49872 562 9.499I6 594 9.49960 626 9.50004 658 9.50048 31690 9.50092 722 9.50i36 754 9.50180 786 9.50223 8i8 9.50267 31850 9.503II 882 9.50355 914 9.50398 946 9.50442 978 9.50485 32010 9.50529 042 9.50572 074 9.506i6 io6 9.50659 139 9.50703 32171 9.50746 203 9.50789 235 9.50833 267 9.50876 299 9.50919 32331 9.50962 363 9.51005 396 9.5I048 428 9.51092 460 9.5II35 492 9.5II78 45 45 45 45 45 45 45 45 45 45 45 44 45 45 44 45 44 45 44 45 44 45 44 44 45 44 44 44 44 44 44 44 44 44 44 44 44 43 44 44 44 43 44 43 44 43 44 43 44 43 43 44 43 43 43 43 43 44 43 43 0.51466 3.2709 0.51421 675 0.51376 641 0.5I33i 607 0.5I286 573 0.5124I 3.2539 0.51196 506 0.5II5I 472 0.5II06 438 0.5Io6I 405 0.51016 3.2371 0.50971 338 I 0.50927 305 0.50882 272 0.50837 238 0.50793 3.2205 0.50748 172 0.50704 I39 I0.50659 io6 0.50615 073 0.50570 3.2041 0.50526 oo8 0.50481 3.1975 0.50437 943 0.50393 9IO 0.50348 3.1878 0.50304 845 0.50260 813 0.50216 780 0.50172 748 0.50I28 3.I7i6 0.50084 684 0.50040 652 0.49996 620 0.49952 588 o.49908 3.1556 0.49864 524 0.49820 492 0.49777 460 0.49733 429 0.49689 3.I397 0.49645 366 0.49602 334 0.49558 303 0-495I5 271 0.4947I 3.1240 0.49428 209 0.49384 178 0.4934I 146 0.49297, 115 0.49254 3.i084 0.49211 053 0.49167 022 0.49124 3.099I 0.4908i 961 0.49038 3.0930 0.48995 899 0.48952 868 0.48908 838 0.48865 807 0.48822 777 60 59 58 57 56 -&5 54 53 52 5' 50 49 48 47 46 451 44 43 42 4'L 401 39 38 36 35' 34 33 32 3I 30 29 28 27 261 2' 24 23 22 21J 20 '9:c8 '7 i6 '5 I4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 40 40 39 40 40 39 40 40 39 40 39 40 39 39 39 40 39 39 39 39 39 39 39 39 39 INat. COS Log. dL INat. Sin Log. d. ]Nat. Cot Log. cA. Log.Tan Nat.l - I 72 0 180 INat. Sin Log. d. INat. COS Log. d. lNat.TanLog.!c.d. Log. Cot Nat.I 1 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 -f5 i6 '7 18 19 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 56 57 58 59 60 30902 9.48998 929 9.49037 957 9.49076 985 9.4911S 3IOI2 9.49I53 3I040 9.49I92 o68 9.4923I 095 9.49269 I23 9.49308 151 9.49347 31178 9.49385 206 9.49424 233 9.49462 261 9.49500 289 9.49539 31316 9.49577 344 9.496i5 372 9.49654 399 9.49692 427 9.49730 3I454 9.49768 482 9.49306 5IO 9.49844 537 9.49882 565 9.49920 31593 9.49958 620 9.49996 648 9.50034 675 9.50072 703 9.50II0 31730 9.50I48 758 9.50185 786 9.50223 813 9.50261 841 9.50298 31868 9.50336 896 9.50374 923 9.50411 951 9.50449 979 9.50486 32006 9.50523 034 9.5056i o6i 9.50598 089 9.50635 ii6 9.50673 32144 9.50710 171 9.50747 199 9.50784 227 9.50821 254 9.50858 32282 9.50896 309 9.50933 337 9.50970 364 9.5I007 392 9.5I043 32419 9.5i080 447 9.51117 474 9-51I54 502 9.51191 529 9.51227 557 9.51264 39 39 39 38 39 39 38 39 39 38 39 38 38 39 38 38 39 38 38 38 38 38 38 38 38 38 38 38 38 38 37 38 38 37 38 38 37 38 37 37 38 37 37 38 37 37 37 37 37 38 37 37 37 36 37 37 37 37 36 37 95106 9.97821 097 9.978I7 o88 9.978I2 079 9.97808 070 9.97804 95061 9.97800 052 9.97796 043 9.97792 033 9.97788 024 9.97784 95015 9.97779 oo6 9.97775 94997 9.9777I 988 9.97767 979 9.97763 94970 9.97759 961 9-977$4 952 9.97750 943 9.97746 933 9.97742 94924 9.97738 915 9.97734 906 9.97729 897 9.97725 888 9.97721 94878 9.977I7 869 9.97713 86o 9.97708 85I 9.97704 842 9.97700 94832 9.97696 823 9.9769i 814 9.97687 805 9.97683 795 9.97679 94786 9.97674 777 9.97670 768 9.97666 758 9.97662 749 9.97657 94740 9.97653 730 9.97649 721 9.97645 712 9.97640 702 9.97636 94693 9.97632 684 9.97628 674 9.97623 665 9.976i9 656 9.976i5 94646 9.976io 637 9.97606 627 9.97602 6i8 9.97597 609 9.97593 94599 9.97589 590 9.97584 580 9.97580 571 9.97576 561 9.9757I 552 9.97567 4 5 4 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 4 5 4 4 5 4 4 5 4 4 5 4 4 5 4 32492 9.5II78 524 9.51221 556 9.5I264 588 9.5I306 621 9.5I349 32653 9.51392 685 9-5I435 7I7 9.5I478 749 9.51520 782 9.5I563 32814 9.5i606 846 9.51648 878 9.5169i 9II 9.5I734 943 9.51776 32975 9.5i8i9 33007 9.5i86i 040 9.5I903 072 9.51946 104 9.5i988 33136 9.52031 169 9.52073 201 9.52II5 233 9.52I57 266 9.52200 33298 9.52242 330 9.52284 363 9.52326 395 9.52368 427 9.52410 33460 9.52452 492 9.52494 524 9.52536 557 9.52578 589 9.52620 33621 9.52661 654 9.52703 686 9.52745 718 9.52787 751 9.52829 33783 9.52870 8i6 9.52912 848 9.52953 88i 9.52995 913 9.53037 33945 9.53078 978 9.53120 340IO 9.53i6i 043 9.53202 075 9.53244 34108 9.53285 140 9.53327 173 9.53368 205 9.53409 238 9.53450 34270 9.53492 303 9.53533 335 9.53574 368 9.53615 400 9.53656 433 9.53697 43 43 42 43 43 43 43 42 43 43 42 43 43 42 43 42 42 43 42 43 42 42 42 43 42 42 42 42 42 42 42 42 42 42 4' 42 42 42 42 41 0.48822 3.0777 0-48779 746 0.48736 716 0.48694 686 0.48651 655 0.48608 3.0625 0.48565 595 0.48522 565 0.48480 535 0.48437 505 0.48394 3.0475 0.48352 445 0.48309 415 0.48266 385 0.48224 356 0.48181 3.0326 0.48139 296 0.48097 267 0.48054 237 0.48012 208 0.47969 3.0178 0.47927 149 0.47885 120 0.47843 090 0.47800 o6i 0.47758 3.0032 0.47716 003 0.47674 2.9974 0.47632 945 0.47590 916 0.47548 2.9887 0.47506 858 0.47464 829 0.47422 8oo 0.47380 772 0.47339 2.9743 0.47297 714 0.47255 686 0.47213 657 0.47171 629 0.47I30 2.9600 0.47088 572 0.47047 544 0.47005 515 0.46963 487 0.46922 2.9459 0.46880 43I 0.46839 403 0.46798 375 0.46756 347 0.46715 2.9319 0.46673 291 0.46632 263 0.46591 235 0.46550 ' 208 0.46508 2.9180 0.46467 152 0.46426 125 0.46385 097 0.46344 070 0.46303 042 60 59 58 57 56 55 54 53 52 SI 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 19 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 42 4' 42 42 4' 42 4I 4' 42 4' 42 4' 4' 4' 42 4' 4' 4' 4' 4' tNat.CoS Log. I INat. Sin Log. d. INat. Cot Log.cI1.ALog.Tan Nat.I ) 0 710 190 INat. Sin Log. d. INat. COS Log. d. INat.TanLLog. c.d. Log. Cot Nat.l 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 15 I6 17 I8 '9 21 22 23 24 2W 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4I 42 43 44 45 46 47 48 49, 50 SI 52 53 54 "7 56 57 58 59 60 32557 9.51264 584 9.51301 612 9.51338 639 9.5I374 667 9.51411 32694 9.51447 722 9.5I484 749 9.51520 777 9-5I557 804 9.51593 32832 9.51629 859 9.51666 887 9.51702 914 9.5I738 942 9.51774 32969 9.518II 997 9.5i847 33024 9.5i883 051 9.51919 079 9.5I955 33Io6 9.5199I 134 9.52027 i6i 9.52063 189 9.52099 216 9.52135 33244 9.52171 271 9.52207 298 9.52242 326 9.52278 353 9.52314 33381 9.52350 408 9.52385 436 9.52421 463 9.52456 490 9.52492 33518 9.52527 545 9.52563 573 9.52598 6oo 9.52634 627 9.52669 33655 9.52705 682 9.52740 7I0 9.52775 737 9.52811 764 9.52846 33792 9.52881 819 9.52916 846 9.52951 874 9.52986 90I 9.5302I 33929 9.53056 956 9.53092 983 9.53126 340II 9.53i6i 038 9.53196 34065 9.53231 093 9.53266 120 9.53301 147 9.53336 175 9.53370 202 9.53405 37 37 36 37 36 37 36 37 36 36 37 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 35 36 35 36 35 36 35 36 35 35 36 35 35 35 35 35.35 35 36 34 35 35 35 35 35 35 34 35 94552- 9.97567 542 9.97563 533 9.97558 523 9.97554 514 9.97550 94504 9.97545 495 9.9754I 485 9.97536 476 9.97532 466 9.97528 94457 9.97523 447 9.97519 438 9.975i5 428 9.975IO 418 9.97506 94409 9-9750I 399 9.97497 390 9.97492 380 9.97488 370 9.97484 94361 9.97479 351 9.97475 342 9.97470 332 9.97466 322 9.97461 94313 9.97457 303 9.97453 293 9.97448 284 9.97444 274 9.97439 94264 9.97435 254 9.97430 245 9.97426 235 9.97421 225 9.974I7 94215 9.974I2 206 9.97408 196 9.97403 i86 9.97399 176 9.97394 94167 9.97390 157 9.97385 147 9.97381 137 9.97376 127 9.97372 94118 9.97367 io8 9.97363 098 9.97358 o88 9.97353 078 9.97349 94068 9.97344 058 9.97340 049 9.97335 039 9.9733I 029 9.97326 94019 9.97322 009 9.97317 93999 9.97312 989 9.97308 979 9.97303 969 9.97299 4 5 4 4 5 4 5 4 4 S 4 4 5 4 S 4 S 4 4 5 4 S 4 5 4 4 5 4 5 4 5 4 5 -4 5 4 5 4 5 4 S 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 34433 9.53697 465 9.53738 498 9.53779.530 9.53820 563 9.53861 34596 9.53902 628 9.53943 66i 9.53984 693 9.54025 726 9.54065 34758 9.54I06 791 9.54I47 824 9.54I87 856 9.54228 889 9.54269 34922 9.54309 954 9.54350 987 9.54390 35020 9.5443I 052 9.54471 35085 9.54512 Ii8 9.54552 150 9.54593 183 9.54633 216 9.54673 35248 9.54714 281 9.54754 314 9.54794 346 9.54835 379 9.54875 35412 9.54915 445 9.5495$ 477 9-54995 5I0 9.55035 543 9.55075 35576 9.55II1 6o8 9.55I5$ 641 9.55195 674 9.5523$ 707 9.55275 35740 9.553I5 772 9.55355 805 9.55395 838 9.55434 871 9.55474 35904 9.55514 937 9.55554 969 9.55593 36002 9.55633 035 9.55673 36068 9.557I2 IOI 9.55752 134 9.55791 167 9-5583I 199 9.55870 36232 9.559IO 265 9.55949 298 9.55989 331 9.56028 364 9.56067 397 9.56I07 0.46303 2.9042 41 0.46262 015 4I 0.46221 2.8987 4' 0.46180 960 4' 0.46139 933 0.46098 2.8905 41 o.46057 878 41 0.46016 85I 40 0.45975 824 40 0.45935 797 0.45894 2.8770 40 0.45853 743 40 0.45813 716 I 0.45772 689 40 04573i 662 40 0.4569i 2.8636 40 0.45650 609 40 0.45610 582 40 0.45569 556 40 0.45529 529 0.45488 2.8502 40 0.45448 476 4' 40 0.45407 449 40 0.4$367 423 0045327 397 4I 4' 0.45286 2.8370 40 0.45246 344 4I 0.45206 318 40 0.45i65 291 40 0.45125 265 40 0.45085 2.8239 40 0.45045 213 40 0.45005 187 40 0.44965 i6i 40 0.44925 135 40 0.4488g 2.8109 40 0.44845 083 40 0.44805 057 40 0.44765 032 40 0.44725 oo6 40 278 40 04468$ 2-798o 4 0.44645 955 40 0.44605 929 34 0.44566 903 40 0.44526 878 40 0.44486 2.7852 40 0.44446 827 40 0.44407 8oi 40 0.44367 776 0044327 751 0.44288 2.7725 40 0.44248 700 39 0.44209 675 40 0.44169 650 39 0.44130 625 40 0.44090 2.7600 9044051 575 40 0.440II 550 0.43972 525 39 0.43933 500 40 0.43893 475 1 L I -- - 60 59 58 57 56 55 54 53 52 5' 50 49 48 47 46 45 44 43 42 4' 46 39 38 37 36 35 34 33 32 3' 36 29 28 27 26 '25 24 23 22 21 20 '9, i8 '7 i6 15 '4 13 12 II 10 9 8 7 6 5 4 3 2 I I 0 Nat. Co 0Log. L. Nat. Sin Log. d. Nat. C OtLog. I - c.d.lLog. I anNat.1 T i 700 20~ Nat. Sin Log. d. Nat. C Lg. d. NatTanLog.c.d.Log. CotNat. 034202 9.53405 9396 9.97299 36397 9.56I07.43893 27475 60 229 953440 5 959 997294 430 95646 39.43854 450 59 229 35 5 0 959 9~97439 2 257 9.53475 3 949 9.97289 463 9.5685 390.43815 425 58 3 284 9.53509 34 939 997285 4 496 9.56224 3 0.43776 400 57 4 311 9.53544 929 9.97280 529 9-56264 o.43736 376 56 34 _ 39 5 34339 9-53578 3 93919 9.97276 36562 9-56303 o.43697 2.7351 55 6 366 9-536I3 5 909 9.9727I 5 595 9.56342 3 o.43658 326 54 7 393 9.53647 34 899 9.97266 5 628 9.56381 39 0.43619 302 53 8 42I 9.53682 35 889 9.97262 4 661 9.56420 39 0.43580 277 52 9 448 9-53716 34 879 9.97257 5 694 9.56459 39 0.43541 253 51 35 5 - 39 10 34475 95375 93869 9.97252 36727 9.56498 390.43502 2.7228 50 II 503 9.53785 34 859 9.97248 4 760 9.56537 39 0.43463 204 49 12 530 9.5389 3 849 9.97243 793 9.56576 0.43424 17948 I3 557 9.53854 839 9.972385 826 9.56615 39 0.43385 155 47 14 584 9.53888 34 829 9.97234 4 859 9.56654 39 043346 130 46 15 34612 9.5325 39 15346I2 9.53922 93819 9.97229 36892 9.56693 0.43307 2.7I06 45 I6 639 9.53957 35 809 9.97224 925 9.56732 0.43268 082 44 17 666 9.53991 799 9.97220 958 95677I 390.43229 058 43 19 769 9.54029 34 789 9.9/21 5 39 8 694 9.54025 34 789 99725 99I 9.5680 39 0.43190 034 42 19 721 9.54059 779 9-972045 37024 9.56849 39 0.43151 009 41 20 34748 9.54093 34 93769 9.97206 37057 9.56887 0.43113 2.6985 40 34 59 o9970 9.56926 38 96I 39 21 775 9.5427 3 759 9.9720I 90 56926.43074 96 39 22 803 9.546I 4 748 9.97196 I123 9-56965 39 0.43035 937 38 23 830 9-54I95 34 738 9.97I92 5 57 957004 3.42996 913 37 24 857 9-54229 728 9.97187 I90 9.57042 0.42958 889 36 25 s~ 9. / ~2I34 $ 39 _ 25 34884 9.54263 937I8 9.97I82 37223 9.57081 0.42919 2.6865 35 26 912 9.54297 34 708 9.97178 4 256 9.57120 3 0.42880 841 34 27 939 9-5433I 34 698 99773 5 289 9.571583 0.42842 818 33 28 966 954365 3 688 9.97168 5 322 9.579 042803 79 32 3 33 4 38 0.42803 794 32 29 993 9.54399 34 677 9-97I63 355 9-57235 0.42765 770 31 3035021 9.54433 3 93667 9.97I59 37388 9.572743 0.42726 2.674630 31 o48 9.54466 33 657 9.9754 5 422 9.57312 3 0.42688 723 29 32 075 9.54500 34 647 9.97149 455 39 0.42649 699 28 455 9-57351 0.42638 699 28 33 I02 9.54534 34 637 9.97145 4 488 9.57389 3 0.42611 675 27 34 130 9.54567 33 626 997I40 5 521 9.57428 39 0.42572 652 26 35 35-57 95460 i 9364 5 38 5 35I57 9.5460 3 93616 9.97135 5 37554 957466 8 0.42534 2.6628 25 36 I84 9-54635 606 9.97130 588 9.57504 3 0.42496 605 24 37 211 9.54668 33 596 9.97126 4 62I 9.57543 0.42457 58I 23 38 239 9.54702 34 585 99712I 5 654 9.5758I 3 0.42419 558 22 39 266 9.54735 33 575 9-9II6 5 687 9.57619 3 0.42381 534 21 40 35293 9.54769 93565 9-97II 37720 9576583 0.42342 2.6511 20 4 320 9.54802 33 555 997107 754 9.57696 38 0.42304 488 19 42 347 9.54836 3 544 9-9702 787 9-57734 38 0.42266 464 8 43 375 9.54869 3 534 9.97097 820 9.57772 38 0.42228 44I 17 44 402 9.54903 3 524 9.97092 853 9.57810o 0.42190 418 i6 95496 3269 9. 395 15 45 35429 9.54936 935I4 9.97087 37887 9.57849 3 425 2639515 46 456 9-54969 3 503 9.97083 920 9-57887 0.42II3 37 I4 6 9.538.429I6 373 14 47 484 9.55003 3 493 9.97078 5 953 957925 38 0.42075 348 13 48 511 9.55036 483 9.97073 5 986 9.57963 38 0.42037 325 I2 49 538 9-55069 33 472 997068 38020 9.58001 38 0.4999 302 II 33 5 38 50 35565 9.55102 93462 9.97063 38053 958039 38 0.41961 2.6279 10 51 592 95536 452 9.97059 86 958077 0.41923 256 9 3o86 9.58077. 38 0.492 256 52 619 95569 33 44 9.9754 20 9.58115 38 0.4885 233 53 647 9.55202 431 9-97049 I53 9-58I53 38.4847 20 7 54 674 9-55235 3 420 9.97044 5 1 86 9.5819I 3 0.41809 187 6 55 36701 9.55268 93410 9.97039 38220 9.58229 38 0.4177I 2.665 5 56 728 9-5530I 400 9.97035 4 253 9.58267 0.41733 142 4 57 755 9-55334 33 389 997030 286 9.58304 3 0.41696 119 3 58 782 9.55367 3 379 997025 5 320 958342 38 0.41658 096 2 59 8o0 9.55400 33 368 9.97020 35 3 9 3 0 0 74 60 837 9.55433 33 358 9.97015 386 9.58418.41582 051 0 386 9.58418 O.41582 O51 0 Nat. COS Log. d. INat. Sin Log. d. Nat. Cot Log. c.d. Log.TanNat. ' 69~ 210 1 0 f Nat. Sin Log.- d. INat. COS Log. d. INat.TanLog. Ic~d Log. Cot Nat.1 0 I 2 3 4 -w 6 7 8 9 II 12 '3 '4 15 I6 '7 20 21 22 23 24 295 26 27 28 29 30 3' 32 33 34 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 56 57 58 59 60 35837 9.55433 864 9.55466 891 9.55499 918 9.55532 945 9.55564 35973 9.55597 36000 9.55630 027 9.55663 054 9.55695 081 9.55728 36108 9.5576i 135 9.55793i 162 9.55826 190 9.55858 2I7 9.5589i 36244 9.55923 27I 9.55956 298 9.5.5988 325 9.56021 352 9.56053 36379 9.56085 406 9.56i118 434 9.56150 461 9.56182 488 9.56215 365I5 9.56247 542 9.56279 569 9.56311I 596 9.56343 623 9.56375 36650 9.56408 677 9.56440 704 9.56472 73I 9.56504 758 9.56536 36785 9.56568 812 9.56599 839 9.5663I 867 9.56663 894 9.56695 36921 9.56727 948 9.56759 975 9.56790 37002 9.56822 029 9.56854 37056 9.56886 083 9.56917 110 9.56949 137 9.56980 164 9.57012 37191 9.57044 218 9.57075 245 9.57107 272 9.57I38 299 9.57i69 37326 9.5720I 353 9.57232 380 9.57264 407 9.57295 434 9.57326 461 9.57358 33 33 33 32 33 33 33 32 33 33 32 33 32 33 32 33 32 33 32 32 33 32 32 33 32 32 32 32 32 33 32 32 32 32 32 3'I 32 32 32 32 32 3' 32 32 32 3' 32 3' 32 32 3' 32 3' 3' 32 3' 32 3' 3' 32 93358 9.970I5 348 9.970i0 337 — 9.97005 327 9.97001 316 9.96996 93306 9.9699i 295 9.96986 285 9.9698i 274 9.96976 264 9.96971 93253 9.96966 243 9.96962 232 9.96957 222 9.96952 211 9.96947 93201 9.96942 190 9.96937 I80 9.96932 169 9.96927 3159 9.96922 93148 9.969I7 I37 9.969I2 127 9.96907 ii6 9.96903 io6 9.96898 93095 9.96893 084 9.96888 074 9.96883 063 9.96878 052 9.96873 93042 9.96868 031 9.96863 020 9.96858 010 9.96853 92999 9.96848 92988 9.96843 978 9.96838 967 9.96833 956 9.96828 945 9.96823 92935 9. 96818 924 9.968I3 9I3 9.96808 902 9.96803 892 9.96798 92881 9.96793 870 9.96788 859 9.96783 849 9.96778 838 9.96772 92827 9.96767 8i6 9.96762 8o5 9.96757 794 9.96752 784 9.96747 92773 9.96742 762 9.96737 75I 9.96732 740 9.96727 729 9.96722 718 9.96717 S S 4 5 5 5 5 S S 5 4 5 5 S 5 5 5 5 5 5 5 5 4 5 5 5 5 5 5 5 5 S 5 S 5 S 5 S S 5 38386 9.584i8 420 9.58455 453 9.58493 487 9.5853I 520 9.58569 38553 9.58606 587 9.58644 620 9.58681 654 9.58719 687 9.58757 3872I 9.58794 754 9.58832 787 9.58869 82I 9.58907 854 9.58944 38888 9.58981 92I 9.59019 955 9.59056 988 9.59094 39022 9.59I31 39055 9.59168 089 9.59205 122 9.59243 156 9.59280 190 9.59317 39223 9.59354 257 9.5939I 290 9.59429 324 9.59466 357 9.59503 39391 9.59540 425 9.59577 458 9.59614 492 9.5965I 526 9.59688 39559 9-5-9725 593 9.59762 626 9.59799 66o 9.59835 694 9.59872 39727 9.59909 761 9.59946 795 9.59983 829 9.60019 862 9.60056 39896 9.60093 930 9.60I30 963 9.601i66 997 9.60203 40031 9.60240 40065 9.60276 098 9.60313 132 9.60349 i66 9.60386 200 9.60422 40234 9.60459 267 9.60495 301 9.60532 335 9.60568 369 9.60605 403 9.60641 37 38 38 38 37 38 37 38 38 37 38 37 38 37 37 38 37 38 37 37 37 38 37 37 37 37 38 37 37 37 37 37 37 37 37 37 37 36 37 37 37 37 36 37 37 37 36 37 37 36 37 36 37 36 37 36 37 36 37 36 0.41582 2.605i 0.41545 028 0.41507 oo6 0.4I469 2.5983 0.4I431 961 0.4I394 2.5938 0.4I356 916 0.4I3i9 893 0.4I28i 87I 0.4I243 848 0.4I2o6 2.5826 0.4Ii68 804 0.4II3I 782 0.4I093 759 0.4I056 737 0.410I9 2.57I5 0.4098i 693 0.40944 67I 0.40906 649 0.40869 627 0.40832 2.5605 0.40795 583 0.40757 561 0.40720 539 0.40683 5I7 0.40646 2.5495 0.40609 473 0.4057I 45 2 0.40534 430 0.40497 408 0.40460 2.5386 0.40423 365 0.40386 343 0.40349 322 0.403I2 300 0.40275 2.5279 0.40238 257 0.40201 236 0.40165 214. 0.40128 193 0.40091 2.5I72 0.40054 150 0.400I7 129 0.3998i io8 0.39944 o86 0.39907 2.50)65 0.39870 044 0.39834 023 0.39797 002 0.39760 2.498i 0.39724 2.4960 0.39687 939 0.3965i 918 0.39614 897 0.39578 876 0.3954I 2.4855 0.39505 834 0.39468 813 0.39432 792 0.39395 772 0.39359 75I 60 59 57 54 53 52 5' 50 49 48 47 46 45 -44 43 42 4' 40 39 38 37 36 -35 34 33 32 3' 30 29 28 27 26 25 -24 23 22 21 20 '9 '7 15 '4 '3 12 I' 10 9 8 7 6 5 4 3 2 I 0 S 5 S 5 S 5 5 S 6 S S S S S S S S S S S lNat.COS Log. d. Nat. Sin Log. d. INat. Cot Log. c.dl.lLog.TanlNat. f 680 22? fINat. S in Log. d. INat. Co0s Log. d. INat.Tafl Log.Ic.d. lLog. Cot Nat.I 0 I 2 3 4 -i 6 7 8 9 10 II 12 '3 '4 15 '7 I8 20 21 22 23 24 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 56 57 59 60 37461 9.57358 488 9.57389 5I5 9.57420 542 9.5745I 569 9.57482 37595 9-575I4 '622 9.57545 649 9.57576 676 9.57607 703 9.57638 37730 9.57669 757 9.57700 784 9.57731 8ii 9.57762 838 9.57793 37865 9.57824 892 9.5785 -9I9 9.57885 946 9.57916 -973 9.57947 37999 9.57978 38026 9.58008 053 9.58039 o8o 9.58070 107 9.58101 38I34 9.58I31 i6i 9.58162 i88 9.58192 259.58223 241 9.58253 38268 9.58284 295 9.583I4 322 9.5834 -349 9.58375 376 9.58406 38403 9.584.36 430 9.58467 456 9.58497 483 9.58527 510 9.58557 38537 9.58588 564 9.586i8 59I 9.58648 6I7 9.58678 644 9.58709 38671 9.58739 698 9.58769 725 9.58799 752 9.58829 778 9.58859 38805 9.58889 832 9.589i9 859 9.58949 886 9.58979 9I2 9.59009 38939 9.59039 966 9.59069 993 9.59098 39020 9.59128 046 9.59158 073 9.59i88 3' 3' 3' 3' 32 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 3' 30 3' 3' 3' 30 3' 3' 3' 30 3' 30 3' 30 3' 30 3' 30 3' 30 3' 30 30 30 3' 30 30 30 3' 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 927i8 9.967I7 707 9.967I1 697 9.96706 686 9.9670I 675 9.96696 92664 9.9669i 653 9.96686 642 9.9668i 631 9.96676 620 9.96670 92609 9.96665 598 9.9,6660 587 9.96655~ 576 9.96650 565 9.96645 92554 9.96640 543 9.96634 532 9.96629 52I 9.96624 510 9.966i9 92499 9.966I4 488 9.96608 477 9.96603 466 9.96598 455 9.96593 92444 9.96588 432 9.96582 421 9.96577 410 9.96572 399 9.96567 92388 9.96562 377 9.9~6556 366 9.9655I 355 9.96546 343 9.9654I 92332 9.96535' 32I 9.96530 310 9.96525 299 9.96520 287 9.965I4_ 92276 9.96509 265 9.96504 254 9.96498 243 9.96493 231 9.96488 92220 9.96483 209 9.96477 198 9.96472 i86 9.96467 I75 9.96461 92164 9.96456 I52 9.96451 141 9.96445 I30 9.96440 119 9.96435 92107 9.96429 096 9.96424 085 9.96419 073 9.96413 062 9.96408 050 9.96403 6 5 S 5 5 5 S 5 6 5 5 5 5 5 5 6 5 5 5 5 6 5 5 5 5 6 5 5 5 S 6 5 5 5 6 5 5 5 6 5 5 6 5 5 5 6 5 S 6 5 5 6 5 5 6 5 5 6 5 5 40403 9.6064i 436 9.60677 470 9.60714 504 9.160750 538 9.60786 40572 9.60823 6o6 9.60859 640 9.60895 674 9.60931 707 9.60967 4074I 9.6i004 775 9.61040 809 9.61076 843 9.6iiI2 877 9.61148 40911 9.61184 945 9.6I220 979 9.6I256 4I013 9.61292 047 9.61328 41081 9.61364 115 9.6I400 I49 9.6i436 183 9.61472 2I7 9.61508 41251 9.61544 285 9.61579 319 9.61615 353 9.6i651 387 9.61687 4I42I 9.6172-2 455 9.6I758 490 9.6i794 524 -9.6i830 558 9.6186 41592 9.6i901 626 9.6i936 66o 9.6I972 694 9.62008 72-8 9.62043 41763 9.62079 797 9.62114 831 9.621-50 865 9.62i85 899 9.6222I 41933 9.62256 968 9.62292 42002 9.62327 036 9.62362 070 9.62398 42105 9.62433 I39 9.62468 I73 9.62504 207 9.62539 242 9.62574 42276 9.62609 310 9.6264-5 345 9.62680 379 9~.627I5 413 9. 62750 447 9. 62785 36 37 36 36 37 36 36 36 36 37 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 35 36 36 36 35 36 36 36 35 36 35 36 36 35 36 35 36 35 36 35 36 35.35 36 35 35 36 35 35 35 36 35 35 35 35 0.39359 2.475I 0.39323 730 0.39286 709 0.39250 689 0.39214 668 0.39I77 2.4648 0.39141 627 0.39105 6o6 0.39069 586 0.39033 566 0.38996 2.4545 0.38960 525 0.38924 504 0.38888 484 0.38852 464 0.388i6 2.4443 0.38780 423 0.38744 403 0.38708 383 0.38672 362 0.38636 2.4342 0.38600 322 0.38564 302 0.38528 282 0.38492 262 0.38456 2.4242 0.3842I 222 0.38385 202 0.38349 182 0.383I3 162 0.38278 2.4142 0.38242 I22 0.38206 I02 0.38I70 083 0.38I35 063 0.38099 2.4043 0.38064 023 0.38028 004 0.37992 2.3984 0.37957 964 0.3792I 2.3945 0.37886 925 0.37850 906 0.37815 886 0.37779 867 0.37744 2.3847 0.37708 828 0.37673 8o8 0.37638 789 0.37602 770 0.37567 2.3750 0.37532 73I 0.37496 712 0.3746i 693 0.37426 673 0-37391 2.3654 0.37355 635 0.37320 6i6 0.37285 597 0.37250 578 0.372IS 559 60 59 58 57 55 54 53 52 5' 50 49 48 47 46 44 43 42 4' 39 38 37 36 35 34 33 32 3' 29 28 27 26 25 24 23 22 21 '9 'I J6 14 I3I I2 II 10 [Nat. COS Log.7 [.Nat. S in Log) d. INat. Cot Log. Ic~d.Log.TanlNat] 670 230 f Nat. Sin L-og-.-d-. N-a-t. CosLog. d.jNat Tan Log. c.d. Log. CotNat. I 0 I 2 3 4 -5 -6 7 8 9 10 II 12 13 '4 15 '7 '9 21 22 23 24 26 27 28 29 3' 32 33 34 36 37 38 39 -46 4' 42 43 44 45 46 47 48 49 50.5' 52 53 54 56 57 59 60 39073 9.59i88 100 9.592i8 127 9.59247 I53 9.59277 18o 9.59307 39207 9.59336 234 9.59366 260 9.59396 287 9.59425 3I4 9.59455 3934I 9.59484 367 9.59514 394 9. J9543 42I 9.59573 448 9.59602 39474 9.59632 50I 9.5966i 528 9.59690 555 9.59720 58I 9.59749 39608 9.59778 635 9.59808 66I 9.59837 688 9.!59866 715 9.59895 3974I 9.59924 768 9.59954 795 9.59983 822 9.600i2 848 9.6004I 39875 9.60070 902 9.60099 928 9.60I28 955 9.60157 982 g.6oi86 40008 9.60215 035 9.60244 062 9.60273 o88 9.60302 115 9.60331 40I4I 9.60359 i68 9.60388 195 9.604I7 221 9.60446 248 9.60474 40275 9.60503 301 9.60532 328 9.6056i 355 9.60589 381 9.60618 40408 9.60646 434 9.60675 461 9.60704 488 9.60732 514 9.6076i 40541 9.60789 567 9.608i8 594 9.60846 62I 9.60875 647 9.60903 674 9.60931 30 29 30 30 29 30 30 29 30 29 30 29 30 29 30 29 29 30 29 29 30 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 28 29 29 29 28 29 29 29 28 29 28 29 29 28 29 28 29 28 29 28 28 92050 9.96403 039 9.96397 028 9.96392 oi6 9.96387 005 9.9638i 91994 9.96376 982 9.96370 97I 9.96365 959 9.96360 948 9.916354 9I936 9.96349 925 9.96A43 9I4 9.96338 902 9.96333 89I 9.96327 91879 9.916322 868 9.963I6 856 9.963II 845 9.96305 833 9.96300 91822 9.96294 8io 9.96289 799 9.96284 787 9.96278 775 9.96273 9I764 9.96267 752 9.96262 74I 9.96256 729 9.9)625I 718 9.96245 9I706 9.96240 694 9.96234 683 9.96229 67I 9.96223 66o 9.962i8 91648 9.96212 636 9.96207 625 9.9620I 613 9.96196 6oi 9.96i90 91590 9.96185 578 9.96179 566 9.96i74 555 9.96i68 543 9.96162 91531 9.96I157 5I9 9.96151 5o8 9.96I46 496 9.96140 484 9.96135 91472 9.96129 461 9.96I23 449 9.96ii8 437 9.9611I2 425 9.96107 91414 9.96I10 402 9.96095 390 9.96090 378 9.96084 366 9.96079 355 9.96073 6 5 5 6 5 6 5 S 6 5 6 5 S 6 5 6 S 6 S 6 5 5 6 5 6 5 6 5 6 5 6 5 6 5 6 S 6 S 6 42447 9.62785 482 _9.62820 5i6 9.62855 551 9.62890 585 9.62926 42619 9.6296i 654 9.62996 688 9.6303I 722 9.63066 757 9.631Io 42791 9.63I35 826 9.63I70 86o 9.63205 894 9.63240 929 9.6.3275 42963 9.633I0 998 9.63345 43032 9.63379 067.9.63414 101 9.63449 43136 9.63484 I70 9.635i9 205 -9.63553 239 9.63588 274 9.63623 43308 9. 63657 343 9.63692 378 9.63726 412 9.63761 447 9.63796 43481 9.63830 ~516 9.63865 550 9.63899 585 9.63934 620 9.63968 43654 9.64003 689 9.64037 724 9.64072 758 9.64106 793 9.64I40 43828 9.64175 862.9.64209 897 _9.64243 932 9.64278 966 9.643I2 44001 9.64346 036 9.64381 07I 9.64415 105 9.64449 140 9.64483 44175 9.645117 210 9.645,52 244 9.64586 279 9.64620 314 9.64654 44349 9.64688 384 9.64722 418 9.64756 453 9.64790 488 9.64824 523 9.64858 35 35 35 36 35 35 35 35 35 34 35 35 35 35 35 35 34 35 35 35 35 34 35 35 34 35 34 35 35 34 35 34 35 34 35 34 35 34 34 35 34 34 35 34 34 35 34 34 34 34 35 34 34 34 34 34 34 34 34 34 0.372I5 2.3559 0.37180 539 0.37145 520 0.37II0 501 0.37074 483 0.37039 2.3464 0.37004 445 0.36969 426 0.36934 407 0.36899 388 0.36865 2.3369 0.36830 35I 0.36795 332 0.36760 313 0.36725 294 0.36690 2.3276 0.36655 257 0.3662I 238 0.36586 220 0.36551 201 0.36516 2.3i83 0.36481 164 0.36447:146 0.364I2 I27 0.36377 109 0.36343 2.3090 0.36308 072 0.36274 053 0.36239 035 0.36204 017 0.36170 2.2998 0.36135 980 0.36i01 962 0.36066 944 0.36032 925 0.35997 2.290'7 0.35963 889 0.35928 871 0.35894 853 0.35860 835 0.35825 2.281I7 0.35791 799 0.35757 781 0.35722 763 0.35688 745 0.35654 2.2727 0.35619 709 0.35585 691I 0.3555I 673 0-355I7 655 0.35483 2.2637 0.35448 620 0.35414 602 0.35380 584 0.35346 566 0.353I2 2.2549 0.35278 53'I 0.35244 513 0.35210 496 0.35176 478 0.35I42- 460 60 59 57 56 54 53 52 5' 50 49 48 47 46 44 43 42 41 39 38 37 36 34 33 32 3' 30 29 28 27 26 24 23 22 21 20 '9 '7 i -& '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 S 6 5 6 6 S 6 5 6 S 6 6 5 6 5 6 6 5 6 S 6 - Nat. 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CotNa 0 F - - -- - - - - - - - - Z:5 - --- I I- - ------- z5- -I I - - ---- - -- - - - - ID - I 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 175 '7 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 57 58 59 60 40674 9.6093I 700 9.60960 727 9.60988 753 9.6i016 780 9.6i045 40806 9.6I073 833 9.6ii0i 86o 9.61129 886 9.6i1I58 9I3 9.6ii86 40939 q.6I2I4 966 9.6I242 992 9.61270 41019 9.6I298 045 9.61326 4I072 9.6I354 098 9.6I382 125 9.61411 151 9.6i438 178 9.6I466 41204 9.61494 231 9.16I522 257 9.61550 284 9.6I578 310 9.6i606 41337 9.61634 363 9.6i662 390 9.6i689 416 9.61717 443 9.6i745 41469 9.61773 496 9.61800 522 9.61828 549 9.6i856 575 9.6i883 41602 9.61911 628 9.6i939 655 9.6i966 68i 9.61994 707 9.62021 41734 9.62049 760 9.62076 787 9.62I04 813 9.62131 840 9.62159 41866 9.62186 892 9.62214 919 9.6224I 945 9.62268 972 9.62296 41998 9.62323 42024 9.62350 051I 9.62377 077 9.62405 104 9.62432 42130 9.62459 I56 9.62486 183 9.62513 209 9.62541 235 9.62568 262 9.62595 29 28 28 29 28 28 28 29 28 28 28 28 28 28 28 28 29 27 28 28 28 28 28 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 27 28 27 28 27 28 27 28 27 27 28 27 27 27 28 27 27 27 27 28 27 27 91355 9.96073 6 343 9.96067 33I 9.96062 6 3I9 9-96o$6 6 307 9.96050 9I295 9.96045 6 283 9.96039 272 9.96034 6 260 9.96028 6 248 9.96022 91236 9.96017 6 224 9.96011 6 2I2 9.9600~5 200 9.96000 6 i88 9.59 6 911I76 9.95988 6 164 9.95982 5 152 9-95977 6 140 9.95971 6 128 9.9596 5 91116 9.95960 6 104 9-95954 6 092 9.95948 6 o8o 9.95942 o68 9.53 6 91056 9.9593T 6 044 9.95925 032 9.95920 6 020 9.959146 0o8 9.95908 6 972 9.95891 6 960 9.95885 6 948 9.95879 6 90936 9.95873 5 924 9.95868 6 911 9.95862 6 899 9.958$6 6 887 9.95850 6 90875 9.95844 863 9.95839 6 85I 9.95833 6 839 9.95827 6 826 9.95821 90814 9.9581~56 802 9.95810 6 790 9.95804 6 778 9.95798 6 766 9.95792 6 90753 9.95786 6 74I 9.95780 5 729 9.95775 6 7I7 9.95769 6 704 9.95763 6 90692 9.95757 6 68o 9.95751 6 668 9-95745 6 655 9.95739 6 643 9.95733 63I 9.957285 44523 9.64858 55 9.64892 593 9.64926 627 9.64960 662 9.64994 44697 9.65028 732 9.65062 767 9.65096 802 9.65130 837 9.65164 44872 9.65197 907 9.65231 942 9.65265 977 9.65299 45012 9.65333 45047 9.65366 082 9.65400 117 9.65434 I52 9.65467 187 9.65501 45222 9.65535 257 9.65568 292 9.65602 327 9.65636 362 9.65669 45397 9.65703 432 9.65736 467 9.65770 502 9.65803 538 9.65837 45573 9.65870 6o8 9.65904 643 9.65937 678 9.6597I 713 9.65004 45748 9.66038 784 9.66071 819 9.66104 854 9.66138 889 9.66171 45924 9.66204 960 9.66238 995 9.66271 46030 9.66304 o65 9.66337 46101 9.6637I 136 9.66404 171 9.66437 206 9.66470 242 9.66503 46277 9.66537 3I2 9.66570 348 9.66603 383 9.66636 418 9.66669 46454 9.66702 489 9.66735 525 9.66768 56o 9.66801 595 9.66834 631 9.66867 340.35I42 2.2460 340.35I08 443 340.35074 425 340.35040 408 340.35006 390 340.34972 2.2373 340.34938 355 340.34904 338 34 0.34870 320 33 o.3486 303 0.34803 2.2286 34 0.34769 268 340.34735 251 340.3470I 234 34 0.34667 216 33 0.34634 2.2I99 34 0.34600 182 34 0.34566 i65 330.34533 I48 340.34499 130 34 0.34465 2.2113 330.34432 096 34 0.34398 079 34 0.34364 062 33 0.34331 045~ 330.34297 2.2028 0.34264 OIl 340.34230 2.1994 33 0.34197 977 34 0.34163 960 330.34I30 2.1943 34 0.34096 926 33 0.34063 909 340.34029 892 33 0.33996 876 34 0.33962 2.1859 33 0.33929 842 33 0.33896 825 34 0.33862 8o8 33 0.33829 792 33 0.33796 2.I775 34 0.33762 758 33 0.33729 742 33 0.33696 725 33 0.33663 708 34 0.33629 2.1692 33 0.33596 675 33 0.33563 659 33 0.33530 642 33 0.33497 625 34 0.33463 2.i609 33 0.33430 592 33 0.33397 576 33 0.33364 56o 33 0.3333I 543 33 0.33298 2.I527 33 0.3.3265 510 33 0.33232 494 33 0.33199 478 333 0.33i66 461 33 0.33133 445 60 59 58 57 -56 54 53 52 SI 50. 49 48 47 46_ -45 -44 43 42 4' 40 39 38 37 -36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 17 15 '4 '3 12 II 10 9 8 7 6 S 4 3 2 I 0 Nat.COS Log. d. Nat. Sin Log. d. Nat. Cot Log. c~d. Log.TaflNat. p j I f INat. Sin Log. d. INat. COS Log. d. lNat.Taan Log.Ic.d. Log. Cot Nat.j Oj 42262 9.62595 I, 288 9.62622 2 3I5 9.62649 3 341 9.62676 4 367 9.62703 5 42394 9.62730 6 420 9.62757 7 446 9.62784 8 473 9.628ii 9 499 9.62838 10 42525 9.62865 II 552 9.62892 12 578 9.62918 13 604 9.62945 14 631 9.62972 15 42657 9.62999 i6 683 9.63026 17 709 9.63052 i8 736 9.63079 19 762 9.63Io6 20 42788 9.63I33 21 8i5 9.63159 22 841 9.63186 23 867 9.632I3 24 894 9.63239 25 42920 9.63266 26 946 9.63292 27 972 9.633I9 28 999 9.63345 29 43025 9.63372 30 43051' 9.63398 31 077 9.63425 32 104 9.63451 33 130 9.63478 34 I56 9.63504 35 43182 9.6353I 36 209 9.63557 37 235 9.63583 38 261 9.636i0 39 287 9.63636 40 43313 9.63662 41 340 9.63689 42 366 9.63715 43 392 9.6374I 44 418 9.63767 45 43445 9.63794 46 471 9.63820 47 497 9.63846 48 523 9.63872 49 549 9.63898 50 43575 9.63924 5I 602 9.63950 52 628 9.63976 53 654 9.64002 54 68o 9.64028 55 43706 9.64054 56 733 9.64080 57 759 9.64106 58 785 9.64I32 59 8ii 9.64I58 60 837 9.64i84 27 27 27 27 27 27 27 27 27 27 27 26 27 27 27 27 26 27 27 27 26 27 27 26 27 26 27 26 27 26 27 26 27 26 27 26 26 27 26 26 27 26 26 26 27 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 90631 9.95728 6i8 9.95722 6o6 9.95716 594 9.957IO 582 9.95704 90569 9.95698 557 9.95692 545 9.95686 532 9.95680 520 9.95674 90507 9.95668 495 9.95663 483 9.95657 4-70 9.9565I 458 9.95645 90446 9.95639 433 9.95633 42I 9.95627 408 9.95621 396 9.956i5 90383 9.95609 371 9.95603 358 9.95597 346 9.9559I 334 9.95585 90321 9.95579 309 9.95573 296 9.95567 284 9.95561 271 9.95555 90259 9.95549 246 9,95543 233 9.95537 221 9.95531 208 9.95525 90196 9.955I9 183 9.95513 171 9.95507 158 9.95500 146 9.95494 90133 9.95488 120 9.95482 io8 9.95476 095 9.95470 082 9.95464 90070 9.95458 057 9.95452 045 9.95446 032 9.95440 019 9.95434 90007 9.95427 89994 V-9542I 981 9.95415 968 9.95409 956 9.95403 89943 9.95397 930 9.9539I 918 9.95384 905 9.95378 892 9.95372 879 9.95366 6 6 6 6 6 6 6 6 6 6 S 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 6 6 6 6 6 7 6 6 6 6 6 6 7 6 6 6 46631 9.66867 666 9.66900 702 9.66933 737 9.66966 772 9.66999 46808 9.67032 843 9.67065 879 9.67098 914 9.6713I 950 9.67i63 46985 9.67I96 47021 9.67229 056 9.67262 092 9.67295 128 9.67327 47163 9.67360 199 9.67393 234 9.67426 270 9.67458 305 9.6749i 4734I 9.67524 377 9.67556 412 9.67589 448 9.67622 483 9.67654 47519 9.67687 555 9.67719 590 9.67752 626 9.67785 662 9.678I7 47698 9.67850 733 9.67882 769 9.67915 805 9.67947 840 9.67980 47876 9.680I2 912 9.68044 948 9.68077 984 9.68109 48019 9.68142 48055 9.68174 091 9.68206 127 9.68239 163 9.68271 198 9.68303 48234 9.68336 270 9.68368 306 9.68400 342 9.68432 378 9.68465 48414 9.68497 450 9.68529 486 9.6856i 521 9.68593 557 9.68626 48593 9.68658 629 9.68690 665 9.68722 701 9.68754 737 9.68786 773 9.68818 33 33 33 33 33 33 33 33 32 33 33 33 33 32 33 33 33 32 33 33 32 33 33 32 33 32 33 33 32 33 32 33 32 33 32 32 33 i32 I33 32 32 33 32 32 33 32 32 32 33 32 32 32 32 33 32 32 32 32 32 32 0.33I33 2.1445 0.33I00 429 0.33067 413 0.33034 396 0.33001 380 0.32968 2.1364 0.32935 348 0.32902 332 0.32869 315 0.32837 299 0.32804 2.1283 0.32771 267 0.32738 251 0.32705 235 0.32673 219 0.32640 2.1203 0.32607 287 0.32574 171 0.32542 155 0.32509 139 0.32476 2.1123 0.32444 I07 0.32411 092 0.32378 076 0.32346 o6o 0.32313 2.1044 0.32281 028 0.32248 013 0.32215 2.0997 0.32183 981 0.32150 2.0965 0.32118 950 0.32085 934 0.32053 918 0.32020 903 0.31988 2.0887 0.31956 872 0.3I923 856 0.31891 840 0.31858 825 0.31826 2.0809 0.31794 794 0.31761 778 0.3I729 763 0.31697 748 0.31664 2.0732 0.31632 717 0.31600 701 0.31568 686 0.31535 671 0.3I503 2.0655 0.3147i 640 0.3I439 625 0.31407 609 0.31374 594 0.31342 2.0579 0.3I3I0 564 0.3I278 549 0.31246 533 0.31214 518 0.31182 503 60 59 58 57 56 55 54 53 52 5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 3 34 33 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 I INat. UOS Log. d. INat. Sin Log. d. Nat. COt Log. c.d. Log.TanlNat.]' I 640 260 f Nat. Sin Log. d. jNat. COS Log. d. jNat.TanLog.1c.d. Log. Cot Nat.j 0 I 2 3 4 5 6 7 8 9 10 II 12 13 '4 15 i6 17 188 19 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 5' 52 53 54 56 57 58 59 60 43837 9.64184 863 9.64210 85 9 9.64236 916 9.64262 942 9.64288 43968 9.643I3 994 9.64339 44020 9.64365 046 9.6439i 072 9.64417 44098 9.64442 124 9.64468 I5I 9.64494 177 9.645I9 203 9.64545 44229 9.6457I 255 9.64596 281 9.64622 307 9.64647 333 9.64673 44359 9.64698 385 9.64724 411 9.64749 437 9.64775 464 9.64800 44490 9.64826 5i6 9.64851 542 9.64877 568 9.64902 594 9.64927 44620 9.64953 646 9.64978 672 9.65003 698 9.65029 724 9.65054 44750 9.65079 776 9.65I04 802 9.65I30 828 9.65I55 854 9.65i80 44880 9.65205 906 9.65230 932 9.65255 958 9.6528i 984 9.65306 450IO 9.6533I 036 9.65356 062 9.6538I o88 9.65406 II4 9.6543I 45140 9.65456 i66 9.65481 192 9.65506 218 9.6553I 243 9.65556 45269 9.65580 295 9.65605 321 9.65630 347 9.65655 373 9.65680 399 9.65705 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 89879 9.95366 867 9.95360 854 9.95354 841 9.95348 828 9.9534I 89816 9.95335 803 9.95329 790 9.95323 777 9.953I7 764 9.953IO 89752 9.95304 739 9.95298 726 9.95292 713 9.95286 700 9.95279 89687 9.95273 674 9.95267 662 9.95261 649 9.95254 636 9.95248 89623 9.95242 6io 9.95236 597 9.95229 584 9.95223 57I 9.95217 89558 9.95211 545 9.95204 532 9.95I98 519 9.95192 506 9.95185 89493 9.95179 480 9.95173 467 9.95167 454 9.95160 441 9.95I54 89428 9.95148 415 9.95I4I 402 9.951135 389 9.95129 376 9.95122 89363 9.951i6 350 9.95II0 337 9.95103 324 9.95097 311 9.95090 89298 9.95084 285 9.95078,272 9.95071 259 9.95065 245 9.95059 89232 9.95052 219 9.95046 206 9.95039 193 9.95033 i8o 9.95027 89167 9.95020 153 9.950I4 140 9.95007 127 9.95001 I14 9.94995 101 9.94988 6 6 6 7 6 6 6 6 7 6 6 6 6 7 6 6 6 7 6 6 6 7 6 6 6 7 6 6 7 6 6 6 7 6 6 7 6 6 7 6 6 7 6 7 6 6 7 6 6 7 6 7 6 6 7 6 7 6 6 7 48773 9.688i8 809 9.68850 845 9.68882 88i 9.68914 917 9.68946 48953 9.68978 989 9.69010 49026 9.69042 062 9.69074 098 9.69106 49134 9.69I38 170 9.69I70 206 9.69202 242 9.69234 278 9.69266 49315 9.69298 351 9.69329 387 9.69361 423 9.69393 459 9.69425 49495 9.69457 532 9.69488 568 9.69520 604 9.69552 640 9.69584 49677 9.696i5 713 9.69647 749 9.69679 786 9.697I0 822 9.69742 49858 9.69774 894 9.69805 931 9.69837 967 9.69868 50004 9.69900 50040 9.69932 076 9.69963 II3 9.69995 149 9.70026 185 9.70058 50222 9.70089 258 9.70I21 295 9.70I52 331 9.70184 368 9.702I5 50404 9.70247 441 9.70278 477 9.70309 514 9.7034I 550 9.70372 50587 9.70404 623 9.70435 66o 9.70466 696 9.70498 733 9.70529 50769 9.70560 8o6 9.70592 843 9.70623 879 9.70654 916 9.70685 953 9.707I7 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 3' 32 32 32 32 3' 32 32 32 31 32 32 31 32 32 3I 32 3' 32 32 3' 32 3' 32 3' 32 3' 32 3' 32 3' 3' 32 3' 32 3' 3' 32 3' 3' 32 31 3' 3' 32 0.31182 2.0503 0.31150 488 0.31118 473 0.31086 458 0.31054 443 0.31022 2.0428 0.30990 413 0.30958 398 0.30926 383 0.30894 368 0.30862 2.0353 0.30830 338 0.30798 323 0.30760 308 0.30734 293 0.30702 2.0278 0.30671 263 0.30639 248 0.30607 233 0.30575 219 0.30543 2.0204 0.30512 189 0.30480 I74 0.30448 i6o 0.30416 I45 0.30385 2.0130 0.30353 II5 0.30321 101 0.30290 o86 0.30258 072 0.30226 2.0057 0.30195 042 0.30163 028 0.30132 013 0.30100 1.9999 0.30068 1.9984 0.30037 970 0.30005 955 0.29974 941 0.29942 926 0.29911 1.9912 0.29879 897 0.29848 883 0.29816 868 0.29785 ~ 854 0.29753 I.9840 0.29722 825 0.29691 Sii 0.29659 797 0.29628 782 0.29596 1.9768 0.29565 754 0.29534 740 0.29502 725 0.29471 711 0.29440 1.9697 0.29408 683 0.29377 669 0.29346 654 0.293i5 640 0.29283 626 60 59 58 57 -56 55 54 53 52 5' 50 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 2-5 24 23 22 21 20 '9 I8 '7 i6 15 14 '3 12 I' 10 9 8 7 6 5 4 3 2 I 0 Nat.OCOS Log. d. Nat. Si n Log. d. Nat. Cot Log. cd.jLog.Tan Nat.If 6.30 270 I 0 INat. Sin Log. d. INat. COS Log. d. INatTan Log.1c.d.JLog. COtNat.J 0 I 2 3 4 - 6 7 8 9 10 II I2 '3 '4 15 i6 '7 i8 '9 20 21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5I 52 53 54 55 56 57 58 59 60 45399 9.6570 2 425 9.65729 24 451 9.65754 25 477 9.65779 25 503 9.65804 25 24 45529 9.65828 25 554 9.65853 25 580 9.65878 24 6o6 9.65902 25 632 9.65927 25 45658 9.65952 24 684 9.65976 25 710 9.66001 25 736 9.66025 24 762 9.66050 25 — 25 45787 9.66075 813 9.66099 25 839 9.66124 25 865 9.66148 24 89i 9.66173 25 24 459I7 9.66197 24 942 9.66221 25 968 9.66246 25 994 9.66270 24 46020 9.66295 25 46046 9.663I9 072 9.66343 24 097 9.66368 25 123 9.66392 24 149 9.664i6 24 46I75 9.6644i 25 201 9.66465 24 226 9.66489 24 252 9.665I3 24 278 9.66537 24 46304 9.66562 25 330 9.66586 24 355 9.66610 24 381 9.66634 24 407 9.66658 24 46433 9.66682 24 458 9.66706 24 484 9.6673I 25 510 9.66755 24 536 9.66779 24 46561 9.66803 24 587 9.66827 24 6I3 9.66851 24 639 9.66875 24 664 9.66899 24 46690 9.66922 23 716 9.66946 24 742 9.66970 24 767 9.66994 24 793 9.67018 24 46819 9.67042 24 844 9.67066 24 870 9.67090 24 896 9.67II3 23 921 9.67I37 24 947 9.67i6I 24 I I 89ioi 9.94988 6 087 9.94982. 7 074 9-94975 6 o6, 9.94969 048 9.94962 7 89035 9.94956 02 021 994949 6 008 9.94943 88995 9.94936 7 981 9.94930 88968 9.94923 6 955 9-949I7 6 942 9-949II 928 9.94904 915 9.94898 7 88902 9.94891 6 888 9.94885 875 9.94878 7 862 9.94871 7 848 9.94865 88835 9.94858 7 822 9.94852 8o8 9.94845 6 795 9.94839 782 9.94832 6 88768 9.94826 755 9.94819 6 741 9.948I3 728 9.94806 7 715 9.94799 7 6 88701 9.94793 688 9.94786 7 674 9.94780 66i 9.94773 7 647 9.94767 88634 9.94760 7 620 9.94753 7 607 9.94747 593 9.94740 7 580 9.94734 88566 9.94727 7 7 553 9.94720 6 539 9.947I4 526 9.94707 7 512 9.94700 6 88499 9.94694 6 485 9.94687 7 472 9.94680 6 458 9.94674 6 445 9.94667 7 88431 9.94660 6 417 9.94654 6 404 9.94647 7 390 9.94640 6 377 9.94634 88363 9.94627 7 349 9.94620 6 336 9.946I4 322 9.94607 7 308 9.94600 7 295 9.94593 7 50953 9.707I7 989 9.70748 5126 9.70779 063 9.708io 099 9-7084I 51136 9.70873 173 9.70904 209 9.70935 246 9.70966 283 9.70997 51319 9.71028 356 9.7I059 393 9.7I090 430 9.71121 467 9.7II53 51503 9.7184 540 9.71215 577 9.71246 614 9.71277 65I 9.7I308 51688 9.7I339 724 9.7I370 761 9.71401 798 9.7I431 835 9-7I462 51872 9.71493 909 9.71524 946 9.71555 983 9.7I586 52020 9.7i6I7 52057 9.71648 094 9.71679 131 9.7I709 i68 9.7I740 205 9.7I77I 52242 9.71802 279 9.7I833 316 9.71863 353 9.71894 390 9.7I925 52427 9.71955 464 9.71986 501 9.720I7 538 9.72048 575 9.72078 52613 9.72I09 65o 9.72140 687 9.72170 724 9.72201 761 9.72231 52798 9.72262 836 9.72293 873 9.72323 910 9.72354 947 9.72384 52985 9.7241$ 53022 9.72445 059 9.72476 096 9.72506 134 9.72537 171 9.72567 3' 31 3' 3' 32 3' 3' 3' 3' 3' 3' 3' 3' 32 31 3' 3' 3' 3' 3' 31 3' 30 3' 3' 3' 3' 3' 3' 3' 3' 30 3' 3' 3' 3' 30 3' 3' 30 3' 3' 3' 30 3' 3' 30 3' 30 31 3' 30 31 30 3' 30 3' 30 31 30 0.29283 i.9626 0.29252 612 0.2922I 598 0.29190 584 0.29159 570 0.29127 I.9556 0.29096 542 0.290g6 528 0.29034 514 0.29003 500 0.28972 I.9486 0.28941 472 0.28910 458 0.28879 444 0.28847 430 0.28816 1.9416 0.28785 402 0.28754 388 0.28723 375 0.28692 361 0.2866i I9347 0.28630 333 0.28599 319 0.28569 306 0.28538 292 0.28507 I.9278 0.28476 265 0.28445 251 0.28414 237 0.28383 223 0.28352 I.9210 0.28321 196 0.28291 183 0.28260 169 0.28229 I55 0.28198 I.9I42 0.28167 128 0.28137 115 0.28106 101 0.28075 o88 0.28045 I.9074 0.28014 o6i 0.27983 047 0.27952 034 0.27922 020 0.27891 I.9007 0.27860 i.8993 0.27830 980 0.27799 967 0.27769 953 0.27738 i.8940 0.27707 927 0.27677 913 0.27646 goo 0.27616 887 0.27585 1.8873 0.27555 86o 0.27524 847 0.27494, 834 0.27463 820 0.27433 807 60 59 58 57 56 55 54 53 52 5' 50 49 48 47 46 45 44 43 42 4I 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 iNat. COS Log. d. INat. Sin Log. d. INat. COt Log. c.d.lLog.Taf Nat.I ' 620 280 "rn fNat. Sin Log. d. INat. COS Log. d. jNat.TanLog.lc.d. Log. Cot Nat. I 0 I 2 3 4 5 6 7 8 9 10 II 12 '3 14 15 i6 17 i8 '9 20 21 22 23 24 25 26 27 28 29 30 3' 32 33' 34 35 36 37 38 39 40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 55 56 57 58 59 60 46947 9.67161 973 9.67185 999 9.67208 47024 9.67232 050 9.67256 47076 9.67280 101 9.67303 127 9.67327 153 9.67350 178 9.67374 47204 9.67398 229 9.67421 255 9.67445 281 9.67468 306 9.67492 47332 9.675I5 358 9.67539 383 9.67562 409 9.67586 434 9.67609 47460 9.67633 486 9.67656 511 9.67680 537 9.67703 562 9.67726 47588 9.67750 614 9.67773 639 9.67796 665 9.67820 690 9.67843 47716 9.67866 741 9.67890 767 9.67913 793 9.67936 8i8 9.67959 47844 9.67982 869 9.68006 895 9.68029 920 9.68052 946 9.68075 47971 9.68098 997 9.68121 48022 9.68i44 048 9.68167 073 9.68190 48099 9.68213 124 9.68237 1i50 9.68260 175 9.68283 201 9.68305 48226 9.68328 252 9.68351 277 9.68374 303 9.68397 328 9.68420 48354 9.68443 379 9.68466 405 9.68489 430 9.68512 456 9.68534 481 9.68557 24 23 24 24 24 23 24 23 24 24 23 24 23 24 23 24 23 24 23 24 23 24 23 23 24 23 23 24 23 23 24 23 23 23 23 24 23 23 23 23 23 23 23 23 23 24 23 23 22 23 23 23 23 23 23 23 23 23 22 23 88295 9.94593 281 '9.94587 267 9.94580 254 9.94573 240 9.94567 88226 9.94560 2T3 9.94553 199 9.94546 I85 9.94540 172 9.94533 88158 9.94526 I44 9.945I9 130 9.94513 117 9.94506 103 9.94499 88089 9.94492 075 9.94485 062 9.94479 048 9.94472 034 9.94465 88020 9.94458 oo6 9.9445I 87993 9.94445 979 9.94438 965 9.94431 87951 9.94424 937 9.944I7 923 9.94410 909 9.94404 896 9.94397 87882 9.94390 868 9.94383 854 9.94376 840 9.94369 826 9.94362 87812 9.94355 798 9.94349 784 9.94342 770 9.94335 756 9.94328 87743 9.94321 729 9.943I4 715 9.94307 701 9.94300 687 9.94293 87673 9.94286 659 9.94279 645 9.94273 631 9.94266 617 9.94259 87603 9.94252 589 9.94245 575 9.94238 56i 9.9423I 546 9.94224 87532 9.94217 9i8 9.94210 504 9.94203 490 9.94i96 476 9.94i89 462 9.94182 6 7 7 6 7 7 7 6 7 7 7 6 7 7 7 7 6 7 7 7 7 6 7 7 7 7 7 6 7 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 7 7 6 7 7 7 7 7 7 7 7 7 7 7 7 7 53171 9.72567 208 9.72598 246 9.72628 283 9.72659 320 9.72689 53358 9.72720 395 9.72750 432 9.72780 470 9.72811 507 9.7284I 53545 9.72872 582 9.72902 620 9.72932 657 9.72963 694 9.72993 53732 9.73023 769 9.73054 807 9.83084 844 9.73II4 882 9.73144 53920 9.73I75 957 9.73205 995 9.7323$ 54032 9.73265 070 9.73295 54I07 9.73326 145 9.73356 183 9.73386 220 9.734i6 258 9.73446 54296 9.73476 333 9.73507 371 9.73537 409 9.73567 446 9.73597 54484 9.73627 522 9.73657 56o 9.73687 597 9-737I7 635 9.73747 54673 9.73777 711 9.73807 748 9.73837 786 9.73867 824 9.73897 54862 9.73927 900 9.73957 938 9.73987 975 9-740I7 55013 9.74047 55051 9.74077 089 9.74I07 127 9.74137 165 9.74166 203 9.74i96 55241 9.74226 279 9.74256 317 9.74286 355 9.74316 393 9.7434$ 43I 9.74375 3' 30 3' 30 3' 30 30 3I 30 3' 30 30 3' 30 30 3I 30 30 30 3' 30 30 30 30 3' 30 30 30 30 30 3' 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 29 30 30 30 30 30 29 30 0.27433 1.8807 0.27402 794 0.27372 781 0.27341 768 0.27311 755 0.27280 i.874I 0.27250 728 0.27220 715 0.27189 702 0.27159 689 0.27128 1.8676 0.27098 663 0.27068 65o 0.27037 637 0.27007 624 0.26977 i.86ii 0.26946 598 0.26916 585 0.26886 572 0.26856 559 0.26825 i.8546 0.26795 533 0.26765 520 0.26735 507 0.26705 495 0.26674 1.8482 0.26644 469 0.26614 456 0.26584 443 0.26554 430 0.26524 1.8418 0.26493 405 0.26463 392 0.26433 379 0.26403 367 0.26373 i.8354 0.26343 34I 0.26313 329 0.26283 316 0.26253 303 0.26223 1.8291 0.26193 278 0.26163 265 0.26133 253 0.26103 240 0.26073 1.8228 0.26043 215 0.26013 202 0.25983 190 0.25953 177 0.25923 1.8165 0.25893 152 0.25863 140 0.25834 127 0.25804 115 0.25774 1.8103 0.25744 090 0.25714 078 0.25684 o65 0.25655 053 0.25625 040 60 59 58 57 56 54 54 53 52 -5I 56 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 '9 i8 '7 i6 15 '4 '3 12 II 10 9 8 7 6 S 4 3 2 I 0 0 I Nat.COS Log. d.[Nat. Sin Log. d. Nat.COt Log. c.d. Log.Tanl Nat. F 610 290 INat. Sin Log. d. INat. COS Log. d. INatTan Log.Ic.d. Log. Cot Nat. 0 0 IJ 2 3 4 5 6 7 8 9 10 II 12 '3 '4 15 i6 '7 i8 '9 90 -21 22 23 24 25 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 46. 47 48 49 50 SI 52 53 154 55 56 57 58 59 60 48481 9.68557 5o6 9.68580 532 9.6860o 557 9.68625 583 9.68648 48608 9.68671 634 9.68694 659 9.68716 684 9.68739 710 9.68762 48735 9.68784 761 9.68807 786 9.68829 8ii 9.68852 837 9.68875 48862 9.68897 888 9.68920 913 9.68942 938 9.68965 964 9.68987 48989 9.69010 49014 9.69032 040 9.69055 c)65 9.69077 090 9.6g100 49116 9.69122 141 9.69i44 i66 9.69167 192 9.69i89 217 9.69212 49242 9.69234 268 9.69256 293 9.69279 318 9.69301 344 9.69323 49369 9.69345 394 9.69368 419 9.69390 445 9.694I2 470 9.69434 49495 9.69456 52I 9.69479 546 9.6950I 571 9.69523 596 9.69545 49622 9.69567 647 9.69589 672 9.696i1 697 9.69633 723 9.69655 49748 9.69677 773 9.69699 798 9.6972I 824 9.69743 849 9.69765 49874 9.69787 899 9.69809 924 9.69831 950 9.69853 975 9.69875 50000 9.69897 23 23 22 23 23 23 22 23 23 22 23 22 23 23 22 23 22 23 22 23 22 23 22 23 22 22 23 22 23 22 22 23 22 22 22 23 22 22 22 22 23 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 87462 9.94182 448 9.94I75 434 9.94168 420 9.94161 406 9.94154 87391 9.94I47 377 9.94I40 363 9.94133 349 9.94126 335 9.94II9 87321 9.94112 306 9.94i05 292 9.94098 278 9.94090 264 9.94083 87250 9.94076 235 9.94069 221 9.94062 207 9.94055 193 9.94048 87178 9.9404I 164 9.94034 150 9.94027 136 9.94020 121 9.94012 87107 9.94005 093 9.93998 079 9-9399I 064 9.93984 050 9.93977 87036 9.93970 021 9.93963 007 9.93955 86993 9.93948 978 9.93941 86964 9.93934 949 9.93927 935 9.93920 921 9.939I2 906 9.93905 86892' 9.93898 878 9.9389i 863 9.93884 849 9.93876 834 9.93869 86820 9.93862 805 9.93855 791 9.93847 777 9.93840 762 9.93833 86748 9.93826 733 9.938i9 719 9.938II 704 9.93804 690 9.93797 86675 9.93789 66i 9.93782 646 9.93775 632 9.93768 617 9.93760 603 9.93753 7 7 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 7 7 7 8 7 7 7 7 7 7 7 8 7 7 7 7 -7 8 7 7 7 7 8 7 7 7 8 7 7 7 7 8 7 7 8 7 7 7 8 7 5543' 9.74375 469 9.74405 507 9.74435 545 9-74465 583 9.74494 55621 9.74524 659 9.74554 697 9.74583 736 9.746I3 774 9.74643 55812 9.74673 85o 9.74702 888 9.74732 926 9.74762 964 9.7479I 56003 9.7482I 041 9.74851 079 9.74880 117 9.74910 I56 9.74939 56194 9.74969 232 9.74998 270 9.75028 309 9.75058 347 9.75087 56385 9.75II7 424 9.75146 462 9.75I76 501 9.75205 539 9.75235 56577 9.75264 6i6 9.75294 654 9.75323 693 9.75353 731 9.75382 56769 9.754II 8o8 9.7544I 846 9.75470 885 9.75500 923 9.75529 56962 9.75558 57000 9.75588 039 9.756I7 o78 9.75647 ii6 9.75676 57I55 9.75705 193 9.75735 232 9.75764 271 9.75793 309 9.75822 57348 9.75852 386 9.75881 425 9.759IO 464 9.75939 503 9.75969 57541 9.75998 58o 9.76027 619 9.76056 657 9.76086 696 9.76ii5 735 9.76I44 30 30 30 29 30 30 -29 30 30 30 29 30 30 29 30 30 29 30 29 30 29 30 30 29 30 29 30 29 30 29 30 29 30 29 29 30 29 30 29 29 30 29 30 29 29 30 29 29 29 30 29 29 29 30 29 29 29 30 29 29 0.25625 1.8040 0.25595 028 0.25565 oi6 0.25535 003 0.25506 1.7991 0.25476 I.7979 0.25446 966 0.25417 954 0.25387 942 0.25357 930 0.25327 I.79I7 0.25298 905 0.25268 893 0.25238 88i 0.25209 868 0.25179 1.7856 0.25149 844 0.25120 832 0.25090 820 0.25061 8o8 0.25031 1.7796 0.25002 783 0.24972 77I 0.24942 759 0.24913 747 0.24883 I.7735 0.24854 723 0.24824 711 0.2479$ 699 0.24765 687 0.24736 1.7675 0.24706 663 0.24677 65I 0.24647 639 0.246i8 627 0.24589 1.7615 0.24559 603 0.24530 591 0.24500 579 0.24471 567 0.24442 1.7556 0.24412 544 0.24383 532 0.24353 520 0.24324 5o8 0.24295 1.7496 0.24265 485 0.24236 473 0.24207 461 0.24I78 449 0.24148 1.7437 0.24119 426 0.24090 414 0.24061 402 0.24031 39I 0.24002 I.7379 0.23973 367 0.23944 355 0.239I4 344 0.23885 332 0.23856 32I 60 59 58 57 56 5 5 54 53 521 5' 49 48 47 46 45 44 43 42 4' 40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 201 '9 i8 '7 1i 16 -'4 '3 12 II 10 9 8 7 6 5 4 3 2 I 0 INat. COS Log. d. Nat. Sin Log. d. INat. Cot Log. c~d. Log.TanNat.1 ' I 600 300 Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog.c.d. Log. CotNat 50000 9.69897 22 86603 9.93753 57735 9764429 0.23856 1.7321 60 I 025 9.699I9 22 588 9.93746 774 9-76I73 29 23827 30959 74 9.76173 29 0.23827 309 59 2 50 9.69941 22 573 993738 8I3 9.76202.23798 297 58 3 076 9.69963 2 597 85I 9-7623 9 0.23769 286 57 2i 559 9.9373I 7 9 4 loI 9.69984 22 544 993724 7 89 9.76261 3 0.23739 274 56,,57 29 5 50I26 9.70006 22 86530 9-937I7 8 57929 9.76290 0.23710 1.7262 55 6 I5I 9.70028 22 5I5 9.93709 968 9-763I9 29 0.2368 25 54 7 176 9.70050 22 50I 9.93702 7 58007 976348 29 0.23652 239 53 8 201 9.70072 2I 486 9.93695 7 046 9.76377 29 0.23623 228 52 9 227 9.70093 2 47 9.93667 085 9.76406 0.23594 216 _5 10 50252 9.70I5 1 22 86457 9.93680 58124 9.76435 29 0.23505 1.7205 50 II 277 9-70I37 22 442 9.93673 7 62 9.76464 29 0.23530 I93 49 2 302 9.70159 2i 427 9.93665 20I 9.76493 0.23507 18248 13 327 9.70I80 413 9.93658 8 240 9.76522290.23478 17047 I4 352 9.70202 22 398 9.93650 279 9.7655 29 023449 159 46 1553 9.0.22 22 7 29 1550377 9-70224 2I 86384 9.93643 583I8 9.76580 0.23420 1.7I47 45 I6 403 9-76245 2 369 9.93636 7 357 9.76609 9 2339 3644 22 8936289.76609 30 0.23391 I26 44 I7 428 9.70267 21 354 993628 396 976639 0.23361 24 43 I8 453 9.70288 22 340 9.9362I 7 435 9.76668 290.23332 1I3 42 9 478 9.703I0 22 325 9-936I4 474 9-76697 2 0.23303 I02 4 20 50503 970332 21 86310 9.93606 58513 976725 0.2325 I7090 40 21 528 9.70353 22 29 993599 552 976754 29 0.23246 079 39 22 553 9.70375 21 281 9.93591 59I 9.76783 0.23217 67 38 23 578 9.70396 22 266 9-93584 7 63I 9.76812 29 0.23188 056 37 24 603 9.70418 25I 993577 7 670 9.7684I 9 0.23159 045 36 21 8 29 25 50628 9.70439 2286237 9-93569 58709 9.76870 29 0.23130 1.7033 35 26 654 9-70461 22 222 9.93562 748 9.76899 0.23101 02234 27 679 970482 22 207 993554 787 9.76928 29 0.23072 OII 33 28 704 9-70504 21 I92 9-93547 826 9.76957 0.23043 1.6999 32 29 729 9.70525 178 9.93539 865 9-76986 0.23014 988 22 7239 30 50754 9.70547 2 86I63 9.93532 58905 977015 0.2295 I6977 30 3I 779 9.70568 2 48 9-93525 944 9.77044 29 0.22956 905 29 32 804 9.70590 2I I33 9.935I7 983 9-77073 8 0.22927 954 28 33 829 9.70611 i 9 9.935IO 59022 9.77IOI 0.22899 943 27 22 IT9 9~935I028 -0. 22877I0 2 34 854 9.70633 2I I04 9.93502 06i 9.7730 9 0.22870 932 26 21 8 o7 29 -35 50879 9.70654 2 86089 993495 8 59 9.77159 0.22841 1.6920 25 36 904 9-70675 22 074 9.93487 140 9.77188 29 0.228I2 909 24 37 929 970697 2 059 9.93480 7 I79 97727 29 0.22783 898 23 38 954 9.70718 2I 045 9-93472 2i8 9.77246 28 022754 887 22 39 979 9-70739 030 9.93465 7 258 9-77274 0.22726 875 2I 22 8 29 40 5I004 9-7076I 2I 860I5 9.93457 59297 9.77303 0.22697 I.6864 20 4I 029 9.70782 2I 0oo 9.93450 7 336 9.77332 0.22668 853 I9 42 054 9.70803 2 85985 9.93442 376 9.77361 0.22639 842 i8 43 079 9.70824 2 970 993435 7 45 9.7739028 0.2260 831 I7 44 104 9-70846 956 9-93427 454 9.774I8 0.22582 820 16 21 7 29 45 5II29 9.70867 2I 85941 9.93420 8 59494 977447 29 0.22553 1.68o8 15 46 I54 9.70888 2I 926 9.934I2 533 977476 290.22524 797 4 47 179 9.70909 22 g9I 9.934058 573 9-77505 28 224 7 48 204 9-7093 21 896 9.93397 6I2 977533 0.22467 775 12 49 229 9.70972 88I 9.93390 7 65I 9.77562 29 0.22438 764 II 50 51254 9.70973 2I 85866 9.93382 59691 9.7759 28 0.22409 1.6753 10 51 279 970994 2 85 993375 8 730 9-7769 29 0.2238I 742 9 52 304 9.7IO15 21 836 9.93367 770 9.77648 0.22352 73I 8 53 329 9.7036 82 993360 7 809 9776772 0.22323 720 7 54 354 9-71058 22 806 9.93352 8 849 9-77706 0.22294 709 6 55 5I379 9-7I079 2I 85792 9-93344 7 59888 9-77734 2 0.22266 1.6698 5 56 404 9.7I0OO 2I 777 9.93337 98 9-77763 280.22237 687 4 57 429 9.712 21 762 9.93329 967 97779 29 0.22209 66 3 58 454 9.7II42 2I 747 9.93322 8 60007 9.77820 29 0.2280 665 2 59 479 9-7II63 732 99334 046 9.77849 28 0.225 654 I 60 504 9.7I184 7I7 9.93307 o86 9.77877 0.22123 643 0 Nat. CoSLog. d. Nat. Sin Log. d. Nat. CotLog. c.d. Log.Tan Nat. ' 659 I M 31~ Nat. Sin Log. d. Nat. Cos Log. c.d.Log. Cot Nat. 0 51504 9.71184 2, 85717 9.93307 8 60086 9.77877 0.22123 1.6643 60 I 529 9.7I205 21 702 9.93299 8 I26 9.77906 29 0.22094 632 59 2 554 9.7I226 21 687 9.93291 I65 9.77935 28 0.22065 621 58 3 579 9.7I247 2I 672 9.93284 78 205 977963 29 0.22037 6 57 4 604 9.71268 657 9.93276 245 9-77992 2 0.22008 599 56 5 51628 9.7I289 21 85642 9.93269 8 60284 9.78020 0.21980 1.6588 55 6 653 9-7I3IO 2I 627 9.93261 8 324 9.78049 28 0.2951 57754 7 678 9-7I33I 2 6I2 9.93253 364 9.78077 2 0.21923 566 53 8 703 9.7I352 21 597 9.93246 7 403 9.78I06 29 0.21894 555 52 9 728 9.7I373 582 9.93238 443 978I35 2 0.2865 545 51 20 8 28 10 5I753 9.7I393 21 85567 9.93230 60483 9-7863 29 0.21837 6534 50 II 778 9.7I44 2I 55 99322 522 9.78192 280.2808 52349 3 8 9.78i92 28 523 49 I2 803 9.7I435 21 536 9.93215 8 562 978220 29 0.2780 512 48 I3 828 9.7I456 21 52I 9.93207 602 9.78249 2 0.21751 50I47 14 852 9.71477 21 5o6 9.93200 o 14 852 9-7I477 21 5o6 993200 78 642 9-78277 2 0.21723 490 46 15 5I877 9-7498 2I 85491 9.93I92 8 6068 9.78306 28 0.21694 I.6479 45 I6 902 9-71519 20 476 9.93I84 721 9.78334 29 0.2I666 469 44 I7 927 9-7I539 21 46I 9-93177 7 76I 9-78363 28 0.21637 458 43 I8 952 9.7I560 21 446 9.9369 8 80 97839 28 0.2609 447 42 I9 977 9-7158I 43I 9.93I6I 84I 9.784I9 0.21581 436 4 _ 21 - 7 29 20 52002 9.7I602 20 85416 9.9354 8 6088i 9-78448 28 0.21552 1.6426 40 21 26 9.7I622 21 40I 993I46 8 92I 9-78476 29 0.21524 415 39 22 051 9.7I643 21I 385 9-93I38 96 97805 28 0.21495 404 38 23 o76 9.7I664 2 370 99313 6iooo 9-78533 29 0.21467 393 37 24 II 9.7I685 1 355 993I23 040 9.78562 28 0.2438 383 36 25 52126 9.7I705 21 85340 9-93II5 b o080 9.78590 28 0.2I4o 1.6372 35 26 I51 9.71726 21 325 9.93I08 78 20 9.78618 2 0.21382 361 34 27 I75 9.-7747 20 3I0 9-93IOO 8 i6o 9-7847 28 0.21353 35I 33 28 200 9-71767 21 294 9.93092 8 200 978675 0.21325 340 32 29 225 9.7I788 279 9-93084 240 9.78704 0.2I296 329 3I o21 7 28 3052250 9.71809 0 85264 993077 8 6280 9.78732 28 0.21268.63I9 30 31 275 9.7I829 2I 249 9-93069 8 320 978760 0.2240 308 29 32 299 9.780 20 234 9-9306 8 360 9-78789 28 0.2I2 297 28 33 324 9.7I870 2I 2I8 993053 400 9-78817 28 0.21183 287 27 34 349 9.71891 203 9.93046 7 44 978845 0.21155 276 26 35 52374 9.79I 2 8588 9.93038 8 6480 97887428 0.21126 1.626525 36 399 9.7I932 20 I73 9-93030 8 520 9-78902 28 0.21098 255 24 37 423 9.71952 2I 157 9.93022 8 561 978930 2 0.2I070 244 23 38 448 9-7I973 2I I42 9-930I4 6 9789592 0.2I04 234 22 39 473 9.71994 27 9.93007 7 28.2IOI3 223 2 39_ 473 9 308127993007 641 9.78987 28 0.21013 223 21 4052498 9-720I4 20 85112 9.92999 8 6i68i 9.790I5 28 0.20985 I.6212 20 41 522 9.72034 21 096 9.92991 8 72I 9-79043 2 0.20957 202 I9 42 547 9.7205$ 20 8 9.92983 76i 979072 28 0.20928 191I 8 43 572 9.72075 2 66992976 8o 9.79I0 28 0.20900 i8i 7 44 597 9.72096 051 9.92968 842 9.79I28 o.20872 170 i6 552220 8 28 45 5262I 9-72II6 21 85035 9-92960 8 61882 9.79156 0.20844 i.6i6o 15 020 9.92952 922 291728& 46 646 9-7237 20 8 28 0.2085 149 I4 572020 9992944 962 207 21 47 67I 9-7257 20 005 9929448 962 9-792I3 28.20787 I39 3 48 696 9.72177 21 84989 9.92936 62003 9.79241 8 0.20759 28 12 49 720 9.72I98 974 9.92929 7 43 9.79269 8 0.2073I 1I8 II 20 8 28 50 52745 9.7228 20 84959 9.9292I 8 62083 9.79297 2 0.20703.6107 10 1 7700 9972238 0.20674 097 9 5I 770 9-72238 2I 943 9.929I3 8 124 9.79326 2897 52 794 9.72259 20 928 9.92905 8 164 979354 28 0.20646 87 8 53 8I9 9-72279 20 9I3 9-92897 8 204 9.79382 28 0.20618 076 7 54 844 9-72299 2 897 992889 245 9.794IO 28 0.20590 o66 6 55 52869 9.72320 20 84882 9.9288I 62285 9-79438 28 0.20562 1.6055 5 56 893 9.72340 20 866 9.92874 7 325 9.79466 2 0.2053 045 4 57 918 9.72360~i 0.20505 034 3 57 9I8 9-72360 20 85I 9.92866 8 366 979495 28 20505 34 58 943 9.72381 20 836 9.92858 8 406 9.79523 28 0.20477 024 2 59 967 9.72401 20 820 9.92850 8 446 97955 28 0.20449 014 60 992 9.72421 805 9.92842 487 9.79579 0.20421 003 0,,,.....~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Nat. Co Log. d. Nat. Sin Log. d. Nat. COtLog. c.d.lLog.TanNat.1 ' I Il l 'I 320 PON F Nat. Sin Log. d. INat. COS Log. d. INat.TanLog.1c.d.1 Log. Cot Nat.1 I I 0 I 2 3 4 6 7 8 10 II 12 '3 '14 15 '7 20 21 22 23 24 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 40 4' 42 43 44 45 [46 47 48 49 50 ~5' 52 53 54 55 56 57 59 60 52992 9.7242I 53017 9.7244I 041 9.7246i o66 9.72482 09I 9.72502 53II5 9.72522 140 9.72542 164 9.72562 I89 9.72582 2I4 9.72602 53238 9.72622 263 9.72643 288 9.72663 312 9.72683 337 9.72703 53361 9.72723 386 9.72743 411 9.72763 435 9.72783A 460 9.72803 53484 9.72823 509 9.72843 534 9.72863 558 9.72883 583 9.72902 53607 9.72922 632 9.72942 6_;6 9.72962 68i 9.72982 705 9.73002 53730 9.73022 754 9.730411 779 9.7306i 804 9.7,308I 828 9.73i01 53853 9.73I2I 877 9.73I40 902 9.73i60 926 9.73i80 951 9.73200 53975 9.73219 54000 9.73239 024 9.73259 049 9.73278 073 9.73298 54097 9.73318 122 9.73337 146 9.73357 17I 9.73377 I95 9.73396 54220 9.73416 244 9.73435 269 9.73455 293 9.73474 3I7 9.73494 54342 9.735I3 366 9.73533 39I 9.73552 4I5 9.73572 440 9.735911 464 9.736ii 20 20 21 20 20 20 20 20 20 20 21 20 20 20 20 20 20 20 20 20 20 20 20 '9 20 20 20 20 20 20 '9 20 20 20 20 '9 20 20 20 '9 20 20 '9 20 20 '9 20 20 '9 20 '9 20 '9 20 '9 20 '9 20 '9 20 84805 9.92842 789 9.92834 774 9.92826 759 9.928i8 743 9.928i0 84728 9.92803 712 9.92795 697 9.92787 68i 9.92779 666 9.927711 84650 9.92763 635 9.92755 619 9.92747 604 9.92739 588 9.92731 84573 9.92723 557 9.92715 542 9.92707 526 9.92699 511 9.9269i 84495 9.9268,3 480 9.92675 464 9.92667 448 9.92659 433 9.9265I 84417 9.92643 402 9.92635~ 386 9.92627 370 9.926i9 355 9.926II 84339 9.92603 324 9.92595 308 9.92587 292 9.92579 277 9.9257I 84261 9.92563 245 9.92555 230 9.92546 214 9.92538 198 9.92530 84182 9.92522 167 9.925I4 151 9.92506 135 9.92498 I20 9.92490 84104 9.92482 o88 9.92473 072 9.92465 057 9.92457 041 9.92449 84025 9.9244I 009 9.92433 83994 9.92425 978 9.924i6 962 9.92408 83946 9.92400 930 9.92392 9I5 9.92384 899 9.9-2376 883 9.92367 867 9.92359 8 8 8 8 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 8 8 9 8 8 8 8 8 8 9 8 8 8 8 8 9 8 62487 9.79579 527 9.79607 568 9.79635 6o8 9.79663 649 9.7969I 62689 9.797I9 730 9.79747 770 9.79776 8II 9.79804 852 9.79832 62892 9.79860 933 9.79888 973 9.79916 63014 9.79944 055 9.79972 63095 9.80000 136 9.80028 I77 9.80056 2I7 9.80084 258 9.80112 63299 9.80140 340 9.80168 380 9.80195 421 9.80223 462 9.8025I 63-503 9.80279 544 9.80307 584 9.80335 625 9.80363 666 9.80391 63707 9.804I9 748 9.80447 789 9.80474 830 9.80502 871 9.80530 639I2 9.80558 953 9.80586 994 9.806I4 64035 9.80642 o76 9.80669 64117 9.80697 158-9 9.80725 199 9.80753 240 9.8078I 281 9.80808 64322 9.80836 363 9.80864 404 9.80892 446 9.80919 487 9.80947 64528 9.80975 569 9.8I003 6io 9.8i030 652 9.8i058 693 9.8i086 64734 9.8II13 775 9.&1141 8I7 9.8ii69 858 9.81196 899 9.8I224 941 9.8I252 28 28 28 28 28 28 29 28 28 28 28 28 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 28 28 28 27 28 28 28 28 28 28 27 28 28 28 28 27 28 28 28 27 28 28 28 27 28 28 27 28 28 27 28 28 0.20421 1.6003 0.20393 I.5993 0.20365 983 0.20337 972 0.20309 962 0.2028i 1.5952 0.20253 94I 0.20224 93I 0.20196 92I 0.20168 911 0.20140 I.5900 0.20112 890 0.20084 88o 0.20056 869 0.20028 859 0.20000 I.5849 0.19972 839 0.I9944 829 0.i9916 8i8 0.i9888 8o8 0.19860 I.5798 0.19832 788 0.9805 778 0.19777 768 0.19749 757 0.19721 I.5747 0.19693 737 0.i9665 727 0.9637 717 0.9609 707 0.9581i I.5697 0.:19553 687 0.19526 677 0.9498 667 0.19470 657 0.19442 i.5647 0.19414 637 0.19386 627 0.19358 6I7 0.933i 607 0.19303 I.5597 0.19275 587 0.19247 577 0.19219 567 0.I9192 557 0.19164 I.5547 0.19I36 537 0.19i08 527 0.19081 517 0.19053 507 0.19025 I.5497 0.o18997 487 0.18970 477 0.i8942 468 0.18914 458 0.18887 1.5448 o.i8859 438 0.883I 428 0.18804 418 0.i8776 408 0.i8748 399 60 59 58 57 56 54 53 52 5'I 49 48 47 46 -45 44 43 42 41 40 39 38 37 36 35 34 33 32 -3' 29 28 27 26 f57 24 23 22 21 20 '9 I8 '7 i6 15 -'4 '3 12 I' 10 9 8 7 6 5 4 3 2 I 0 i Ii jNat.COS Logf dL Nat. Sin Log.. [Nat. CotLog.cA.dLog.TanlNat.L f I 670 33 0 [ m fINat. Sin Log. d. INat. COS Log. d. I Nat T afl Log.Ic.d.ILog. Cot Nat.I 0154464 9.736ii I' 488 9.73630 2 5I3 9.73650 3 537 9.73669 4 561 9.73689 5 54586 9.73708 6 6io 9.73727 7 63 9.73747 8 659 9.73766 9 683 9.73785 10 54708 9.73805 II 732 9.73824 12 756 9.73843 13 78I 9.73863 14 8o5 9.73882 15 54829 9.7390I i6 854 9.739211 17 878 9.73940 i8 902 9.73959 19 927 9.73978 20 54951 9.73997 21 I 975 9.74017 2.2 999 9.74036 23 55024 9.74055 24 048 9.74074 2555072 9.74093 26 097 9-74II3 27 I21 9.74I32 28 I45 9.74151 29 169 9.74I70 -3055194 9.74i89 31 2I8 9.74208 32 242 9.74227 33 266 9.74246 34 291 9.74265 35 55315 9.74284 36 339 9.74303 37 363 9.74322 38 388 9.7434i 39 412- 9.74360 -4-0 55436 9.74379 41 460 9.74398 42 484 9.74417 43 509 9.74436 44 533 9.74455 45 55557 9.74474 46 581 9.74493 47 605 9.745I2 48 630 9.74531 49 654 9.74549 50- -55678 9.74568 51 702 9.74587 52 726 9.74606 53 750 9.74625 54 775 9.74644 55 55799 9.74662 56 823 9.74681 57 847 9.74700 58 871 9.747I9 59 895 9.74737 60 919 9.74756 '9 20 '9 20 '9 20 '9 20 '9 20 '9 20 '9 20 '9 20 '9 '9 '9 '9 '9 '9 '99 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 '9 83867 9.92359 85I 9.9235I 835 9.92343 8i9 9.92335 804 9.92326 83788 9.923i8 772 9.923110 756 9.92302 740 9.92293 724 9.92285 83708 9.92277 692 9.92269 676 9.92260 66o 9.92252 645 9.92244 83629 9.92235 6I3 9.92227 597 9.922119 581 9.92211 565 9.92202 83549 9.92I194 533 9.92i86 517 9-92I77 501 9.92i69 485 9.92i6i 83469 9.92152 453 9.92I44 437 9.92I36 42I 9.92I27 405 9.92II9 83389 9.92III 373 9.92I02 356 9.92094 340 9.92086 324 9.92077 83308 9.92069 292 9.92060 276 9.92052 260 9.92044 244 9.92035 83228 9.92027 212 9.92018 195 9.92010 179 9.92002 163 9.91993 83I47 9.911985 "'13I 9.9I976 115 9.91968 098 9.9I959 082 9.9I951 83066 9.91942 050 9.9I934 034 9.91925~ 017 9.9I917 001 9.9i908 82985 9.91900 969 9.9i891 953 9.9i883 936 9.91874 920 9.9i866 904 9.9I857 8 8 8 9 8 8 8 9 8 8 8 9 8 8 9 8 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 8 9 8 9 8 8 ' 9 i8 6494I 9.8I252 2 982 9.8I279 2 65024 9.8I307 28 o65 9.8I335 2 io6 9.8I362 27 651I48 9.8I390 28 189 9.814i8 2 23I 9.8I445 27 272 9.81473 2 314 9.81500 2 65355 9.8I528.28 397 9.8I556 2 438 9.81583 27 480 9.8i6ii12 52I 9.8i638 27 655,63 9.8i666 128 604 9:8i6_93 2 646 9.8172I12 688 _9.8I748 27 729 9.81776 27 6577I 9.8i803 28 813 9.811831 2 854 9.8i858 27 896 9.8i886 27 938 9.8I913 2 65980 9.8194I 28 6602I 9.8i968 27 063 9.8i996 2 105 9.82023 27 147 9.82051 27 66189 9.82078 28 230 9.821062 272 9.82133 27 314 9.82i6i 2 356 9.82188 27 66398 9.822I5,,7 440 9.82243 2 482 9.82270 27 524 9.8229 28 566 9.82325 27 66608 9.82352 28 650 9.82380 2 692 9.82407 28 734 9.82435 2 776 9.82462 27 668i8 9.82489 27 86o 9.82517 27 902 9.82544 27 944 9.825711 27 986 9.82599 2 67028 9.82626 27 07I 9.82653 28 113 9.8268i12 I55 9.82708 2 197 9.82735 27 67239 9.82762 28 282 9.82790 27 324 9.82817 27 366 9.82844 27 409 9.82871I2 45I 9.82899 0.i8748 I.5399 0.i8721 389 0.i8693 379 om866~ 369 0.i8638 359 o.i86io I.5350 0.18582 340 0.i8555 330 0.i8527 320 0.8500 311 0.18472 I.530I 0.18444 29I 0.18417 282 0.i8389 272 0.18362 262 0.18334 I.5253 0.i8307 243 0.i8279 233 0.18252 224 0.18224 214 0.8197 I.5204 0.i8i69 I95 0.i8142 185 0.18iI4 I75 0.i8087 i66 0.18059 I.5I56 0.180,32 I47 0.18004 I37 0.17977 127 0.117949 II8 0.17922 I.5io8 0-I7894 0991 0.I7867 089 0.17839 o8o 0.17812 070 o.i7785 I.5o6i 0.17757 05I 0.17730 042 0.17702 032 0.17675 023 0.17648 I.50I3 0.17620 004 0.I7593 1-4994 0.I7565 985 0.17538 975 0.75ii 1.-4966 0.I7483 957 0.I7456 947 0.I7429 938 0.1740I 928 0.I7374 I.49I9 0.17347 910 0.17319 900 0.17292 891 o.i726S 882 0.17238 I.4872 0.17210 863 0.17i83 854 0.I7156 844 0.17129 835 0.171101 826 60 59 58 57 55 54 53 52 S5I 50 49 48 47 46 44 43 42 4'0 39 38 37 34 33 32 3' 30 29 28 27 26 24 23 22 210 i6 -15 '4 '3 12 II 9 8 9 8 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 8 9 6 5 4 3 2 I A I I-,., I-,, -. I IT — r — IL,- I IN at. %p U5L0g. a. IIN at. a III Log. 0. I N at. % VLuL0g. IC.G. I og. I aEI IiN a-L. I I 560 34~ I I ' INat. Sin Log. d. INat. COS Log. d.|Nat.TanLog.|c.d.Log.Cot Nat. 0 559i9 9.74756 82904 99I857 8 67451 9.82899 27 O.I710I 1.4826 60 I 943 974775 9 887 9.9I849 493 9.82926 27 17074 86 59 2 968 974794 i8 871 9.9840 536 9.82953 27 07047 807 58 3 992 9.74812 855 9.91832 578 9.82980 280.7020 798 57 4 56016 9.74831 I 839 9.91823 620 9.83008 0.6992 788 56 T 56040 9.74850 I 82822 9.91815 67663 9.83035 0.6965 I-477955 6 64 9.74868 806 9.9806 705 9.83062 27 0.16938 770 54 7 o88 9.74887 I 79 99798 748 983089 27 0.16911 761 53 8 112 9.74906 I 773 9.9789 790 9.83II7 270.I6883 75I 52 18 773 9.97802 9IC27 9 I36 9-74Q24 1I 757 9.9I78I 832 9.83144.I6856 742 51 105660 9.74943 18 82741 9-9I772 67875 9.8317I 27.I6829 I-473350 II I84 9.7496 I 724 9.91763 97 9.83I98 27 0.6802 724 49 I2 208 9.74980 I 708 99755 960 9.83225 27.I6775 75 48 I3 232 9.749998 692 9.91746 9 68002 9.83252 28 0.6748 705 47 I4 256 9.75i07 675 9.9738 045 9.83280 0.16720 696 46 _6 9830 27 _ 15 56280 9.75036 82659 9.9I729 68088 9.83307 0.16693 I1.4687 45 i6 305 9.75054 89 643 9.9172 0 I30 9.83334 27 o.i6666 678 44 I7 329 9-75073 I8 626 9.9I712 9 73 9.83361 27 0.6639 669 43 I8 353 9.7509I I9 6io 9.91703 8 2I5 9.83388 27 0.I6612 659 42 I9 377 9-75IIO 8 593 9.91695 258 9.834I5 2 0.6585 65o 41 20 56401 9.75I28 I9 82577 9.91686 68301 9.83442 0.6558 464I 40 2I 425 9-75I47 i8 56i 9.9I677 343 983470 2 0.16530 632 39 22 449 9-75I65 I9 544 9.9I669 386 9.83497 27 0.16503 623 38 23 473 9-75I84 I8 528 9.9660 9 429 9.83524 27 0.16476 6I4 37 24 497 9.75202 511 9.91651 8 47 9.8355 270.16449 605 36 25 56521 9.7522I i8 82495 9.91643 685I4 9.83578 0.16422 1.4596 35 26 545 9-75239 I 478 9.9I634 9 557 9.83605 27 0.16395 586 34 27 569 9.75258 8 462 9.9I625 600 9.83632 27 0.6368 577 33 28 593 9-75276 I8 446 9.91617 642 9.83659 270.1634 56832 29 6I7 9.75294 I 429 9.91608 9 685 9.83686 27 0.I6314 559 31 30 5664 9.75313 I 824I3 9.9599 68728 983713 27o.628760 4550 30 31 665 9.7533I I 396 9.9I59I 77I 9.83740 2 0.16260 54 29 32 689 9.75350 I 380 9.91582 9 8I4 9.83768 28 0.16232 532 28 33 7I3 9-75308 I8 363 9.91573 8 857 9.83795 27 0.I620 523 27 34 736 9.7=386 347 9.9I565 900 9.83822 27 0.16178 5I4 26 35 56760 9.75405 882330 9-9I556 68942 9.83849 O.i6151 1.4505 25 36 784 9.75423 I8 314 9.9I547 9 985 9.83876 27 O.I6124 496 24 37 808 9.7544 8 297 9.91538 69028 9.83903 27 O.6097 487 23 38 832 975459 I9 28 9.91530 07 9.83930 2 0.6070 47822 39 856 9.75478 I8 264 9.9152I 9 114 9.83957 27 0.16043 469 2I 40 56880 9-75496 i8 82248 9.91512 8 69I57 9.83984 2 o.I6o6 1.4460 20 41 904 9.75514 9 231 9.9I504 200 9.84011 27 0.15989 45I 19 42 928 9-75533 I 8 2I4 9-9I495 243 9.84038 7 0.15962 442 i8 43 952 9-7555I I8 I98 9.9I486 9 286 9.84065 27 0.15935 433 I7 44 976 9.71.6o I8 i8i 9.9I477 9 329 9.84092 27 0.15908 424 i6 45 57000o 9.75587 i8 82I65 9.91469 69372 9.84II9 0.1I588i 1.44I5 15 46 024 9-75605. I48 9.91460 46 9.8446 270.1I5854 406 14 47 047 9.75624 I8 I32 9.91451 9 459 9.84173 270.5827 397 13 48 071 9.75642 I8 II5 9.9I442 9 502 9.84200 27 0.15800 388 I2 49 095 9.7566o0 098 9-9I433 545 9.84227 0.1.5773 379 II 18 82o82 9 - 6958 8 27 50 57119 9-75678 I8 82082 9.9I42 69588 9.8425 4 2 0.15746 1.4370 10 51 143 9.75696 i8 o65 9.914I6 9 631 9.84280 0.15720 36I 9 52 I67 9-757I4 9 048 9.-9407 9 675 9.84307 27 0.15693 352 8 53 19I 9-75733 I8 032 9-9I398 9 78 9.84334 270.15666 344 7 54 215 9.757I I8 OI5 9.91389 9 761 9.8436I 27 0.I5639 335 6 55 57238 9.75769 18 81999 9.9I38I 69804 9.84388 70.15612 1.4326 5 56 262 9-75787 i8 982 9.9I372 9 847 9.84415 270.15585 37 4 57 286 9.75805 i8 965 9.91363 9 89I 9.84442 27 0.15558 308 3 58 310 9-75823 I8 949 9.9I354 9 934 9.84469 27 0.I553I 299 2 59 334 9.75841 i8 932 9.9I345 977 9.84496 0.1.5504 290 I 60 358 9.75859 I 915 9.9I336 9 7002I 9-84523 0.-I5477 28I 0 60 389789 5.9133 Nat. COS Log. d. Nat. Sin Log. d. Nat. COtLog. c.d. Log.TanNat. ' 55~ 360 0 f Nat. Sin Log. d. INat. COS Log. d. jNat.TanlLog.1c.d.ILog. Cot Nat.I I I 0 'I 2 3 4 5 6 7 8 9 10 II 12 '3 '4 '7 20 21 22 23 24 26 27 28 29 30 3' 32 33 34 35 36 37 38 39 -40 4' 42 43 44 45 46 47 48 49 50 5' 52 53 54 -55 57 58 59 60 57358 9.75859 381 9.75877 405 9.75895 429 9.75913 453 9.7593' 57477 9.75949 501 9.75967 524 9.75985 548 9.76003 572 9.7602I 57596 9.76039 619 9.7605p7 643 9.76075 667 9.7609,3 691 9.76iii 577I5 9.76I29 738 9.76146 762 9.76i64 786 9.76i82 8io 9.76200 57833 9.762i8 857 9.76236 881 9.76253 904 9.76271 928 9.76289 57952 9.76307 976 9.76324 999 9.76342 58023 9.76360 047 9.76378 5807o 9.76395 094 9-764I3 ii8 9.7643I 141 9.76448 i65 9.76466 58189 9.76484 2I2 9.7650I 236 9.765119 260 9.76537 283 9.76554 58307 9.76572 330 9.76590 354 9.76607 378 9.76625 401 9.76642 58425 9.76660 449 9.76677 472 9.76695 496 9.76712 519 9.76730 58543 9.76747 567 9.76765 590 9.76782 614 9.76800 637 9.76817 58661 9.76835 684 9.76852 708 9.76870 73I 9.76887 755 9.76904 779 9.76922 i8 i8 i8 i8 i8 '7 i8 i8 '7 i8 '7 i8 '7 '7 i8 '7 i8 i8 i8 i8 '7 '7 '7 i8 '7 i8 '7 '7 i8 '7 '7 '7 81915 9.91336 899 9.9I328 882 9.91319 865 9.91310 848 9.91301 81832 9.91292 815 9.91283 798 9.91274 782 9.91266 765 9.9I257 81748 9.91248 73I 9.91239 7I4 9.91230 698 9.91221 68i 9.91212 81664 9.91203 647 9.91194 631 9.91I85 614 9.911I76 597 9.9ii67 8I58o 9.91I58 563 9.911I49 546 9.91 141 530 9.91I32 513 9.91123 81496 9.91114 479 9.9II05 462 9.9i096 445 9.91087 428 9.9I078 81412 9.91069 395 9-91t060 378 9.9I05I 361 9.91042 344 9.91033 81327 9.91023 310 9.91014 293 9.91005 276 9.90996 259 9.90987 8I242 9.90978 225 9.90969 208 9.90960 191 9.90951 I74 9.90942 81157 9.90933 140 9.90924 123 9.90915 io6 9.9090o6 089 9.90896 8I072 9.90887 055 9.90878 038 9.90869 02I 9.90860 004 9.90851 80987 9.90842 970 9.90832 953 9.90823 93-6 9.90814 919 9.90805 902 9.90796 8 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 Io 9 9 9 9 9 9 9 9 9 9 9 9 9 Io 9 9 9 9 9 9 Io 9 9 9 9 7002I 9.84523 064 9.84550 I07 9.84576 151 9.84603 I 94 9.84630 70238 9.84657 281 9.84684 325 9-847I11 368 9.84738 412- 9.84764 70455 9.84791 499 9.84818 542 9.84845 586 9.84872 629 9.84899 70673 9.84925 7I7 9.84952 760 9.84979 804 9.85006 848 9.85033 7089I 9.850-59 935 9.85086 979 9.8511I3 71023 9.85I40 o66 9.85166 71110 9.85193 I54 9.85220 198 9.85247 242 9.85273 285 9.85300 7I329 9.85327 373 9.85354 4I7 9.85380 461 9.85407 505 9.85434 7I549 9.85460 593 9.85487 637 9.85514 68i 9.85540 725 9.85567 7I769 9.85594 8I3 9.85620 857 9.85647 901 9.85674 946 9.85700 7I990 9.85727 72034 9.85754 078 9.85780 I22 9.85807 167 9.85834 722II 9.85860 255 9.85887 299 9.85913 344 9.85940 388 9.85967 72432 9).85993 477 9.86020 521 9.86046 565 9.86073 6io 9.86i00 654 9.86I26 27 26 27 27 27 27 27 27 26 27 27 27 27 27 26 27 27 27 27 26 27 27 27.26 27 27 27 26 27 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 27 26 27 26 27 27 26 27 26 27 27 26 0.15477 I.4281 0.15450 273 0.15424 264 0.15397 255 0.15370 246 0.15343 I.4237 0.153i6 229 0.15289 220 0.15262 211 0.15236 202 0.15209 I.4193 0.15i82 i85 0.5I5~ 176 0.15I28 167 0.15101 158 0.I5075 I.4I50 0.15048 '41 0.15021 132 0.14994 124 0.14967 115 0.14941 1.4106 0.14914 097 0.14887 089 0.14860 o8o 0.14834 071I 0.14807 I.4063 0.14780 054 0.14753 045 0.14727 037 0.14700 028 0.14673 1.4019 0.14646 oii 0.14620 002 0.14593 I.3994 0.14566 985 0.I4540 I.3976 0.145I3 968 0.i4486 959 0.4460 951 0.14433 942 0.14406 I.3934 0.14380 925 0.14353 916 0.14326 908 0.i4300 899 0.14273 I.389i 0.14246 882 0.14220 874 0.14193 865 0.1466 857 0.14I40 I.3848 0. 1411I3 840 0.14087 831 0.14060 823 0.14033 814 0.14007 I.38o6 0.13980 798 0.13954 789 0.13927 781 0.13900 772 0.13874 764 59 57 56 54 53 52 5' 50 49 48 47 46 45, 44 43 42 4' -40 39 38 37 36 35 34 33 32 3' 30 29 28 27 26 25 24 23 22 21 '9 -'7 15 '4 '3 12 'I To9 8 7 6 -5 4 3 2 I 0 - Nat. COS Log. d. INat. siflLog. dcLNat. Cot Log. 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COS Log. d. Nat. Sin Log. d. Nat.COtLog. c.d.lLog.TanNat.j 46~ 440? Nat. sin Log. dj at. COS Log. d. r Nat. Tafl Log. Ic.d. ILog. 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Co0s Log. d. JN'at. S in Log. d. INat. C Ot Log.I cn. o.T Nat.i w I 460