PLANE TRIGONOMETRY WITH PRACTICAL APPLICATIONS BY LEONARD E. DICKSON, PH.D. CORRESPONDANT DE 'INSTITUT DE FRANCE PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF CHICAGO BENJ. H. SANBORN & CO. CHICAGO NEW YORK BOSTON COPYRIGHT, 1922 BY BENJ. H. SANBORN & CO. PREFACE Distinctive features of this book are its immediate justification of the study of trigonometry, its emphasis on the practical applications, its sensible problems, its model solutions of sample problems, its concreteness, simplicity, and clearness, and its use of a traverse table in addition to the usual tables. The great majority of students of trigonometry, whether Trigonometry in the high school or the college, take it as their final here justifies itself course in mathematics. Hence the course should justify itself at the time, and not be merely a stepping stone to further mathematical subjects. Without overlooking the needs of the few who will go further in mathematics, we may justify trigonometry to the others by demonstrating its great utility by means of simple applications to various subjects which are vital in the practical world today. Practical This book introduces at an early stage concrete applicaapplications tions of trigonometry to the elementary parts of navigation and surveying, which are the two simplest exact sciences, as well as to the two elementary topics of physics which are known as composition of forces and refraction of light. There is, too, a full explanation of the theory and construction of a Mercator map, a subject of great importance also in geography. Three separate chapters are devoted to these subjects. The necessary terms and ideas are explained at length and illustrated concretely. We thereby obtain an abundance of simple problems whose importance is so convincing that they cannot fail to arouse real interest. Actual experience with classes has firmly convinced the author that these practical applications offer the best means to drive home the principles of trigonometry and to make the subject truly vital. Sensible The problems are simple and sensible. Puzzle problems problems have been discarded, as well as those serving no purpose beyond the scourge of endless computation. Instead of the usual dull problems calling for the solution of a triangle in which certain sides and angles are given, the problems here proposed are real and reflect some activity of actual life. iii iv PREFACE Sample The problems which present continuity of thought are colproblems lected into a set of exercises and given an appropriate solved descriptive heading. This plan will greatly aid the instructor in his selection of problems for assignment. Before each such collection of problems are inserted examples worked out in detail which together illustrate all of the different types of problems occurring in that collection. This feature will commend itself to both student and teacher. The harder problems (marked *) may be proposed for extra credit. Concr s Several informal illustrations of the tangent and sine are Concreteness given prior to the formal definitions. The concrete information about angles of any size and their measurement, which is acquired in the chapters on navigation and surveying, furnishes a desirable background for the introduction of general angles. And the same is true as to familiarity with latitude and longitude before coordinates are introduced for the sake of defining the trigonometric functions of a general angle. But above all, the book is concrete on account of the practical applications included and the practical nature of the problems. Simplicity The development of the subject is leisurely and the student Clearness is given ample time in which to digest each idea. There are given full and lucid explanations of all new terms and ideas. Lack of the precise knowledge of the mathematical meaning of terms is one of the chief sources of difficulty in the study of mathematics. Various terms which should already be familiar to students are re-defined. On the basis of careful readings both of the manuscript and proof sheets by various experienced teachers in high schools and colleges, it is believed that the presentation is throughout both simple and clear. Tables The tables are as simple as possible, and accurate for computation to four significant figures, which are ample for all ordinary practical purposes. It is true that some delicate astronomical measurements justify computations with 5, 6, or 7 place tables; but no new theory is involved. The traverse table, which is necessary for navigation and surveying, is really a systematic list of the sides and angles of all right triangles of moderate size. Its additional headings aid in making the present exposition of navigation much simpler than was possible heretofore. The traverse table is extremely useful in all parts of trigonometry and its applications, partly by relieving the monotony of logarithmic computation, but chiefly for the instantaneous checking of computations. PREFACE v A suggestion The chapters on navigation and surveying are each divided to teachers into two parts, this making possible either a brief, wholly untechnical, introduction to those applications, or a fuller treatment. When these chapters are reached, it is suggested that henceforth two hours a week be devoted to these applications and the remaining class periods to general trigonometry, which begins with page 103. Under this plan the student will be applying the theory of the right triangle, which he has already learned, while he is acquiring the theory of the oblique triangle, and will complete the former applications just when he is ready for the applications of the latter theory. Under such a program the student will understand at all times why he is doing what he is doing, will have real respect for the subject, and will take a genuine interest in it. Acknowledg- Valuable suggestions were made, after reading the entire ments manuscript, by Dr. J. M. Kinney and Professor O. M. Miller, both of the Hyde Park High School, Chicago, by the author's colleague, Dr. Mayme I. Logsdon, and by Dr. E. J. Moulton of Northwestern University, while the latter read also the proof sheets critically. An earlier form of the chapter on navigation was read by Professors Moulton, R. G. D. Richardson of Brown University, and the author's colleague, J. W. A. Young. The chapter on surveying was read by the author's colleagues, Professors K. Laves and G. W. Myers, and by G. D. Tompkins of the Bureau of Maps and Plats of the City of Chicago; while the proof sheets were carefully read by Professor B. F. Yanney of the College of Wooster, Ohio. The author is greatly indebted to these experienced teachers, and especially to Professor Moulton for his generous help at all three stages of the book. Plates for the cuts of the surveyor's compass and transit were kindly loaned by the instrument makers, W. and L. E. Gurley, of Troy, New York. L. E. DICKSON UNIVERSITY OF CHICAGO DECEMBER 3, 1921 CONTENTS CHAPTER I Trigonometric Functions of Acute Angles ARTICLE PAGB 1. Nature of Trigonometry........... 1 2. Drawings to scale........ 1 3. Absolute and relative errors.... 3 4. Horizontal and vertical lines, planes, angles.. 4 5. Angles of elevation and depression.... 5 6. Illustrations of the tangent of an acute angle... 5 7. Definitions of the trigonometric functions of an acute angle. 6 8. Problems on heights and distances... 9 9. Given one trigonometric function, to find the others.. 11 10. Relations between the six trigonometric functions of an acute angle....... 12 11. Further formulas true for every acute angle.... 14 12. Relations between the functions of complementary angles. 16 CHAPTER II Solution of Right Triangles by Means of Tables of the Natural Functions 13. How to use tables of natural functions....... 19 14. Solution of right triangles by means of tables of natural functions... 21 15. Cases when an angle cannot be accurately found by Table II 23 CHAPTER III Traverse Table; Solution of Right Triangles by Inspection; Problems on Forces and Refraction of Light 16. Description of Traverse Table VI...... 26 17. When and how to use Traverse Table VI.... 26 Exercises on heights and distances..... 28 vii viii *. Vlll CONTENTS ARTICLE PAGE 18. Force......... 29 19. Resultant of two forces.... 29 20. Parallelogram of forces..... 29 21. Component of a force...... 30 22. Refraction of light....... 31 CHAPTER IV Logarithms, Slide Rule 23. Powers of 10, index laws...... 34 24. Logarithms........ 36 25. Significant digits..... 38 26. Mantissa and characteristic of a logarithm.. 39 27. To find the logarithm of a number by Table VII. 40 28. To find the number with a given logarithm by Table VII. 41 29. Extraction of roots by logarithms... 41 30. Logarithmic scale....... 43 31. Slide rules........ 44 32. Logarithms of trigonometric functions... 46 CHAPTER V Solution of Right Triangles by Logarithms 33. Results in Chapter II recalled....... 48 34. Solution by logarithms.. 48 35. Given the hypotenuse and a leg..... 49 36. Errors of computation..... 51 37. Area of a right triangle..... 53 38. Isosceles triangles and regular polygons... 53 39. Problems on heights and distances.... 55 CHAPTER VI Navigation: Dead Reckoning 40. Navigation and its subdivisions... 59 41. Geographical terms. 59 42. Nautical mile....... 60 CONTENTS ix ARTICLE PAGE 43. How distance is measured......... 60 44. Ship's course, compass card.... 61 PART I. THE SAILINGS (TRUE COURSE ASSUMED) 45. Plane Sailing....... 62 46. Unfavorable case in the use of a traverse table.. 63 47. Traverse Sailing...... 64 48. Parallel Sailing....... 66 49. Middle Latitude Sailing..... 68 50. The Mercator chart, meridional parts.... 71 51. Angle and distance on a Mercator chart..... 73 52. Mercator's Sailing....... 73 PART II. FINDING THE TRUE COURSE; COMPASS CORRECTIONS 53. The mariner's compass..... 76 54. Variation and deviation of the compass.. 76 55. Leeway........ 76 56. Courses, compass course, true course, corrections.... 77 57. Dead reckoning..... 79 CHAPTER VII Land Surveying 58. Branch of surveying treated..... 82 59. Chains, tapes, area.......... 82 60. Course....... 83 61. True bearing........... 83 PART I. BALANCING A SURVEY, AREA (TRUE BEARINGS ASSUMED) 62. Latitude and departure..... 84 63. Balancing a survey, error of closure.... 84 64. Double meridian distances..... 86 65. Area of a field. 86 66. Plotting............... 88 PART II. SURVEYING INSTRUMENTS; FINDING TRUE BEARINGS 67. Verniers........ 89 68. Surveyor's compass..... 91 x, CONTENTS ARTICLE 69. Bearing with respect to any course 70. Magnetic bearing 71. Magnetic declination, variation charts 72. Surveyor's transit 73. Measuring angles with a transit 74. Traverse 75. Direct angle 76. Deflection angle 77. Azimuth 78. Balancing a transit survey PAGE ~ ~~.........93..... 94 94...... 94 ~ ~~.........96 ~ ~~.........99 **...... 99...... 100...... 100..........100...... 101 CHAPTER VIII Trigonometric Functions of Any Angle 79. Rectangular coordinates, plotting 80. Radius vector 81. Generalized notion of angle. 82. Trigonometric position of an angle; the four quadrants 83. Trigonometric functions of any angle 84. Trigonometric identities 85. Reduction of the trigonometric functions of any angle to functions of an acute angle 103 104 105 106 107 110 111 CHAPTER IX Solution of Oblique Triangles 86. Altitude and area of any triangle. 115 87. Law of sines.............. 115 88. Law of tangents, Mollweide's equations; solution of a triangle, given two angles and a side or two sides and the included angle....... 116 89. Solution of a triangle, given two sides and the angle opposite to one of them...... 121 90. Law of cosines....... 126 91. Area of a triangle in terms of its sides...... 128 92. Radius of the inscribed circle.... 129 93. To compute the angles of a triangle, given the sides.. 129 Exercises on resultants and components of forces.... 31 CONTENTS xi CHAPTER X Relations Between Functions of Several Angles ARTICLE PAGE 94. The addition theorem for sine.... 133 95. Functions of A+90~..... 135 96. The addition theorem for cosine. 136 97. The subtraction theorems for sine and cosine.. 136 98. Heights and distances.... 137 99. The addition and subtraction theorems for tangent and cotangent........ 138 100. Functions of double angles......... 140 101. Functions of multiple angles..... 141 102. Trajectories. 142 103. Functions of half angles........ 144 104. Sum or difference of two sines or two cosines expressed as a product....... 145 105. Trigonometric equations......... 147 CHAPTER XI Graphs of the Trigonometric Functions and Their Inverses, Radians 106. Line representations of the trigonometric functions...150 107. The sine and cosine curves..........152 108. The tangent curve............. 153 109. Graphical solution of trigonometric equations; harmonic curves....... 156 110. The radian unit of angle......... 157 Reduction of degrees, minutes, and seconds to radians..159 111. Approximate values of sines and tangents of small angles.161 112. Equations involving both an angle and its trigonometric functions...... 162 113. The inverse trigonometric functions........ 164 LIST AND INDEX OF FORMULAS......... 169 INDEX, INCLUDING INDEX TO DEFINITIONS... 171 ANSWERS TO CERTAIN OF THE FIRST FIVE EXERCISES OF EACH SET 173 TABLES, SEPARATELY PAGED PLANE TRIGONOMETRY WITH PRACTICAL APPLICATIONS CHAPTER I TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES 1. Nature of trigonometry. The word trigonometry means literally the measurement of triangles. For example, if we are given the lengths of two sides of a triangle and the size of their included angle, we can, as we shall learn, compute the size of each remaining angle, the length of the third side, and the area of the triangle. Computations are made by means of tables and checked against gross errors either by a drawing made to scale or more quickly by a traverse table (see Table VI). Trigonometry is a prerequisite to engineering, physics, astronomy, and other exact sciences. Applications to navigation, surveying, and a few simple topics of physics, are introduced early in this text, with a full explanation of the terms involved. 2. Drawings to scale. Angles of a drawing are measured by means of a protractor, which in its simplest form (Fig. 1) consists of a semicircle graduated to degrees. It \ is recalled that a A right angle is divided into 90 degrees (90~), A each degree into 60 minutes (60'), and each minute into 60 c seconds (60"). Thus FIG. 1 one-sixteenth of a right angle equals 5~37'30". To measure Z A CB, in Fig. 1, place the protractor with its center at C and with its diameter along one arm CA of the angle, and note the 1 2 TRIGONOMETRY [Art. 2 reading (about 41~) at the intersection of the other arm CB of the angle with the semicircle. There will be given in Chapter VII (on Surveying) a description of instruments used to measure angles between lines, and lengths of lines, on the earth's surface. Angles may be measured with a transit correctly to minutes or even to within 20 seconds. The length of a straight line of a drawing can be estimated by means of a graduated ruler, or by counting divisions in case the drawing has been made on square-ruled plotting paper (Fig. 2). EXAMPLE 1. Draw to scale a right triangle whose legs are 15.4 ft. and 12.7 ft., and measure its acute angles and hypotenuse. Solution. Select a point C _ I I I I _- I _ I_ I __of intersection of two heavy A _ _ _ i lines of the square-ruled ___- _____ __ _______.paper to represent the ver_XiN -- I- I__ _ -- -- tex of the right angle of our E XI ____ I ____ I/ _-triangle ABC (Fig. 2). Let E,v / _ -_ __ __ __ __ L- a small division represent 1 ^, —^^ -- - - ~~_~xL~~ / foot. Locate the point A by _.\.-_S _ -- _l- 4 / _________ counting off 151 small divi_ _ _ _ A__ --- - _ _ _ ___sions from C on one arm of _ _ __Sv_^___ -Z___/ 1 _ the right angle. Locate B on _- _ _ _ _ 117 _ _the other arm of the right _ - angle. To measure AB, - -N_, _N,/-: —._- _ _ __transfer it to a position AD C -._- I ftr- - F B parallel to a ruling line by FIG. 2 means of a pair of compasses (or by placing a strip of paper along the line AB and marking the points opposite to A and B, and then moving the strip of paper into the desired position AD). Or we may measure AB by means of a strip of the square-ruled paper cut as a permanent ruler. The approximate results are AB = AD = 20 ft., ZA = 39I~, ZB = 50~1. In any triangle ABC, the side BC opposite to angle A is denoted by a, the side opposite to B by b, and the side opposite to C by c. Ch. I] FUNCTIONS OF ACUTE ANGLES 3 EXAMPLE 2. Construct a triangle in which a = 12, b = 9, B = 25~, and measure side c and angles A and C. Solution. Construct ZABC = 25~ and lay off BC = a = 12. With C C as a center and 9 as a radius, draw an arc of a circle cutting BA at A and cutting BA produced at, / A' (Fig. 3). Thus both of the tri- B-\ angles BAC and BA'C satisfy the A\' requirements. By measurement, the first triangle has the required IG. 3 parts BA ='c = 3.5, ZBAC = A = 148~, ZBCA =C = 8~, approximately, and the second triangle BA'C has the required parts BA' = = 18.2, ZBA'C = A' = 35~, ZBCA' = C' = 119~~. 3. Absolute and relative errors. The scales used in making measurements are not perfect and the reading of them involves an estimate by the eye. Hence all measurements involve errors. If, in Fig. 2, we measure the diagonal EF of one of the larger squares, whose side represents 10 ft., we obtain a number representing 14.1 ft., approximately. The length of the hypotenuse EF of the right triangle CEF can be found exactly by applying the theorem of Pythagoras that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs of the right triangle. Thus EF2= 102+102, EF = 10 V2. The true length of the diagonal is therefore 10 2 = 14.14... ft. Its length by measurement was 14.1 ft., so that the (absolute) error just exceeds 0.04 ft. The ratio of the absolute error to the true value is called the relative error, and in this instance is 4/1414, approximately; or, if we prefer, 0.28 per cent. EXERCISES ON DRAWINGS TO SCALE 1. Draw a triangle at random, measure its angles and find their sum. What is the error? What is the relative error? 4 TRIGONOMETRY [Art. 4 2. Draw a quadrilateral at random, measure its angles and find their sum. What is the error? What is the percentage of error? 3. What is the simplest way to draw, on square-ruled paper, a right triangle whose acute angles are each 45~? Draw such a triangle having a leg equal to 10 small divisions, measure the hypotenuse and compute the error. 4. On square-ruled paper select a line AB along a ruling and containing 10 small divisions. What is the simplest way to locate the vertex C of an equilateral triangle having AB as its base? Measure the altitude from C and compute the error. 5. At the points A and B in Ex. 4, construct angles each of 60~ by use of a protractor and hence find C. What is the present error in the altitude? If in a triangle ABC the side opposite angle A is called a, etc., construct the triangles in which the following parts are given and measure the parts not given: 6. a = 7.4, b = 9.8, c = 5.1. 7. a = 11.3, b = 13.4, C = 24~. 8. a = 8.5, B = 36~, C = 68~. 9. a = 8.5, b = 10.5, A = 54~. 4. Horizontal and vertical lines, planes, angles. The vertical line through a point P is the straight line determined by a plumb line or cord one of whose ends is at P, while the other end is attached to a suspended weight. Any plane which contains a vertical line is called a vertical plane. A horizontal line or plane is one which is perpendicular to a vertical line. A more convenient test for horizontal position is furnished by a spirit level, which is composed of a tube filled so nearly full of alcohol that a single air bubble is left. A line is horizontal if, when the level is placed along the line, the bubble is at the middle of the tube. To test whether a plane is horizontal or not, place the spirit level along it in two different directions in turn. A horizontal angle is one whose arms are in a horizontal plane; a vertical angle' is one whose arms are in a vertical plane. 1 This term is not to be confused with the usage in elementary geometry, where two opposite angles formed by two intersecting lines are called vertical angles. Ch. I] FUNCTIONS OF ACUTE ANGLES 5 5. Angles of elevation and depression. The angle which the line (of sight) from an observer's eye to an object makes with a horizontal line in the same vertical plane is called the angle of elevation or angle of de- D pression of the object, according as the object is above or below the horizontal plane of the observer. Thus, in Fig. 4, if the observer is at A and the object is at B, the angle of eleva- tion is the angle E which AB makes with F E the horizontal line AF in the same vertical plane with AB. But if the observer is at FIG. 4 B and the object is at A, the angle of depression is the angle D which BA makes with the horizontal line BC in the same vertical plane with BA. 6. Illustrations of the tangent of an acute angle. A sufficiently short portion of a railroad track may be regarded as having a constant grade or slope, as 3/100 if it rises 3 feet for each 100 feet in a horizontal direction. A wagon road or the roof of a house may have a greater slope, as 2/5. In trigonometry, we speak of the slope of the road or of the roof as the tangent of its angle A of inclination with the horizontal plane, and write it "tan A." Thus, for the roof, tan A = 2/5; while, for the railroad, tan A = 0.03. Now a roof which rises 2 feet per 5 feet horizontal will rise 4 feet per 10 feet horizontal. Whether in Fig. 5 we use the right triangle with the legs 2 and 5, or the larger right triangle with the legs 4 4 and 10, each triangle having A as an acute angle, we obtain the same value -—. l10 --- - for tan A, viz., the ratio of the oppoFIG. 5 site side to the adjacent side. It is clear that if we use a right triangle with the legs 2n and 5n, we again have tan A = 2/5. In mathematics, when the value of one quantity is determined 6 TRIGONOMETRY [Art. 7.by the value of another, the first quantity is said to be a function of the second. For example, the area of a circle is a function of the radius. Again, cost is a function of the quantity bought. It is seen that tan A is determined by the size of the angle A, and hence tan A is a function of A. The same ideas are involved in the ancient method of finding the height h of a tower by measuring the lengths of its shadow f h t. FIG. 7 e ________ 122 ft. \s FIG. 6 and of the shadow of a vertical pole at the same instant. Suppose that the shadows are of lengths 122 and 20 feet, while the pole is 10 feet high. Since the sun's rays are parallel, we have two similar right triangles (Figs. 6, 7), so that h: 122 = 10: 20, whence h = 61 feet. The tangent of the angle of elevation of the sun at the moment is 10/20 = 61/122. 7. Definitions of the trigonometric functions of an acute angle. In the first illustration in Art. 6 of the tangent of an angle, we made use of the horizontal distance beneath a railroad track. But that distance is not so easily or accurately measured as a distance along the track. If we find that the track rises 2 feet for every 100 feet along the track, we speak of 2/100 as the sine of the angle which the track makes with the horizontal plane. In a right triangle there are, in addition to the two ratios of sides which have been called tangent and sine, four more ratios of Ch. I] FUNCTIONS OF ACUTE ANGLES 7 sides. The six ratios occur in pairs, and the ratio b/a is called the reciprocal of a/b. The six ratios are given names as follows: The sine is the ratio of the opposite side to the hypotenuse. The cosecant is the reciprocal of the sine. The cosine is the ratio of the adjacent side to the hypotenuse. The secant is the recipro- a 1 cal of the cosine. The tangent is the ratio of the opposite side A b to the adjacent. The cotangent is the recip- adacentide FIG. 8 rocal of the tangent. These definitions should be memorized in these words. The student should repeat the definitions until he also "knows them backwards" and hence can answer the following questions and the two analogous questions of each type: Which function is the ratio of the adjacent side to the hypotenuse? Which function is the reciprocal of the tangent? In symbols we may express these definitions as follows (Fig. 8): a b a sin A - cos A=-, tan A = c c b 1 c 1 c 1 b csc A = sec A = 1= cot A = -- = sinA a' cos A b tanA a Hence the cosecant is the ratio of the hypotenuse to the opposite side, the secant is the ratio of the hypotenuse to the adjacent side, and the cotangent is the ratio of the adjacent side to the opposite side. As in Art. 6, each of these six trigonometric functions of an acute angle A is a definite number, completely determined by angle A, and that number does not depend on the size of the right triangle, containing angle A, which we may select in defining the function. While the numbers which express the lengths of the sides of such a triangle depend upon the unit of length used, their ratios (i.e., the trigonometric functions of A) are wholly independent of the choice of the unit of length, but depend solely upon the size of A. 8 TRIGONOMETRY [Art. 7 By the theorem of Pythagoras, proved in plane geometry, (1) a2 + b2 = c2. EXAMPLE. Find the six functions of the least angle L in the right triangle having the hypotenuse 17 and a leg 8. Solution. By (1), the other leg is 15. Hence, by Fig. 9, 9 sin. 8.8 15 8 5^ 1 sin L=,7 cos L=-,, tan L=-5 L 17 17 15 FIG. 9 s 8 se EXERCISES ON THE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS1 1. Find the six functions of 45~. Take the adjacent side to be of unit length (Fig. 10). Since the remaining acute angle is 45~ (why?), the opposite side is 1 (why?), and the hypotenuse is </2 (why?). 4.50 1 FIG. 10 2. Find the six functions of 60~. In an equilateral triangle each of whose sides is of length 2, drop a perpendicular from the vertex (Fig. 600 1 \ 11). Then each segment of the base is of length 1 FIG. 11 (why?) and the perpendicular is of length -/3 (why?). 3. From Fig. 11 read off the six functions of 30~. 4. Draw a right triangle with an angle of 15~ and hypotenuse 10. After measurement of A Sin A Cos A the legs, calculate sin 15~ and cos 15~. By draw- 15~ 026 0.97 300 0.50 0.87 ing more triangles, calculate from measurements 45~ 0.71 0.71 the values required to fill out the accompanying 60~ 0.87 0.50 table. 750 0.97 0.26 table. 5. Find the six functions of the least angle in the right triangle having the legs 3 and 4. 1The first three exercises are needed for many later exercises; the figures should be kept in mind. Ch. I]~ FUNCTIONS OF ACUTE ANGLES 9 6. Find the six functions of the larger acute angle in the right triangle having the hypotenuse 13 and a leg 5. 7. Find the six functions of the least angle in the right triangle having the hypotenuse 41 and a leg 40. 8. Show that neither the sine nor the cosine of an acute angle is greater than unity, while neither the secant nor the cosecant is less than unity. 9. By means of right triangles whose bases have the length unity, construct angles A, B, C for which tan A = 0.3, tan B = 3, tan C = p, where p is any positive number. Hence show that the tangent of an acute angle may have any positive value whatever. Why is this true also of the cotangent? 8. Problems on heights and distances. For the following problems involving only the angles 30~, 45~, and 60~, exact answers in terms of square roots may be found. Similar problems involving other angles will be assigned later to be solved by use of the traverse table or logarithms. EXAMPLE 1. When the angle of elevation of the sun is 60~, a vertical pole casts a shadow 50 feet long. What is the height h of the pole? Solution. We have tan 60~ = h= 50tan60~=50AV3. 50 EXAMPLE 2. With an instrument I held 6 feet above the level of a pond, a man observes that the angle of elevation of a tree at the edge of the pond is 45~ and that the angle of depression of its reflection is 60~. Find the height h of the tree. 450 Solution. The reflection BR of the tree BT is also 1 60o A of length h (Fig. 12). Since AB = 6, we have AT = 6 h- 6, AR = h + 6. Since ZITA = 45~, IA =AT B = h - 6. Thus in triangle IRA, -- AR h+G 6 /3= tan 60 = I =h-6 \ Solving for h, we get \ 6(i/3+1) _ 6(V3+1)2 \ h 6(/~-3+) F 12 3- 1 (V3-1) (A/3+1) \R h = 6(2 + V3). FIG. 12 10 TRIGONOMETRY [Art. 8 EXAMPLE 3. The angle of elevation of a balloon B from a station P due south of it is 60~, and from another station Q due west from P and 2 miles from it the angle of elevation is 45~. Find the height h of the balloon. Solution. Let F be the foot of the perpendicular from B to the ground (Fig. 13). In triangle BFP, / h \ Z F is a right angle and 3 = tan 60~ = - FP /whence FP = hll/3. In triangle BFQ, Z F is a right angle, and hence ZB = 45~, so that QF = h. H i~ Finally, in triangle FPQ, ZP is a right angle, so Q l- that ___ h2 FIG. 13 Q2 = FP2 + 4, h2 = -h + 4, h2 = 6, h = 6. EXERCISES ON HEIGHTS AND DISTANCES (Give exact answers, not computing square roots.) 1. At the top of a house 80 feet high, the angle of elevation of the top of a tower is 45~; on the ground floor it is 60~. Find the height of the tower. 2. A man 6 feet tall observes that the angle of elevation of the top of a tree is 45~, while that of the point where the branches begin is 30~. The latter point is 17 feet above the ground. How high is the tree? 3. The angle of ascent of a road is 30~. If a man travels 500 feet up the road, how many feet has he risen? 4. The shadow of a flagpole standing on level ground is 50 feet longer when the angle of elevation of the sun is 30~ than when it is 45~. Find the height of the pole. 5. The upper part of a tree broken over by the wind makes an angle of 30~ with the ground and its top rests on the ground at a point 75 feet from the root. What was the height of the tree? 6. From a boat at sea the angles of elevation of the top and base of a tree 100 feet high on the top of a bluff are found to be 60~ and 45~ respectively. Find the height of the bluff. 7. At a point half way between two buildings the angles of elevation of their tops are 30~ and 60~. Prove that one building is three times as high as the other. Ch. I] FUNCTIONS OF ACUTE ANGLES 11 8. A ladder 24 feet long leans against a house on one side of a street, making an angle of 30~ with the street. After the ladder is turned about its foot until the top touches the house on the opposite side of the street, the angle is 60~. Find the width of the street. 9. From the top of a lighthouse of height 120 feet, the angle of depression of an object at sea is 60~, and from the base 30~. How high is the base above the sea? 10. Two persons stand at opposite sides of a pond in positions such that the eyes of one are 6 feet above the pond and those of the other 7 feet. When each person looks toward the pond in a direction making an angle of 60~ with the vertical, the reflection of an eye of either is visible to the other person. How wide is the pond? 11. From two points 400 feet apart in a horizontal straight line with the foot of a tower which lies between the points, the angles of elevation of its top are 30~ and 60~. How high is the tower? 12. From the top of a tower of height 93 f6et, the angles of depression of the top and base of a tree standing on a level with the base of the tower are observed to be 30~ and 60~. Find the height of the tree. 13.* A square tower stands upon a horizontal plane. From a point in this plane from which three of its upper corners are visible, their angles of elevation are 45~, 60~, and 45~ respectively. What is the ratio of the height of the tower to the breadth of one of its sides? 14.* The angle of elevation of a balloon B from a station P due south of it is a, and from another station Q due west from P and 1 miles from it the angle of elevation is f. If its height is h, prove that h2 (cot2 f -cot2 a) = 12. 9. Given one trigonometric function, to find the others. If the value of one of the six functions of an acute angle A is known, we can find the values of the remaining five functions of A either graphically as in the following example or algebraically by means of the formulas in Art. 10. While the latter method has the advantage that we can always compute the required functions to as many decimal places as desired, the graphical method is less liable to gross errors especially when applied later on to angles which are not acute. 12 TRIGONOMETRY [Art. 10 EXAMPLE. Construct and measure an acute angle whose cosecant is equal to 9/8. Complete a right triangle and read off the values of the remaining functions. Solution. Given that csc A = 9/8, we have sin A = 8/9. Hence in a right triangle one of whose angles is A, the ratio of the opposite side to B the hypotenuse is 8/9. To avoid fractions, we therefore take the hypotenuse equal to 9. Then the opposite side is 8. On square-ruled paper, take a vertical ruling line BC containing 8 divisions, and BD containing 9 divisions 9_/_ C (Fig. 14). Let the circle with B as center and BD as _ _ _ 8 radius cut the horizontal ruling line through C at the _ -- / e__ point A. Then ABC is the desired triangle. Since the A 4 _ number of divisions in AC is 4 (more nearly 4.1), we have C- -4 9 8 4 A — cosA = 4, sec A = tan A =, cot A =48 9 4 4 8 FIG. 14 approximately, while A = 63~, nearly. EXERCISES ON DRAWINGS TO SCALE 1. Construct and measure an acute angle whose tangent is equal to 4/3. Complete a right triangle and read off the remaining functions. 2. Construct and measure an acute angle whose cosine is 8 /17. Complete a triangle and read off the remaining functions. 3. Construct and measure an acute angle whose secant is 13/5, and find its cotangent and cosecant. 4. Construct and measure an acute angle whose sine is 2/3, and find its cosine and tangent. 5. What is the angle of elevation of the sun when a yardstick casts a shadow of length 27 inches? How high is a flagpole whose shadow is then of length 15 feet? 10. Relations between the six trigonometric functions of an acute angle. For any acute angle A, we shall prove that (1) sin2 A + cos2 A = 1, where sin2 A denotes the square of sin A. By geometry, we have a2 + b2 = C2 Ch. I] FUNCTIONS OF ACUTE ANGLES 13 in the right triangle of Fig. 15. Dividing each member by c2, we get (a)2 (b)2 1. Since a/c = sin A, b/c = cos A, this proves formula (1). A- ^ b Next, to prove the useful formulas FIG. 15 sin A cos A (2) tan A = cot A =cos A sin A note that a sin A c a c a 1 T == -X = -= tanA, cotA cos A b c b b tan A c Finally, we shall need also the formulas (3) sec2 A = 1 + tan2 A, csc2 A = 1 + cot2 A, which are proved as follows: sin2 A cos2A cos2 A sin2 A 1 1 + tan2 A = 1 + - = = sec2 A c+OS2 A cos2 A os2 e A, cos2 A sin2 A + cos2A 1 1 + cot2 A = 1 + csc2 A. +cot +sin2 A sin2 A sin2 A Formulas (1)-(3) should be memorized. They enable us to find algebraically all the trigonometric functions of angle A when the value of one function of A is known - a problem treated graphically in Art. 9. EXAMPLE 1. If A is an acute angle whose cosine is 8/17, find by formulas (1)-(3) the five remaining functions of A. Solution. By (1), sin2A= 1-os2A= -( 1 -8)= ( 15) 14 TRIGONOMETRY [Art. 11 Since sin A is positive, we take the positive square root and get sin A = 15/17. Then, by (2), 15 8 15 tan A = 17 -8 By definition, cot A = 8/15, csc A = 17/15, sec A = 17/8. EXAMPLE 2. Given the value v of cot A, find sin A and cos A. Solution. By the second formula of (3), 1 1 csc A = + 1 + v2, sin A= = csc A 1 -+ v2 While cos A could be found from (1), it is simpler to use the second formula of (2), which gives cos A = cot A sin A = -. i + v2 EXERCISES ON THE RELATIONS BETWEEN THE FUNCTIONS OF AN ANGLE In each of the following 18 exercises, the value of one function of an acute angle A is given. Find by formulas (1)-(3) the values of the remaining functions of A. 4 1. sinA =. 7. cotA = 1. 13. tanA = 1. 5 12 15 2. cos A.= - 8. cot A = -. 14. sin A = s. 13 8 8 17 3. tan A = - 9. sec A =- 15. cos A = c. 15 15 3 13 4. sin A =. 10. sec A = 16. tanA = t. 5 12 5 5 5. cos =- 11. csc A = - 17. sec A = m. 13 4 4 5 6. tan A =- 12. cs A = 18. cscA = n. 3 3 19. Prove formulas (3) directly from Fig. 15. 11. Further formulas true for every acute angle. An excellent way to learn formulas (1)-(3) thoroughly is to use them in proving various new formulas. Ch. I] FUNCTIONS OF ACUTE ANGLES 15 EXAMPLE 1. Prove that, for every acute angle A, (tan A + sec A)2 = 1- sin A 1 - sin A Solution. Replace tan A by its value sin A/cos A from (2), and replace sec A by 1/cos A (definition). Then the left member of the proposed equation becomes /sin A (s in A +1)2 (1 + sin A)2 \ Aos A os A cos2 A (1 + sin A) (1 - sin A) since cos2 A = 1 - sin2 A by (1). Finally, we cancel 1 + sin A. EXAMPLE 2. Prove that, for every acute angle x >0, sin x 1 - cos x, 1 -cos x sin x Solution. Express each fraction as a new fraction having the common denominator (1 - cos x) sin x. The new fractions will be equal if their numerators are equal, i.e., if sin2 x = 1 - cos2 x, which is true by (1). Note that this proof might easily be twisted into the illogical one of manipulating the proposed equation (treated as if known to be a true equation) into the equation sin2x = 1 - cos2x by multiplying both members of the former by (1 - cos x) sin x. By such a false kind of argument we could "prove" that 4 = 2 by subtracting 3 from each member to obtain 1 = - 1 and then squaring to obtain the correct equation 1 = 1. It is safer for the student to draw a vertical line separating the two members of the proposed equation and manipulate each member separately by use of formulas (1)-(3) until he obtains two expressions which are identical. EXAMPLE 3. Prove that, for every acute angle B, sin B 1 + cos B -- = 2 csc B 1 + cos B sin B Solution. sin2 B + (1 + cos B)2 2 Solution. = (1 + cos B) sin B sin B sin2 B + cos2 B + 1 + 2 cos B (1 + cos B) sin B 2 + 2 cos B 2 (1 - cos B) B sin B sin B 16 TRIGONOMETRY [Art. 12 EXERCISES ON IDENTITIES Prove that the following equations are true for every acute angle: 1. (tan A + cot A) sin A cos A = 1. 2. cos2A (1 + tan2A) = 1. 3. tan2xcos2x + cot2xsin2 = 1. 4. cos x + tan x sin x = sec x. 5. sin y sec y cot y = cos y csc y tan y. 6. esc 2 ytan2y -1 = tan2y. 7. tan B + sec4B = 2sec 2Btan2B + 1. 8. cot2B - cos2B= cot2Bcos2B. 9. sin x cos x (sec x + cse x) = sin x + cos x. 10. sec 2 + csc 2 = sec 2 X CSC 2 X. 11. (1 + tan P)2 + (1 + cot P)2 = (sec 3 + csc 3)2. 12. sec 2 3csc2 = tan2 + cot2 P + 2. tan A - tan B, 13. cot A + cot B =tan A tan B. 1 - tan2A 2. 14. = cos 2 A -'sin 2A. 1 +- tan 2 A 1 sec z- cse _ tan z- 1 sec z + csc z tan z + 1 tan x - cot x 16. = 2 sin 2 x - 1. tan x + cot x 12. Relations between the functions of complementary angles. Two angles are called complementary if their sum is a right angle or 90~. Hence the two acute angles A and B B (Fig. 16) of any right triangle are complementary. We have.. a >b sin A= - =cos B, cos A =- = sin B, A C C C a b tan A =cot B. Passing to reciprocals, we get csc A = sec B, sec A = csc B, cot A =tan B. Ch. I] FUNCTIONS OF ACUTE ANGLES 17 Since B = 90~ -A, these formulas may be rearranged and written in the form sin (90 - A) = cos A, csc (90~ -A) = sec A, cos (90~ -A) = sin A, sec (90~ -A) = csc A, tan (90 - A) = cot A, cot (90 - A) = tan A. These six useful formulas, which are true for every acute angle A, may be combined into a single formula which is more easily remembered than the six. To this end, let us designate the cosine as the co-function of the sine, the cotangent as the cofunction of the tangent, the cosecant as the co-function of the secant, and, vice versa, the sine as the co-function of the cosine, etc. Thus always the co-function is obtained by annexing or deleting the prefix "co" before the function, according as the name of the function does not already start or starts with "co." Hence any trigonometric function of the complement of an acute angle A is equal to the co-function of A. The values of the trigonometric functions of the acute angles greater than 45~ can therefore be expressed in terms of functions of angles less than or equal to 45~. For example, cos 70~ = sin 20~, cot 65~ = tan 25~. Hence tables of the trigonometric functions need extend only as far as 45~. Moreover, we can now understand the origin of the names cosine, cotangent, and cosecant. In Latin, the cosine was initially complementi sinus, i.e., the sine of the complement, and this term was later abbreviated to cosinus. EXAMPLE. Find an acute angle x for which tan 2x = cot (45~ + x). Solution. Substitute for tan 2x the equal value cot (90~ - 2x). Since two acute angles with the same cotangent are equal, we have 90~ - 2x = 45~ + x, 45~ = 3x, x = 15~. EXERCISES ON FUNCTIONS OF COMPLEMENTARY ANGLES 1. Express as functions of their complementary angles sin 65~, cos 20~, tan 55~ 20', cot 82~ 12', sec 67~, csc 67~. 2. Express as functions of angles less than 45~ all the functions of 70~. 3. Show that sin (45~ + x) = cos (45 - x), tan (60~ + x) = cot (30~ - x). 18 TRIGONOMETRY [Art. 12 Find the acute angles x for which 4. cot x = tan (45~ + x). 5. cos 2x = sin (45~ - x). 6. tan 3x = cot 2x. 7. sin 2x = cos 4x. 8. cosx = cos (90 - x). 9. sin (2x - 30~) = cos (30~ + x). 10. Prove that tan (90~ - A) + cot (90~ - A) = csc A csc (90~ -A). CHAPTER II SOLUTION OF RIGHT TRIANGLES BY MEANS OF TABLES OF THE NATURAL FUNCTIONS 13. How to use tables of natural functions. Tables I-V give the values of the trigonometric functions of every acute angle expressed in degrees and sixths of a degree (10', 20', 30', etc.). In the headings of these tables, the word natural is inserted to distinguish them from the tables of the logarithms of the trigonometric functions, which will be explained in Chapter IV. Tables I and II together give the values of the (natural) sine and cosine to four decimal places of the angles from 0~ to 90~ at intervals of 10 minutes. When the number of minutes in an angle is not a multiple of 10, we resort to interpolation, as explained in Examples 4-6. EXAMPLE 1. Find sin 25~20'. Since we desire the sine of an angle less than 45~, we use Table I, marked natural sines at the top of its page, and look in the left-hand column for 25~ and then along its horizontal row until we reach the entry 4279 which occurs in the column marked 20' at its top. Supplying the decimal point (as indicated at various places in the second column), we have sin 25~20' = 0.4279. (The words in italics have the opposite sense to the corresponding words in Ex. 2; for example, less than and greater than, or top and bottom.) EXAMPLE 2. Find cos 64040'. Since we desire the cosine 'of an angle greater than 45~, we use Table I, marked natural cosines at the bottom of its page, and look in the right-hand column for 64~ and then along its horizontal row until we reach the entry 4279 which occurs in the column marked 40' at its bottom. Thus cos 64040' = 0.4279. This result agrees with that in Ex. 1 since the sine of an acute angle equals the cosine of its complement (Art. 12). We now see why a table of sines serves also as a table of cosines when properly labeled. EXAMPLE 3. Find cos 25020' and sin 64~40'. 19 20 TRIGONOMETRY [Art. 13 Using Table II and following the instructions given in Ex. 1 in the case of 25020', and those in Ex. 2 for 64~40', we get 0.9038 in either case. EXAMPLE 4. Find sin 25~23'. By Table I, sin 25~20' = 0.4279, sin 25030' = 0.4305. Our angle lies between these two angles and is three-tenths of the way from the smaller angle toward the larger. It is assumed that its sine will lie three-tenths of the way from 0.4279 toward 0.4305. The difference of the latter numbers is 0.0026, three-tenths of which is 0.0008 to four decimal places. Hence sin 25~23' = 0.4279 + 0.0008 = 0.4287. This work is abbreviated by making use of the tablettes, headed P. P. (proportional parts), given at the right of our table. Opposite to the marginal number 3 (for 3 tenths), we find in the tablette headed 26 the desired correction 8 (to the fourth decimal place). EXAMPLE 5. Find cos 64~58'. In Table I, using the labels at the bottom of the page and the angles in the right-hand column, we see that the entry 4253 for cos 64~50' exceeds the entry 4226 for cos 64~60' by 27. The P. P. of 27 for 8 (tenths) is 22. This correction is to be subtracted from 4253 to obtain cos 64058/ = 0.4231, since we noticed that the cosine decreases when the angle increases. The use of Tables III-V of natural tangents and cotangents is entirely similar to that of the tables just explained (see Ex. 6). The converse problem of finding an acute angle when one of its functions is given will be explained by the following example. EXAMPLE 6. Find the acute angle x whose cotangent is 2. Since Table V does not contain an entry exactly equal to 2, we employ the adjacent numbers between which 2 lies: cot 26~30' = 2.006 1 6 cot x = 2.000 15 cot 26~40' = 1.991 J Hence x is six-fifteenths of the way from 26~30' toward 26~40'. But 6 2 X 10'= X 10' = 4'. Hence x = 26~34'. Or we may use the P. P. tablette headed 15, look for 6 underneath 15, and at the left of 6 read off the marginal number 4 of minutes in the correction. Ch. II] TABLES OF NATURAL FUNCTIONS 21 EXERCISES ON THE USE OF TABLES OF NATURAL FUNCTIONS 1. Find the sines, cosines, tangents, and cotangents of (a) 18~24'. (b) 36045'. (c) 71~36'. 2. Find tan A + cot A when A = 55~25'. 3. Find cos B + cos -B + cos lB when B = 73~6'. 4. Find the acute angles for which sin x = 0.4567, cos y = 0.7654, tan z = 2.123, cot w = 0.1234. 5. Given cos 2C = 0.3456, find tan C. 6. A ladder 20 feet long leans against the side of a house, its foot being 8 feet from the house. What angle does the ladder make with the ground? 7. What is the angle of elevation of the sun when a vertical pole of height 12 feet casts a shadow 25 feet long? 8. Find the grade (Art. 6) of a railroad track which rises 4 feet per 100 feet along the track. 9. The Grand Canyon of the Colorado River has in places a depth of 1 mile and a width of 8 miles from rim to rim. Assuming that the walls or banks are planes intersecting at the bottom of the canyon, find the angles of inclination of the walls with the horizontal plane. 14. Solution of right triangles by means of tables of natural functions. Not counting the right angle, there are five parts in a right triangle, viz., its three sides and two acute B 'angles. If only the angles are given, the shape, but not the size, of the triangle is determined. But if C a we are given any two parts at least one of which A is a side, we proceed to show how we can solve b the triangle, i.e., find the remaining three parts. G. 17 First, let the two given parts be sides. Then (Fig. 17) in one of the three relations a b a (1) sin A = cos A =c, tanA = angle A is the only unknown, so that we can find A by looking in the tables. There remain two relations (1), of which one determines the third side and the other serves as a check. Finally, B = 90 - A. 22 TRIGONOMETRY [Art. 14 Second, let the two given parts be an angle and a side. Since A + B = 90~, we have both angles. Then two of the relations (1) determine the two unknown sides and the third relation serves as a check. Besides the check mentioned, a further check against gross errors may be secured by inspecting Traverse Table VI, to be explained in Chapter III, or by making a drawing to scale. EXAMPLE 1. Given ZA = 43~15' and the hypotenuse c = 13.20, find the remaining parts of the right triangle. Solution. B = 90~ - A = 46~45'.. a A.. b sin A = a c sin A cos A =,= c cos A c c sin A = 0.6852 cos A = 0.7284 c= 13.2 c = 13.2 13704 14568 20556 21852 6852 7284 a = 9.0446 b = 9.6149 Since only four decimal places of sin A and cos A were given by our tables, we have suppressed the fifth decimal places in the products. Check: tan A = 0.9408 = a/b. EXAMPLE 2. Given the hypotenuse c = 63.12 and a leg a = 47.04, find the remaining parts of the right triangle. Solution. First find ZA from sin A = a/c: 47.04 sin A = 6312 0.7452, A = 48~10~'. Next find b from cos A = b/c: b = c cos A = 63.12 X 0.6669 = 42.09. As a check, compute b directly from the given parts, using (2) b2= (c +a) (c-a), which is a way of writing a2 + b2 = c2. EXAMPLE 3. Given a = 27, b = 40, find the remaining parts. Solution. tan A = 27/40 = 0.6750, A = 34~1', B = 55~59'. c2 = a2+ b2 =729 + 1600 = 2329, c =48.260. Check: cos A = 0.8288 = b/c. The method of Ex. 3 is not recommended when either leg is expressed by a number containing more than two digits. Ch. II] TABLES OF NATURAL FUNCTIONS 23 15. Cases when an angle cannot be accurately found by Table II. If cos A = 0.9994, we cannot tell by Table II which of the angles between 1~50' and 2~10' is to be taken as A. Given the cosine of an angle greater than 9~, we can find the angle correctly1 to within 1', but not if the angle is less than 9~. Similarly, we may make an error of 2' or more if we determine by Table II an angle greater than 81~ whose sine is given. Such an error can arise in only two cases of the solution of right triangles and may be avoided as follows. Given leg b and hypotenuse c, with b nearly equal to c, so that angle A is small, do not attempt to find A from cos A = b/c, but first compute a by means of a = (c + b) (c- b), and then determine A by means of its tangent or sine. Similarly, given a and c, with a nearly equal to c, so that angle A is nearly 90~, do not attempt to find A from.sin A = a/c, but first compute b by means of formula (2) and then determine A by means of its tangent or cosine. In brief, when the hypotenuse and one leg are given, first find the other leg and then find an angle from its tangent. The computation is best made by logarithms (Art. 35). For example, let c = 4.345, b = 4.331. Then c + b = 8.676, c - b = 0.014, a2 = 0.121464, a = 0.3485, tan A = = 0.0805, A = 4~36'. However, in the practice of trigonometry we are concerned with the results of measurements. Let us therefore assume that c and b are correct only to the third decimal place, so that each may be in error by 4=0.0005. The correct value of c- b is between 0.013 and 0.015, and its product a2 by2 c + b is between 0.1127 and 0.130. Thus a is between.335 and.361. Then tan A is between.077 and.0834, whence A is between 4~24' and 4~46'. If we attempt to find A from cos A = b/c, allowing an error of =-.0005 in both b and c, we find that b/c lies between.9965 and.9970, whence A lies between 4~47' and 4~25'. 1As is shown by a five-place table of natural sines. 2In practice, we retain only the same number of significant figures in each factor. The product of 8.7 by.014 is.12 to two places, which alone are reliable. 24 TRIGONOMETRY [Art. 15 Either of our two methods of finding angle A leads to a conclusion which involves the same doubt. This doubt is not due to imperfections in our tables, but is due to the impossibility of finding angle A accurately from our badly chosen measurements. In fact, our first solution (which is theoretically the best one) shows clearly that a small absolute error in b and c may cause double that absolute error in the very small difference c-b and hence a large relative error in c - b, and consequently a large error in A. The same doubt occurs in the geometrical determination of angle B in Fig. 14, given nearly equal values of BC and AB. A small error in either evidently causes a large error in finding the intersection A of line AC with the arc AD of the circle, since the arc runs practically parallel to AC when AB and BC are nearly equal. The long multiplications, divisions, and extractions of square roots, involved in the solution and checking by tables of natural functions, are avoided in the solution by logarithms (Chapter IV) or by a traverse table (Chapter III). Hence only relatively few problems are proposed here for solution by tables of natural functions. EXERCISES ON SOLVING RIGHT TRIANGLES BY NATURAL FUNCTIONS Solve the right triangles having the following given parts: 1. c = 300, B = 20~. 2. A = 15~, b = 40. 3. A = 25, a = 15. 4. a= 115, c = 176. 5. b = 17, c = 18. 6. a= 3, b = 4. 7. From one milestone a house is seen in a direction making 60~4' with a straight road, and at the next milestone the angle is 30~2', the road between the milestones being an arm of each angle. How far is the house from the road? 8. Two forts are 4 miles apart and one is due west of the other. From one a ship is observed due north and from the other it is 15~ west of north (i.e., the line to the ship is to the west of the line to north and the angle between is 15~). How far is the ship from the nearest fort? 9. A tunnel slopes downward at an angle of 10~ with the horizontal. How far below the level ground is a point which is 220 feet down the tunnel? 10. A vertical pole 45 feet high casts a horizontal shadow 75 feet long. What will be the length of the shadow when the sun is 10~ higher? Ch. II] TABLES OF NATURAL FUNCTIONS 25 11. From a point P in the fifth floor of a building the distances between whose successive stories are all equal, the angle of depression of a point in the street is 48~. What is the angle of depression from a point, directly below P, in the third floor? Hint: Assume a convenient value, as 100, for the height of the fifth floor. 12.* From a point 5 feet above a lake, the angle of elevation of the top of a tree standing at the edge of the water is 50~, while the angle of depression of the lowest point of its reflection in the water is 55~. How high is the tree? 13.* From a pier 8 feet above the water, the angle of elevation of the top of the mast of a boat is 48~ and the angle of depression of the lowest point of its reflection in the water is 56~. What is the height of the top? CHAPTER III TRAVERSE TABLE; SOLUTION OF RIGHT TRIANGLES BY INSPECTION; PROBLEMS ON FORCES AND REFRACTION OF LIGHT 16. Description of Traverse Table VI. Table VI gives at sight the values to two decimal places of the legs of each right triangle in which an angle is 1~, 2~,.., or 89~ and the hypotenuse is 1, 2,.., 100, 200,.., or 900. For example, if the angle is 30~ and the hypotenuse is 20, we see by the table that the opposite leg is 10.00 (it is exactly half of the hypotenuse), and the adjacent leg is 17.32 (which is the value of 10 - 3 to two decimal places). In this chapter we ignore the further headingsl in Table VI, dis., lat., dep., D, D cos, D sin, which will be needed in the chapters on navigation and surveying. 17. When and how to use Traverse Table VI. In navigation, farm surveying, and in most practical problems, results correct to only three or four figures are desired, and then the traverse table may be used with a decided saving of time. When greater accuracy is necessary, the computation should be made by logarithms and checked against gross errors by the traverse table, which furnishes a more accurate and more rapid check than a drawing made to scale. When checking by the traverse table, we may omit all interpolation or interpolate roughly by inspection. There is one case in the solution of right triangles in which the use of the traverse table is inconvenient (though possible, Art. 46), viz., when we are given the two legs and seek the hypotenuse and the angles. None of the exercises in this chapter falls under this case. IOnly one set of headings has hitherto been printed in traverse tables. Bowditch's American Practical Navigator (U. S. Hydrographic Office) gives, in 90 pages, the values to one decimal place of latitude and departure for 1~,.., 89~ and distance 1, 2,.., 600. There exist other traverse tables of the same length which read to quarters of a degree and distances up to 100. Much more bulky is Boileau's Traverse Table, Computed for Every Minute of Arc and for Distances 1 to 10, published by Van Nostrand & Co. 26 Ch. III] TRAVERSE TABLE 27 The method of using the traverse table is similar to that for tables of natural functions (Chapter II) and will be explained by examples. EXAMPLE 1. Given the angle 44~ and the hypotenuse 385, find the adjacent leg. Solution. Divide the hypotenuse by 5 to obtain the hypotenuse 77 of a similar triangle which is directly within the limits of Table VI and has the adjacent leg 55.39 by the table. Its product by 5 gives the adjacent leg 276.95 of the original triangle. This result may be in error 0.02, or 5 times the possible error, 61.1 which is at most 0.005, in the reading from the table. To avoid such a material increase in the error, we ex- 300 press 385 as the sum of 300 and 85. For hypotenuse 300, the adjacent leg is 215.80. For hypotenuse 85, the 2l5.8 adjacent leg is 61.14. The sum of these legs gives the adjacent leg 276.94 of the original triangle (Fig. 18). FIG.18 EXAMPLE 2. Find the legs of a triangle whose hypotenuse is 385 and one angle is 44~20'. Solution. As in Ex. 1, we read from Table VI the numbers in the first and third lines of the following tablette: hyp. angle adj. leg opp. leg 385 44~ 276.94 267.45 385 44~ 20' 275.37 269.04 385 450 272.23 272.23 314.71 314.78 — 1.57 +1.59 We use 20 or 3 of the differences to interpolate the values in the second line. EXAMPLE 3. Given the angle 77~ and the opposite leg 55.80, find the hypotenuse and the adjacent leg. Solution. Table VI gives the first and third lines below: opp. leg hyp. adj. leg.26 J 55.54 57 12.82.97 '55.80 57.27 12.88.23 56.51 58 13.05 26X1=27, 26 -XI=.27 2 X.23 =.06 97 97 28 TRIGONOMETRY [Art. 17 EXERCISES ON HEIGHTS AND DISTANCES Solve the following problems by means of Traverse Table VI. 1. When the angle of elevation of the sun is 55~, how high is a flagpole which casts a horizontal shadow 42 feet long? 2. How high is a kite if held by a string 1100 feet long making 35~ with a vertical line? 3. From the top of a cliff 320 feet above water, the angle of depression of a boat is 15~. How far is the boat from the foot of the cliff? 4. Find the height of a kite whose angle of elevation is 40~ if held by a string 500 yards long. 5. From the top of a hill 1240 feet above a level plane, the angles of depression of two houses in the plane, both due south of the observer, were found to be 25~ and 35~. Find the distance between the houses. 6. The centers of two circles, whose radii are 16 and 24 inches, are 6 feet apart. Find the length of an exterior common tangent and its angle of inclination with the line of centers. 7. Two stations on a level plane are 6 miles apart. The angle of depression of one from a balloon directly above the other is 8~. How high is the balloon? 8. A man sees a fort 26~ north of east, and after walking 500 yards in the direction 40~ south of east he sees it due north. How far is the fort from his new position? 9. A spherical balloon 15 yards in diameter subtends an angle of 2~ from a point P at which the angle of elevation of the point of contact of the lower tangent from P is 50~. How high is the center of the balloon? 10. The angle of elevation at a point A of the center of a spherical balloon 40 feet in diameter was found to be 65~. At the same instant the angle subtended at A by the balloon was 2~. Find the height of the center of the balloon above the horizontal plane through A. 11. The angle of elevation of the top T of an inaccessible fort observed from a point P in the level plane is 12~. At a point Q, in the plane, 440 feet from P and on a line PQ perpendicular to PT, the angle PQT is 62~. Find the height of the fort. 12. From the ground floor of a building, the angle of elevation of the top of a hotel is 24~, while from a floor 30 feet higher up the angle of depression of the floor of the hotel is 28~. How high is the hotel? Ch. III] TRAVERSE TABLE 29 13.* There are two routes ABC and ADC from A to a place C on the opposite shore of a lake and 44 miles from A, each route being along portions of two intersecting roads. Given the angles CAB = 65~, BCA = 25~, ACD = 61~, CAD = 29~, find the length of the shorter of the two routes. RESULTANTS AND COMPONENTS OF FORCES 18. Force. Examples of forces are the exertions of a man in lifting, pulling, or pushing an object; gravity, which causes an unsupported object to fall to the ground; and a stroke which drives a base ball. In general, force is that which tends to change the state of rest or uniform motion of a body. We speak not only of the magnitude of a force (telling how large or small it is), but also of its direction. Hailstones descending vertically cause little damage in a city, but when descending obliquely often break window panes. Since a force is completely defined when its magnitude, its direction, and the point at which it is applied, are all given, a force is appropriately represented by a directed (arrowed) segment of a straight line (as OB in Fig. 19) 19. Resultant of two forces. Under the force of | hailstones descend vertically. But when acted upon also by the force of a north wind, hailstones descend obliquely in a southerly direction, just / 4 as if acted upon by a single A force in that oblique line. So always, if two forces are act- ing simultaneously upon a body, there is a single force, called the resultant of the two 4 forces, which will produce the FIG. 19 same effect upon the body as is produced by the joint action of the two forces. gravity alone, 20. Parallelogram of forces. If two given forces are represented 30 TRIGONOMETRY [Art. 21 in magnitude and direction by the lines OA and OB, their resultant will be represented in magnitude and direction by the diagonal OR of the parallelogram of which OA and OB are sides (Fig. 19). This law has been justified by many experiments such as the following. Choose any two numbers, as 2 and 3, and select any third number, as 4, which is less than their sum, but exceeds their difference (whence a triangle OBR can be drawn having 2, 3, 4 as sides). At one end of a cord passing over two pulleys attach a weight of 2 lbs. and at the other end a weight of 3 lbs. At a point 0 of the cord between the pulleys attach a weight of 4 lbs. After some motion, the system of weights will come to rest. Then, if OA represents 2 units of length, OB 3 units, and OR 4 units, the quadrilateral AOBR will be found to be a parallelogram. The same principle applies to velocities as well as to forces. 21. Component of a force. When a boat is towed by a horse walking along a bank of a canal, only a part of the force OR ex4 erted by the horse is effective in pulling O ~- -... 4 the boat in the direction OA of the I canal (Fig. 20). This part OA, where B ^ I angle OAR is a right angle, is called the B t_ i R component of OR in the direction OA. The component OB, in a direction perpendicular to OA and such that OBR is a right angle, tends to pull the boat toward the bank and is. balanced by the resistance of the water on the rudder. EXAMPLE. A ball weighing 80 pounds rests on a smooth plane inclined at the angle of 20~ to the horizontal plane. What force acting parallel to the inclined plane is necessary to keep the ball from rolling down the plane? Solution. We desire the component c, along a line making 70~ with a vertical line, of a force of 80 pounds acting vertically downwards. Hence c is the side adjacent to angle 70~ in a right triangle whose hypotenuse is 80. By Table VI, c = 27.36 pounds. Ch. III] TRAVERSE TABLE 31 EXERCISES ON RESULTANTS AND COMPONENTS OF FORCES 1. Two forces of 5 and 12 pounds act at right angles to each other. Find the resultant force (without using tables) and the angle which it makes with the second force. 2. A sled is pulled along a level road by a force of 65 pounds, the direction of the pull making 50~ with the horizontal. Find the forward pull on the sled. 3. Two men lift a weight by means of ropes in the same vertical plane. One man pulls 42 pounds in a direction making 22~ with the vertical and the other 63 pounds in a direction making 43~ with the vertical. How heavy is the weight? 4. A ship, always headed due east, steams at a speed which would carry it 12 miles per hour in still water. But, on account of a current running due south, its actual speed is 13.5 miles per hour and its track in the water makes an angle of 27~ with an easterly direction. Find the velocity of the current. 5.* Each half of a bridge is inclined 8~ to the horizontal. Find the vertical pressure and the horizontal thrust (i.e., pressure) upon the two supporting piers at the ends when the bridge bears 40 tons at its middle. 6. A horse pulls with a force of 400 pounds along a towline making 7~ with the direction of a canal. Find the effective force on the boat. 7. Find without tables the magnitude of the resultant of a force of 30 pounds acting southwest with a force of 40 pounds acting southeast. 8.* Resolve a force of 100 pounds into two equal components whose directions make an angle of 70~ with each other. 22. Refraction of light. When a ray of light AB passes from the air into water, it does not continue in its former direc- tion ABE, but is said to be refracted into a new direc- tion BC (Fig. 21). The angle G i which the incident ray AB \ makes with the perpendicular PB to the surface of the Q C r water is called the angle of -SD incidence. The angle r which FIG. 21 32 TRIGONOMETRY [Art. 22 the refracted ray BC makes with this perpendicular QBP is the angle of refraction. Experiments show that the quotient sin i - index of refraction sin r has the same value, approximately 4/3, whatever be the angle of incidence i. Hence, if i = 30~, 3 3 sin r = sin 30~ =- r = 22~. 4 8' The index of refraction from air to crown glass is about 3/2. To find the index of refraction from water to air, let the bottom of the tank (Fig. 21) containing the water be horizontal and transparent. For the same ray of light passing through the bottom into the air, let i' be the angle of incidence and r' the angle of refraction. Then i' and r are equal, being alternate angles, while, as shown by experiments, r' = i, so that the ray CD after passing through the water, with its upper and lower surfaces parallel, is exactly parallel to the ray AB before entering the water. Thus the index of refraction from water to air is sin i' sin r 3 sin r' sin i 4 and is the reciprocal of the index of refraction from air to water. EXAMPLE. A pebble lies at the bottom of a pool of water 4 feet deep. How far below the surface will the pebble appear to be to a man above if his line of vision makes an angle of 5~ with the perpendicular to the water? Solution. The ray of light CBA (Fig. 21) proceeds from the pebble C to the eye at A, the angle of incidence being r and the angle of refraction i = 5~. Thus sin r 3 sin 5 = sin 5~ = 0.0872, sin r = 0.0654, r = 3~45'. In triangle QBC, adj. side BQ is 4 ft., or 40 tenths of a foot. Table VI shows that when adj. side is 40 and the angle is 3~ or 4~, the opp. side is 2.09 or 2.80 respectively, and by interpolation is 2.62 for 3~45'. Thus QC = BH = 2.62. Since L GBH = 900-i = 85~, the opp. side HG is 30 (tenths) by Table VI. Hence the answer is 3 ft. Ch. III] TRAVERSE TABLE 33 EXERCISES ON REFRACTION OF LIGHT 1. A pebble lies at the bottom of a pool of water 6 feet deep. How far below the surface will the pebble appear to a man above if his line of sight makes 8~ with the perpendicular to the water? 2. Find the displacement CF (Fig. 21) of a ray of light passing through a crown glass plate 0.258 in. thick at the angle ABP = 55~50' with the perpendicular to the plate. 3. A man whose eye is at a point A, 10 feet above the level of the water BH (Fig. 21), sees at B the image of the foot C of a pile HC driven in the water. His horizontal distance from B is 24 feet, and from the pile 78 feet. Find the length HC of the pile below water. 4. If a ray of light passes through water and makes angle i with the perpendicular to the surface of the water, the largest value of i for which the ray will emerge into the air is C, where sin C = 3/4; while, for i>C, the ray is wholly reflected by the surface back into the water. Find this. critical angle C. Will a ray having i = 50~ be refracted or all reflected? 5. If the critical angle (Ex. 4) for crown glass is 42.5~, what is the index of refraction from crown glass to air? A crown glass prism, whose cross section is an isosceles right triangle, acts as a total reflector for rays entering at right angles to either perpendicular face. CHAPTER IV LOGARITHMS, SLIDE RULE 23. Powers of 10. The symbol 102 denotes the square of 10 and equals 10 X 10 = 100. Again, 103 denotes the cube of 10 and equals 10 X 10 X 10 = 1000. The product 102 X 103, being the product of two factors 10 by three factors 10, equals 105, which is the product of five factors 10. It follows that the quotient of 105 by 102 equals 103. These facts are expressed by the following equations: 102 X 103 = 105, 105 - 102 = 103. Similarly, if m and n are any two positive whole numbers, (1) 1om X 10n = 0 + n, (2) 10m -+ 10 = 10o- (if m>n). Just as (103)2 = 103 X 103 = 106, so always (3) (10m)n = 10n. Formulas (1), (2), (3) are known as the first, second, and third index laws, respectively. We next inquire what meaning must be given to symbols like 10, 10%, 10-2 if also these symbols shall obey the index laws. Using the third law (3), we have n (101)2 = 101 = 10, (10))3 = 10, (10d)d= 10= so that 10- is the (positive) square root of 10, 103 is the cube root of 10, and, in general, 10l/d is the dth root of 107: n (4) 10 = a/10, 103 =, 10. = We agree to employ the second law (2) also in the new case m = n = 2, whence 102 - 102 = 100. 34 Ch. IV] LOGARITHMS 35 But when 102 is divided by 102, the quotient is 1. Hence we make the definition that (5) 100 =. Finally, we employ (2) when m = 0 and obtain 100 + 10 = 1.0-. By our preceding result, 10~ = 1. Hence we have 1 (6):-= Ion To illustrate these new definitions, let us take the number 103 and divide it by 10, then divide the quotient 102 by 10, etc. We obtain the powers of 10 written in the upper of the following two rows: 103 102 101 100 10-1 10-2 10-3 1000 100 10 1 0.1 0.01 0.001. Similarly, starting with 1000, we obtain the numbers in the lower row by successive divisions by 10. This process makes clear that each power of 10 in the upper row should have the value written beneath it, and hence that our definitions (5) and (6) are justified. We have now attached a definite meaning to 10e when the exponent e is any whole number or fraction, whether positive, negative, or zero, i.e., when e is rational. We may also find a suitable meaning for 10e when e is any real number, like V2, which is not rational. In fact, if we employ the successively closer approximations 1.4, 1.41, 1.414, 1.4142,... to -/ 2, the corresponding numbers 101.4 = ' 104 4 10 101.414 101.4142... are successively closer approximations to a definite number' which is designated by 10'2. Conversely, any given positive number can be expressed in the form 10e by choice of a suitable exponent e. 1 Its value to two decimal places is 25.95, since this is the number whose logarithm is 1.4142, as the reader will soon be able to see by the table of logarithms. 36 TRIGONOMETRY [Art. 24 24. Logarithms. If 10o = N, we call e the (common) logarithm of N and write it "log N." Hence, by definition, (7) N = 10logN. Thus the exponent of the power of 10 which gives rise to a positive number N is called the logarithm of N. Expressed otherwise, the logarithm of a positive number N is the number which indicates the power to which 10 must be raised to yield N. Negative numbers do not have logarithms. From the following results in Art. 23: 1 102=100, 103=1000, 10=, 10= 10, 10 0=1, 10~=1 10-2 - 102 we therefore have log 100 = 2, log 1000 = 3, log IlO = i, log T 10 = 3, log 1 = 0, log 0.01 =-2. Since logarithms were defined as exponents (of powers of 10), the three index laws in Art. 23 lead at once to corresponding theorems on logarithms, which we proceed to state, prove, and illustrate. THEOREM 1. The logarithm of a product equals the sum of the logarithms of its factors. Let the factors be M and N. By definition (7), (8) M = 10 log M N = 10 log N. Hence, by the first index law (1), MN = 10log M +log N Thus, by the definition (7) of logarithms, (9) log MN = log M + log N. For example, let M = 1000, N = 0.01. By the above values, we get log MN = log 10 = 1, logM + logN = 3 + (- 2) = 1. THEOREM 2. The logarithm of a quotient equals the logarithm of the dividend minus the logarithm of the divisor. Expressed otherwise, the logarithm of a fraction equals the logarithm of the numerator minus the logarithm of the denominator. Ch. IV] LOGARITHMS 37 By (8) and the second index law (2), we have M lolo0g M- log N N Hence, by the definition of logarithms, M (10) log = log M - log N. For example, let M = 100, N = 10000. Then M log = log 0.01 = -2, log M -log N = 2 - 4 - 2. THEOREM 3. The logarithm of the pth power of a number equals p times the logarithm of the number. Raising the two members of the second equation (8) to the power p and applying the third index law (3), we get NP = (10 log N)p = 10p log N Hence, by the definition of logarithms, (11) log NP = p log N. For example, let N = 100 and p = 2. Then log Np = log 10000 = 4, p log N = 2 X 2 = 4. If in Theorem 3 we take p equal to 1/n, we obtain the COROLLARY. The logarithm of the nth root of a number is obtained by dividing the logarithm of the number by n. For example, let the number be 10000 and let n = 2. Then log 10000 4 log V10000 = log 100 = 2, 2 = 2 EXERCISES ON THE PROPERTIES OF LOGARITHMS 1. Find the logarithms of 100000, 0.1, 0.001, 4/100. 2. Find log V10 X V1000. 3. Given log 2 = 0.30103, log 3 = 0.47712, find (a) log 8. (b) log 24. (c) log A. 3 (d) log V3. (e) log V6. (f) log - (g) log 5. (h) log 60. (i) log 45. 38 TRIGONOMETRY [Art. 25 4. Express in terms of log N and log M the following: log N M2, ogN3 /M2, log IN-3/M-4. 5.* The logarithms of numbers a, ar, ar2, ar3,... in geometrical progression are in arithmetical progression, i.e., have a common difference. 25. Significant digits. The distance from the earth to the sun is not the same at different times of the year, but is said to be 93 millions of miles in round numbers. Since we do not mean to imply that the digits (or figures) which follow 3 are here actually all zeros, it is preferable to write the number in the form 93 X 106 or 9.3 X 107, rather than as 93,000,000. Again, the first measurement of a star's diameter was made by Michelson in 1920, who found that the giant star Betelgeuze has a diameter of about 3 X 108 miles. Since the sun's diameter is less than 106 miles, that star is 27 X 106 times as large as the sun. Similarly, many very small numbers are employed in modern physics, and it is customary to employ the notation 9.3 X 10-6, rather than 0.0000093. This number and 0.93 and 93 are all said to have the same two significant digits, 9 being the first significant digit and 3 the second. In both 93 and 3.7 the digit 3 is said to be in units' place. The results 93,000,000 = 9.3 X 107 and 0.0000093 = 9.3 X 10-6 furnish two illustrations of the following PRINCIPLE. If the first significant digit of a positive number lies p places to the left of units' place, the number can be expressed in the form N X 10P, where N lies between 1 and 10. But if the first significant digit lies p places to the right of units' place, the number can be expressed in the form N X 10-p, where again N lies between 1 and 10. As further examples of this evident principle, consider the numbers 0.93 and 930 in each of which the digit in units' place is 0. In 930, the first significant digit 9 lies two places to the left of units' place, and 930 = N X 102, where N = 9.3. But in 0.93, the first significant digit 9 lies one place to the right of units' place, and 0.93 = N X 10-1, where again N = 9.3. Ch. IV] LOGARITHMS 39 26. Mantissa and characteristic of a logarithm. We have just seen that any given positive number can be written in the form N X 10, where c = -p is a whole number which may be positive, negative, or zero, while N lies between 1 and 10. Hence the logarithm of the number equals the sum of log N and log 10 = c. Further, log N lies between log 1 = 0 and log 10 = 1 and hence is a positive decimal. This positive decimal is called the mantissa and c the characteristic of the logarithm of the given number. Thus the positive decimal part of a logarithm is called its mantissa, while the integral part is called its characteristic, which may be positive, negative, or zero. For example, the two numbers 10 = Vi = 3.162..., 10 = 10VTi = 31.62... have the logarithms ~ = 0.5 and ~ = 1.5, whose mantissas are each.5, and whose characteristics are 0 and 1 respectively. By the principle stated in Art. 25, we have the following THEOREM. The characteristic of the logarithm of a (positive) number is +p if the first significant digit of the number lies p places to the left of units' place, but is -p if it lies p places to the right of units' place. If two numbers have the same significant digits, so that they difer only in the position of the decimal point, their logarithms have the same mantissa and differ only as to their characteristics. For example, the logarithm of 31.62 has the characteristic 1 since the first significant digit 3 lies one place to the left of units' place. This agrees with the result log 31.62 = 1.5 given just above. Next, we have 31.62 log 0.03162 = log -0 = log 31.62 - log 1000 = 1.5 - 3. We do not express this difference in the form - 1.5, which is the sum of - 1 and the negative decimal -.5, since the latter is not a mantissa, mantissas being positive by definition. On the contrary, we write the above difference 1.5 - 3 in the form 2.5, which means - 2 + 0.5, and does not mean - 2.5. Hence we have log 0.03162 = 2.5, whose mantissa is the positive decimal.5 and characteristic is - 2. This is an illustration of our theorem, since the first significant digit 3 lies two 40 TRIGONOMETRY [Art. 27 places to the right of units' place. Some computers prefer to add and subtract 10 in such cases and write log 0.03162 = 8.5 - 10. In view of the last part of the above theorem, we may ignore the decimal point in a number when seeking, in a table of logarithms, the mantissa of its logarithm. Thus if the number is 31.62 or 3.162 or 0.03162 we enter the table with 3162 and read off the mantissa. 27. To find the logarithm of a number by Table VII. EXAMPLE 1. Find the logarithm of 16.17. When seeking the mantissa, we ignore the decimal point. In the left-hand column (headed N for Number), we look for the number 161 formed of the first three digits of 1617 and find it in the twelfth line. Since the fourth digit is 7, we glance along this twelfth line until we reach the entry 2087 in the column headed 7. Hence.2087 is the mantissa of log 16.17. Since the first significant digit 1 lies one place to the left of units' place, the characteristic is 1. Hence log 16.17 = 1.2087, to four decimal places. EXAMPLE 2. Find log 256.2. Since our number exceeds 199.9, it occurs in the second part of Table VII and interpolation is necessary. Our number lies between 256 and 257 whose logarithms occur in the table: log 256 = 2.4082 } 0017 = difference. log 257 = 2.4099 Since 256.2 is two-tenths of the way from 256 toward 257, we add.2 X.0017 =.0003 to log 256 to obtain log 256.2 = 2.4085. This correction 3 to the final digit may be obtained by inspection from the Proportional Parts tablette at the end of Table VII: In the column headed 17 and opposite to the marginal number 2 (for.2), we find the entry 3. EXERCISES ON FINDING LOGARITHMS 1. Find the logarithms of 56.78, 0.3456, 0.08765. 2. Verify by Table VII that log 35 = log 5 + log 7, log 23 = 3 log 2, log2 + log 5 = 1, log8 + log 5 - 2 log2 = 1. 3. Verify that log 1849 = 2 log 43, log 1728 = 3 log 12. 4. Given /To1 = 3.162, find log 3162, log 316.2, log 0.3162. Ch. IV] LOGARITHMS 41 5.* Show that the principle of interpolation is not valid if applied to numbers whose differences are not relatively small, by finding the error in log 200 when its value is interpolated between log 100 and log 300. 28. To find the number with a given logarithm by Table VII. EXAMPLE. Given log N = 2.5000, find N to four significant digits. Since the mantissa does not occur in the table, we use the adjacent numbers 4997 and 5011 between which 5000 lies: log 316 = 2.4997 3 log N = 2.5000 > 14 log 317 = 2.5011 Thus N exceeds 316 by 3/14 =.2, as may also be seen by inspection of the P. P. tablette for 14, whose entry 3 lies opposite to 2 in the margin. The decimal point must be inserted so that there will be two digits to the left of units' place. Hence N = 316.2. 29. Extraction of roots by logarithms. There is one step in the work which needs explanation. To find S N, given N = 0.1631, we use log N = 1.2125, log N = ~ (1.2125). Just as in the problem to take half of 13 ft., 2 in., we "borrow" 1 ft. from 13 ft. and add 12 in. to 2 in., and see that the answer is 6 ft., 7 in., -so here we "borrow" 1 from the characteristic 1 and add it to the mantissa. Thus log fN = (-2 + 1.2125) = 1.6063, - = 0.4099. Similarly, log -/V = 3 (-3 + 2.2125) = 1.7375, /N = 0.5464. EXERCISES ON COMPUTATION BY LOGARITHMS 1. What numbers have the logarithms 2.1516, 2.2222, 4.3333? Compute to four significant figures: 8124 X 0.00345 qa 2. 0.0006 X 87.42 3. /1234. 4. (84.62). 5. 445.24 X 11234. 6. 220. 7. (8.765)3. 8. Find the number of digits in 2400 and in 25100. 42 TRIGONOMETRY [Art. 29 9. Find the radius r of a circle whose area 7rr2 is 222.2 square feet, where 7r = 3.1416, approximately. 10. Find the radius r of a sphere whose volume is 4 7r r3= 4444 cubic feet. 11. If interest is compounded annually at 5 per cent a year, $1 amounts to $1.05 at the end of one year, to (1.05)2 dollars at the end of two years, etc. Find the amount on $2760 at the end of 15 years. 12. What sum of money put at compound interest at 6 per cent annually will amount to $2000 at the end of 10 years? 13. The combined area of the eastern states is 451900, that of the central states is 1380400, and that of the western states is 1193300, each in square miles. Find the percentage area of each division. 14. The Mississippi carries about 75 X 108 cubic feet of sediment yearly to the Gulf. How thick a layer would this sediment make if spread uniformly over the central states (Ex. 13) drained by that river? In how many years would the erosion be one foot? 15. How many cubic feet of water are held by a tank 125.2 feet long, 106.3 feet wide and 12.4 feet deep? 16. How many cubic feet of gas are held by a cylindrical tank 52.4 feet high, the radius of the circular base being 63.6 feet? 17. Regarding the earth to be a sphere of radius r=3957 miles, find its volume, circumference, and surface 4 7r r2. 18. How many feet per minute does a point on the equator move as a result of the earth's daily rotation (see Ex. 17)? 19. The earth moves about 6 X 108 miles in its yearly revolution about the sun. How far does it move in one hour? 20. The average distance from the center of the earth to the center of the moon is 238900 miles, and the moon revolves about the earth once in 27.32 days. Find the average velocity of the moon in (a) miles per hour and (b) feet per second. 21. The male population of a town is 7646 and the female population is 8534. Find the percentage of each. 22. How much iron is contained in 28.6 grams of pianoforte wire, 99.7 per cent of which is iron? 23. Potassium bromide salt was found by chemical analysis to contain.2463 gram of potassium and.5038 gram of bromine. Find the percentage of each in the salt. Ch. IV] LOGARITHMS 43 30. Logarithmic scale. The multiplications and divisions which we have learned to perform by use of a table of logarithms can be performed geometrically (but less accurately if the drawing is of moderate size) by use of segments representing log 2, log 3, etc. The following segments (Fig. 22) represent log 2,..., log 10 to the same scale, the annexed values of the logarithms to two decimal places having been taken from Table VII: _log -0.30._- ____ log 3=0.48 -_____________.log 4 =0.60,______10 log 5= 0.70..........________ log 6= 0.78 -_______________. log 7 =0.85 log 8 =0.90 log 9=095 log 10-1 FIG. 22 We now transfer these segments to the same line so that their left-hand points coincide, and omit "log" from the labels log 2, log 3, etc., at their right-hand points, thus retaining only the labels 2, 3, etc. We obtain Fig. 23, in which the left-hand point is labeled 1 since log 1 = 0. 2 3 4 5 6 7 8 9 10!. ', I ---,i I I I- l - i - I FIG. 23 Such a logarithmic scale was proposed by E. Gunter in 1620. It may be used to perform multiplication and division. For example, to multiply 2 by 3, transfer with the dividers or a pair of compasses the segment extending from 1 to 3 to a position starting at 2; the right-hand end of the transferred segment is seen to be at 6, in agreement with the fact that log 2 + log 3 = log 6. 44 TRIGONOMETRY [Art. 31 Hence multiplication and division can be performed geometrically by means of a logarithmic scale (Fig. 23). 31. Slide rules. It was noticed by W. Oughtred in 1630 that we may dispense with the use of the dividers in Art. 30 by using two like logarithmic scales E and F (Fig. 24), which slide along 1E 3 4 5 6 7 8 9 10 F _ 3 4 5 6 7 8 910 FIG. 24 each other. The process to multiply 2 by 3 now consists in placing scale F so that its point marked 1 is underneath the point marked 2 on scale E; then above the point marked 3 on scale F we find on scale E the required product 6. To divide 6 by 3, place scale F so that its point 3 falls under point 6 on scale E and read on E the point above 1 on F. The above pair of logarithmic scales enables us to find the products, when not exceeding 10, of numbers having only one digit, and hence would be of no practical use. But we readily extend its usefulness as follows. First, by using also log 1.1, log 1.2,..., log 9.9, we subdivide each main interval in Fig. 23 into ten (unequal) divisions, which are marked by short bars (and every fifth one by a longer bar) and not by numbers (1.1, 1.2,..., 1.9 for the first main division). Such a modified logarithmic scale is shown in the left half of scale A in Fig. 25, where are omitted the subdivisions of the shorter main divisions. By using two such modified scales sliding along each other, as the left halves of scales A and B in Fig. 25, we can find the products of numbers each having a units' digit and one decimal place. Second, we can now readily extend the scale to include all whole numbers not exceeding 100. For example, we obtain a segment representing log 27 = 1 + log 2.7 by adding the length Ch. IV] LOGARITHMS 45 unity of the entire scale to the known segment which represents log 2.7. Thus we have only to annex a copy of our scale at the right of it. At first sight it would appear to be necessary to change the former labels 1, 2,..., 10 at the main division points to the proper labels 10, 20,..., 100 on the annexed scale. But such a change in labels is not usually made on slide rules, so that the two halves of scale A in Fig. 25 have the same labels. The label for the point in common to the two halves of scale A is taken to be 1 instead of 10; the final point is labeled 1 instead of 100. Thus the point which is 7 small divisions to the right of the main division 2 of our annexed scale (the right half of scale A) is read 27, while if on the initial scale (the left half of scale A) it is read 2.7. Scale B (Fig. 25) is exactly like scale A and slides along it. By their use we can multiply numbers each having two significant digits. While the two like scales A and B serve for multiplication and division, we need a new scale D whose comparison with scale A will enable us to compute squares and square roots. We proceed to show in detail how this is accomplished. Scales C and D are alike and each is a single logarithmic scale from 1 to 10 whose length equals the combined lengths of the halves of scale A. Hence the logarithm of any number is represented __ c:COJ_ - lf3 - -V M — ~cl -_ I I U1 ca cc_ — tz, - '10 - os co_-I" - -00 -_c -— d- -V-:-nS^ 7 -_sFHe - cc 46 TRIGONOMETRY [Art. 32 on scale D by a segment twice as long as the segment representing the same number on scale A. Thus the number on scale A vertically above any number on scale D is equal to the square of the latter number. Hence the pair of scales A and D serves to find squares and square roots. To facilitate the comparison of scales A and D which are not in direct contact, there is a small movable runner, or framed rectangular glass, a fixed vertical line on which connects corresponding numbers of scales A and D. Scales A and D are on the rule, while B and C are on the slide. Scales C and D may be used together for multiplication and division just as A and B were used; while giving greater accuracy than the latter, the range of numbers is smaller. To find 6 X 7 by use of C and D, shift the slide C to the left until its right-hand mark 1 is above 6 on D; the reading on D of the point below 7 on C is 4.2, so that the desired product is 42. We have now described the slide rule and shown how to use it to perform multiplication, division, squaring, and extraction of square roots. We next indicate the further devices necessary for performing trigonometric computations. If the slide be entirely withdrawn, its reverse side will usually be found to have along one edge a scale S of logarithmic sines and along the other edge a scale T of logarithmic tangents (Art. 32). To compute a = c sin x, set the beginning point of scale S opposite to c of scale A, and opposite to x on S read a on A (since we have added log sin x to log c). To find c cos x, use c sin (90~x). To find b tan x, use scales T and A. For further details, including descriptions of the more accurate cylindrical slide rules of Thacher and Fuller (which are equivalent to straight rules 720 and 500 inches long), see Raymond's Plane Surveying, pp. 179-198. Directions are usually furnished with each slide rule sold. 32. Logarithms of trigonometric functions. Since sin A and tan A are numbers, they have logarithms denoted by log sin A and log tan A, which are read "log sine A" and "log tangent A." For Ch. IV] LOGARITHMS 47 example, by Table I of natural sines, sin 25~ 20' is equal to 0.4279. By Table VII of the logarithms of numbers, log 0.427 = 1.6304, log 0.428 = 1.6314. Hence, by interpolation, log sin 25~ 20' = log 0.4279 = 1.6313 = 9.6313 - 10. This result may be read off at once from Table VIII of log sines. Opposite to 25~ 20' occurs the entry 6313, to which is to be prefixed the heading 9. of log sin column. We must remember to annex -10, which is not printed in the table. The logarithms, to four decimal places, of the sines, tangents, cotangents, and cosines of each acute angle are given in this Table VIII, except that 10 must be subtracted from every entry read from the table. EXAMPLE. Find log cot 6~12'. By the table, log cot 6~10' = 10.9664 - 10, log cot 6~15' = 10.9605 - 10. 2 From the former we subtract the correction 2 X 59 = 24 (which may be read off from the P. P. tablettes for differences of 5'), and obtain log cot 6~12' = 0.9640. EXERCISES ON LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 1. Find log sin 30013', log cos 42057', log tan 42~57'. 2. Find the acute angles for which log sin x = 9.8334 - 10, log cot y = 0.3362, log cos z = 8.7054 - 10, log tan w = 0.4498. 3. Given log sin A = 9.7480 - 10, find A. and log tan A. 4. Given log tan B = 10.0700 - 10, find B and log cos B. 5. Verify that log sin 13~ - log cos 13~ = log tan 13~. Why is the equation true? 6. Verify that log cos 8023' - log sin 8023' = log cot 8023'. Why true? 7.* Verify that log (1 - cos 22~) + log (1 + cos 22~) = 2 log sin 22~. Why true? CHAPTER V SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 33. Results in Chapter II recalled. We saw in Chapter II that we can solve any right triangle if we are given two parts, at least one of which is a side, by using A + B = 90~ and two of the four formulas sin A cos A = -, tan A = a2+ b2 = c2. c' c b Since we there used tables of the natural functions, long multiplications, divisions, and extractions of square roots, were necessary. But this labor may be spared, and hence the possible number of errors reduced, by using logarithms. 34. Solution by logarithms. Since we require no further theory, we are ready to take up an illustrative example. EXAMPLE. Given angle A = 55~45' and the adjacent side b = 12.02, solve the right triangle. Solution. Since we need a figure to derive the B formulas required in the computation, we draw one to scale on square-ruled paper (Fig. 26) and measure I I - I the unknown parts, thus obtaining a rough check on the later computations. By measurement, a- - a = 17.3, c = 21.3, B = 34, -, - -- approximately. Next, we make a complete outline _ r _ -~- which shows all the formulas to be used and also the, -/ 2 —~~ individual terms of the formulas, leaving blank spaces.Ad b for the later insertion of their numerical values. FIG. 26 Finally we insert the values from the tables and perform the additions, etc. It saves time to look up all the logarithms together. By thus separating the theoretical work of making an outline for the solution from the mechanical work with the tables, we materially reduce the chance of errors. a b b tan A =, a = b tan A cos A c, osA b' - c'~~~~~~~ cos A 48 Ch.V] SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 49 log a = log b + log tan A log c = log b - log cos A log b = 1.0799 log b = 1.0799 log tan A = 10.1670 - 10 log cos A = 9.7503 - 10 loga = 1.2469 log c = 1.3296 a = 17.66 c = 21.36 Check: a2 = c2- b2 = (c + b) (c - b) c + b = 33.38 log (c + b) = 1.5234 c-b = 9.34 log (c-b) = 0.9703 log (c2 - b2) = 2.4937 log -Ic2 - b2 = 1.2469 = log a. Our check formula is preferable to sin A = a/c, since in employing the latter we would use our values of log a and log c and hence not detect possible errors in finding a and c from their logarithms. Traverse Table VI furnishes a check which is more accurate and quicker to apply than that from a drawing to scale. Thus for 56~ and c = 21k, we read off opp. leg a = 17.7, adj. leg b = 11.9. Before applying the computation check, note whether the computed parts agree approximately with their measured values or those from the traverse table; if not, repeat the construction of the drawing and the measurement of the unknown parts, or the readings from the traverse table, and, if the gross error is not thus detected, look for a gross error in the computation (Art. 36). 35. Given the hypotenuse and a leg. Given the hypotenuse c and a leg b, we saw in Art. 15 that it is best first to find a by means of (1) a2= (c+ b) (c-b) and then determine angle A F from its tangent. K EXAMPLE. At what greatest dis- C tance at sea can a mountain 3 miles high be seen, if the earth is regarded as a sphere of radius 3957 miles? \ Solution. In Fig. 27, let C be the point at sea, B the top and F the foot of the mountain, and A the cen- A ter of the earth. Then BC is tangent FIG. 27 50 TRIGONOMETRY [Art. 35 to the earth and hence is perpendicular to the radius CA. We regard a = BC as the required distance, given that b = AC = 3957 and c = BF +FA = 3 + 3957. c + b = 7917 log (c + b) = 3.8985 c - b = 3 log (c - b) = 0.4771 log a2 = 4.3756 log a = 2.1878 a = 154.1 NOTE. If we regard the chord CF as the required distance, we may compute its length after finding angle A. BC a log a = 2.1878 tan AC - b log b = 3.5974 log tan A = 8.5904 - 10 A = 2~13.8'. Draw AM perpendicular to chord CF. In the right triangle ACM, CM sin CAM = sin A = 2 b log CM = log b + log sin 2 A log b = 3.5974 log sin 1~6.9' = 8.2891 - 10 log CM = 1.8865 CM = 77.00 Chord CF = 154.0. Hence the chord is practically of the same length as the tangent CB. When, as in our example, leg b is only slightly less than the hypotenuse c, the small angle A cannot be accurately determined by cos A = b/c. For instance, if log cos A = 9.9999 - 10, we cannot tell by Table VIII which of the angles between 50' and 1~ 40' should be taken as A. While we may proceed as in the above example, we readily obtain a formula which gives A at once and with appropriate accuracy (cf. Art. 15). In Fig. 27 draw BK perpendicular to the bisector AML of angle BA C. Then the right triangles A LB and A LK are equal, and AK = AB = c, CK = AK-AC = c-b. Ch.V] SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 51 Since angles CBK and CAL=A are both complementary to angle AKB, they are equal. Hence, from the right triangle BCK, tanA =tan CK c-b (c-b)2 tan A -tan CBK - b)= = - = CIKCB a (c+b)(c by (1). The desired formula is therefore (2) tanA c-b= c+b' In the above example, we had log (c - b) = 0.4771 log (c + b) = 3.8985 21 4.5786 log tan ~A = 2.2893 = 8.2893 - 10 2A = 1~6.9'. 36. Errors of computation. Gross errors may arise by 1. Reading from the figure the wrong ratio as the value of a trigonometric function, and hence starting from a false formula. 2. Solving the formula incorrectly for the unknown part. 3. Passing erroneously to the logarithmic form of the formula. 4. Taking an entry from a wrong column of a table, perhaps using the label at the top instead of the bottom of the column. 5. Failing to omit the characteristic 1 of log a = 1.2470 and erroneously entering the table with 1247, when seeking a. 6. Adding two logarithms when we should subtract. 7. Supplying the wrong characteristic to the logarithm, perhaps failing to subtract 10 from the tabulated logarithm of a trigonometric function. When an answer is in error only in the position of the decimal point, the error was due presumably to the present cause only. If wide disagreement with the values from the drawing to scale or from the traverse table shows that a gross error still remains undetected, check the additions and subtractions by "casting out nines." To cast out nines from 678, subtract 9 from 6 + 7 to obtain 4 and subtract 9 from 4 + 8 to obtain 3. To cast out nines 52 TRIGONOMETRY [Art. 36 from 406, subtract 9 from 4 + 6 to obtain 1. Casting out nines from the sum 1084 of 678 and 406, we obtain 4. Since 3 + 1 = 4, the addition is checked. Similarly, if we desire to check that the subtraction of 406 from 678 yields 272 correctly, we cast out nines from the result to obtain 2 and note than 3 - 1 = 2. When the computed values agree approximately with the values measured from the drawing or read from the traverse table (so that gross errors are improbable), the check formula may show that the computed values contain a small error. Such an error is usually due to one of the following causes: I. Numerical error in interpolation. II. Adding the correction for interpolation instead of subtracting it or vice versa. III. Failure to read minutes in the proper line of the table. EXERCISES ON SOLVING RIGHT TRIANGLES BY LOGARITHMS Solve by logarithms the right triangles in which the following two parts are given, check by a formula and either by a drawing to scale or by the traverse table. 1. A = 28~5', c = 1140. 2. A = 36~44', a = 97.06. 3. a = 2238, c = 4295. 4. b = 35.89, c = 43.27. 5. a = 879.0, b = 656.3. 6. A = 53030', b = 91.28. 7. a = 6.845, b = 8.463. 8. A = 43~48', c = 147.2. 9. Find the greatest distance at sea at which a mountain 2 miles high can be seen, taking the earth's radius as 3957 miles. 10. At what distance can the pilot of an aeroplane I mile high see an object on the earth? 11. How far has a man rowed from a lighthouse 66 feet high when he observes it disappear on the horizon? 12. How far apart are two ships whose crow's nests are 80 and 95 feet above water and are just visible from each other? 13. What was the slope of a road bed 10000 feet long which, after being made horizontal by leveling, measured 9996 feet? 14. Solve the triangle whose hypotenuse equals the product of a leg by 3. Ch. V] SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 53 37. Area(A) of a right triangle. By geometry, A = ~ ab. If the legs a and b are not both given, we may first compute the unknown leg or legs by partially solving the triangle. But as a number computed by logarithms is usually not quite exact in the last decimal place, it is preferable to find the area from the given parts whenever possible. Given angle A and leg a or hypotenuse c, we may use the first or the second of the following formulas: = a2cA, 2 sin A cos A, A 2 n A. To verify these formulas, insert the values from Art. 33: a2 cot A = a2 = - ab, c2 sin A cos A = c2 * * = ab. Z -2 a 2 2 2 Cb 2 a c c Finally, if a and c are given, we must first compute A or b, preferably b (Art. 35). EXERCISES ON THE AREA OF A RIGHT TRIANGLE Find the areas of the right triangles in which 1. a = 14, A = 18~14'. 2. a = 12.3, c = 140. 3. c = 34, A = 2006'. 4. c = 4.231, A = 86~4'. 5. Solve the triangle, given b = 20, area (A) = 232. 6. Solve the triangle, given A = 61~, area = 48. 38. Isosceles triangles and regular polygons. An isosceles triangle is divided into two equal right triangles by the perpendicular VF = h from the vertex V to the base v AB, whose length will be designated by s (Fig. 28). The solution of the isosceles r triangle is therefore reduced to that of one of its component right triangles. A Consider a regular polygon ABCD... of FI 2S n sides, each of length s, which is inscribed in a circle with the center V and radius VA = r. [In Fig. 29 is shown a regular pentagon ABCDE both inscribed in a circle of radius r and circumscribed about a (dotted) circle of radius h.] 54 TRIGONOMETRY [Art. 38 The radii VA, VB, VC, etc., divide the polygon into n equal isosceles triangles VAB, VBC, etc. Since the sum of the n D I V r i& A F FIG. 29 EXAMPLE. Find the side circle of radius 10. angles AVB, BVC, etc., is 360~, each contains 360/n degrees. The half angle AVF therefore contains 180/n degrees. Problems on inscribed regular polygons therefore reduce to problems on one of the right triangles in Fig. 28. Similar remarks hold for a regular polygon circumscribed about a circle of radius h. A hexagon has 6 sides, an octagon has 8 sides, and a decagon has 10 sides. of a regular pentagon circumscribed about a Solution. We employ Fig. 29 with h = 10, ignoring the outer circle. Angle AVF contains 180/5 degrees. Hence AF tan 36~ = tan AVF = A, AF = 10 tan 36~ = 7.265, AB = 14.530. EXERCISES ON ISOSCELES TRIANGLES AND REGULAR POLYGONS For an isosceles triangle lettered as in Fig. 28, with V = LA VB and area A: 1. Given r = 286, s = 220, find A, V, h, A. 2. Given s = 47.04, V = 69~49', find r, h, A. 3. Given h = 8.4, A = 10.92, find A, V, r, s. 4. Given s = 42.90, V = 121~15', find r, h, A. 5. Given r = 30.01, A = 52010', find s, h, A. 6. Given A = 54.34, h = 14.89, find A, r, s. 7. A barn is 80 feet wide and 160 feet long, while the pitch of the roof is 45~. Find the length of the rafters and the area of the roof. 8. Find the angle at the center of a circle of radius 90 subtended by a chord of length 132. 9. Find the radius of a circle in which a chord of length 20 subtends an angle 133~ at the center. Ch.V] SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 55 10. The perimeter p (sum of sides) of a regular polygon of n sides inscribed in a circle of radius r is equal to 2rn sin (180 /n), and its area is equal to 'hp. 11. Find the radius of the circle inscribed in a regular pentagon, the area and perimeter of the pentagon, when the radius of the circumscribed circle is 4 inches. 12. Find the side of a regular decagon inscribed in a circle of radius 100. 13. Find the side of a regular decagon circumscribed about a circle of radius 100. 14. Find the difference between the perimeters of a regular pentagon and a regular hexagon if the area of each is 4800. 15. Find the side, and the radii of the inscribed and circumscribed circles, of a regular octagon whose area is 96. 16. Find the area of the regular octagon formed by cutting off the corners of a square whose side is 100. 17. Find the area of a regular pentagon if its diagonals are each equal to 6. 18. A wedge measures 20 inches along its side, while the angle at the vertex is 20~. Find the width of the base. 19.* A flagpole 24 feet high, standing on a roof which slopes upward at angle 20~, casts a shadow 40 feet long extending perpendicular to the ridge pole of the roof. What is the angle of elevation of the sun? 20.* The base of the great pyramid of Gizeh is a square whose side is 762 feet long and its top is a square whose side is 32 feet long. Each face makes an angle of 51~51' with the horizontal plane. Find the height. 21.* A regular pyramid of altitude 6.079 has a square base whose side is 4.284. Find the angles which the lateral edges and the slant height make with the plane of the base. 39. Problems on heights and distances. Most of the exercises below are slight variations of the example solved and of Ex. 7, for which convenient formulas are supplied. EXAMPLE. TO find the height x of a hill above the horizontal plane through a point P. set up a pole of known length h on the top of the hill and measure the angles of elevation T and B at P of the top and bottom of the pole. Prove that h tan B (3) ~ x - tan T-tan B 56 TRIGONOMETRY [Art. 39 Solution. Let d be the horizontal distance PQ (Fig. 30) from P to the point Q below the top of the hill. Then I x x+h // jtanB =, tan T = Jx d' d By division, MENV @ tan B x p i d IQ tan T x+h FIG. 30 Solving for x, we obtain (3). Note that we must compute the denominator in (3) by use of the table of natural tangents. The rest of the work is performed conveniently by logarithms. An equivalent formula involving only multiplications and divisions, but no subtraction, and hence more suitable for logarithms, will be derived from (3) in Art. 98. EXERCISES ON HEIGHTS AND DISTANCES 1. Apply formula (3) when h = 20, T = 42~, B = 35~. 2. From a point of a horizontal plane the angle of elevation of a mountain peak 2 miles above the plane is 64~, and that of a cloud directly over the peak is 67~. How high is the cloud above the peak? 3. From the top of a cliff the angles of depression of the top and bottom of a lighthouse 962 feet high are observed to be 23015' and 24020' respectively. How much higher is the cliff than the lighthouse? 4. The angle of elevation of the top of a flagstaff is 36~25' as measured by a transit whose telescope is 5 feet above the ground, the horizontal distance from the transit to the foot of the flagstaff being 125.4 feet. Find the height of the flagstaff. 5. Find the width BC of a river, given the length 310.4 feet of a line AB along one bank, and granted that an object C on the opposite bank is located so that ABC = 90~, CAB = 50025'. 6. A ladder 24 feet long stands at an angle of elevation of 72~ against a wall. If the foot of the ladder is drawn away 1~ feet, what is its new angle of elevation an(d how far down the wall has the top of the ladder moved? 7. To find the height h of a hill above a horizontal plane, measure the angles A and B of elevation of the top of the hill from two points P and Q in our plane such that if the line PQ were extended through Q it would Ch.V] SOLUTION OF RIGHT TRIANGLES BY LOGARITHMS 57 pass through the foot F of the perpendicular dropped from the top of the hill to our plane. If d is the distance between P and Q, show that d cot A - cot B Hint: Employ the length 1 of QF. 8. Apply the formula in Ex. 7 when d = 500 ft., A = 30~, B = 45~. 9. From the top of a hill the angles of depression of two consecutive milestones on a straight horizontal road leading to the hill were observed to be 5~ and 15~. Find the height of the hill. Hint: Show that the formula of Ex. 7 may be applied here. 10. The shadow of a tree lengthened 70 feet while the angle of elevation of the sun decreased from 60~ to 40. Find the height of the tree. 11. A tree stands on the bank of a river. The angle of elevation of its top from a point opposite on the other bank is 37~5', but is 17~3' from a point 80 feet from that bank in a straight line from the first point. Find the width of the river and the height of the tree. 12. A hill is inclined 38~ to the horizon. After walking 600 feet away from the foot of the hill, a man finds that the angle of elevation of a point half way up the hill is 20~. How high is the hill? 13. From two successive positions, d miles apart, of a ship the angles between its direction of sailing and the direction of a rock were found to be 260~ and 45~. How near is the rock when the ship passes it? Find another pair of equally favorable angles. 14.* From a point on level ground the angles of elevation of the foot and of the top of a flagstaff standing on the roof of a building were found to be 40~ and 51~ respectively. From a point 150 feet farther away on the ground the angle of elevation of the top of the flagstaff was found to be 33045'. Find the height of the flagstaff. 15.* The angles of elevation of the top of a tower from the fourth floor of a building and from a point directly below on the second floor were found to be 25~30' and 35012', while the distance between consecutive floors is 12 feet. Find the height of the tower and its horizontal distance from a point of observation in the building. 16.* To an eye 20 inches in front of a mirror, an object appears to be 16 inches back of the mirror, the line of sight making an angle of 33~ 58 TRIGONOMETRY [Art. 39 with the mirror. What is the distance and direction of the object from the eye? 17.* A man observes that the horizontal angle subtended by a cylindrical gas tank is 12~, and after moving 400 feet directly away from the tank the angle is 8~. What is the radius of the tank? 18.* A man in a balloon observes that the angle of depression of a house due north is 33~. After the balloon drifts horizontally 4 miles due west, the angle is 210. How high is the balloon? 19.* A hill is due south of one milestone A and is due east of an adjacent milestone B on the same straight horizontal road. At A the angle of elevation of the top of the hill is 40~ and at B 27~. How high is the hill? CHAPTER VI NAVIGATION: DEAD RECKONING 40. Navigation and its subdivisions. Navigation is the science which enables the mariner to determine with sufficient accuracy the position of his ship at any time. The branch of the science which makes use of observations on the sun or stars is called celestial navigation or nautical astronomy. The branch which makes use only of the measurement of angles and distances between points on the earth is called terrestrial navigation; it comprises piloting and dead reckoning. In piloting, a ship's position is found from visible objects on the earth or from soundings of the depth of the sea. In dead reckoning, which alone is treated in this text, a ship's position at a specified time is computed fyom the distances and directions which the ship has sailed from the port left or from a place whose position was found by celestial observations. 41. Geographical terms. The earth rotates daily about its axis, which intersects the earth's p surface at the north pole P and south pole P'. The earth TK I \ \0 K differs from a sphere by being slightly flattened at the poles. / But for the ordinary purposes I... of navigation the earth is as- -- _- - - E- sumed to be a sphere. Meridians are great circles, like PABP' and PGHP', in \ Fig. 31, in which planes through both poles intersect the earth's ' surface. Fro. 31 — -- 59 60 TRIGONOMETRY [Art. 42 The equator WBHE is the intersection of the earth's surface and the plane through the earth's center C perpendicular to the axis PP'. Thus each point B, H, E of the equator is 90~ from each pole P and P'. A parallel of latitude is a small circle, like TAK, in which the earth's surface is intersected by a plane perpendicular to the axis. The latitude AB of a place A on the earth's surface is the arc of the meridian intercepted between the place A and the equator. All points on the same parallel of latitude have the same latitude; for example, AB =KE = TW. Points between the equator and the north pole P are said to be in north latitude; those between it and the south pole P', in south latitude; for example, the latitude of A is 40~ N. The difference of latitude AF of two places A and G is the arc of a meridian included between their parallels of latitude. The longitude of a place A on the earth's surface is the arc BH of the equator intercepted between the meridian of A and the prime meridian PGH through Greenwich, England, and is measured by the angle BPH at the pole P. Longitude is reckoned from the prime meridian east or west in degrees up to 180~; for example, the longitude of A is 30~ W. The difference of longitude of two places is the arc of the equator included between their meridians. 42. Nautical mile. A nautical or sea mile is defined in the United States of America to be 6080.27 feet in length and equal to one-sixtieth part of a degree of a great circle on a sphere whose surface has the same area as the earth's surface. In England a nautical mile is of length 6080 feet. For the purposes of navigation, a nautical mile is assumed to equal one minute of latitude, so that there are 60 nautical miles in a degree of latitude. Throughout this chapter, mile means nautical mile. A statute mile contains 5280 feet. 43. How distance is measured. The distance sailed by a steamer in a given time is most conveniently determined from the Ch. VI] NAVIGATION: DEAD RECKONING 61 number of revolutions of the screw propeller. It may also be determined by the patent log; this instrument carries at the end of a tow line a rotator which if drawn through the water turns about its axis, causing also the tow line to turn, the motion being transmitted to a register aboard ship. 44. Ship's course. The true course of a ship is the angle which the ship's track in the water makes with the meridian (true north and south line) through the ship's position. It may be measured clockwise from north and have any value from 0~ to 360~. It is so measured on the new1 compass card used in the United States Navy, the circular card being numbered clockwise from 0~ at north completely around the circumference to 360~, which is again at north. For example, south (or S) is 180~, while west (or W) is 270~. Or the course may be measured from north or south toward east or west to give an angle not exceeding 90~. For example, if a ship proceeds from 0 in a direction OA between south and east and making an angle of 75~ with the meridian OS (Fig. 32), its course is south 75~ B east or S 75~ E. Similarly, if the ship's \ track OB is between north and west and makes an angle of 35~ with the meridian E0 ON, its course is N 35~ W. These two courses when read on the new compass card are 105~ and 325~, respectively, FI 32 being the angle NOA = 180~-75~ and the re-entrant angle NOB = 360~ - 35~, each read clockwise from N. 10n the old compass card, courses are measured in points from N or S toward E or W to give a value from 0 to 8 points = 90~, one point being 11~. Each of the 32 points and the intermediate half and quarter points has a name. For example, NE E is 4t points from N toward E and thus is N 47048'45" E in the second system of the text. No new principle is involved in solving problems expressed in this older, more cumbersome notation, which is avoided here in the interests of simplicity, and since a special traverse table is desirable when working with courses expressed in points. 62 TRIGONOMETRY [Art. 45 We shall postpone to Part II consideration of the various corrections to the reading of the ship's compass, and assume for the present that we know the true course. PART I. THE SAILINGS (TRUE COURSE ASSUMED) 45. Plane Sailing. Let the distance D sailed from A to B be so short that the curvature of the earth may be neglected. Let the meridian AF through A and the parallel of latiF -- — B tude BF through B intersect at F (Fig. 33). Then AF is the difference of latitude of A and B, D A while FB is called the departure. Let angle C be the true course. Then Cgy ~ diff. lat. = D cos C, dep. = D sin C. A fEXAMPLE. From latitude 36~ N, a ship sails 243 miles FIG. 33 S 56~ W. Find lat. in (i.e., the latitude of the place arrived at) and the departure. Solution. Divide the distance by 3 to obtain a distance, 81, less than 100. Multiply by 3 the entries in Traverse Table VI which are opposite to distance 81 and above the angle 56~, using the headings at the bottom of the page and retaining only one decimal place. We get dep. = 201.4, diff. lat. = 135.9. Dividing the latter by 60, to convert miles into degrees, we have diff. lat. 2~16'. Since the ship sailed in a southerly direction, we subtract this from 36~ N and get lat. in = 33~44' N. In place of the first step in the solution, we may add the entries for the distances 200 and 43 (as in Ex. 1, Art. 17). EXERCISES ON PLANE SAILING Find the quantities indicated by question marks, using Table VI for Exs. 1-5, 7, and logarithms for Exs. 6, 8. Ex. True Course Dist. Dep. Lat. out Lat. in 1. 5 188? 40~ 33' N? 2. 124~ 488? 1~ 45' N? 3. S 17 W?? 400 17' N 37~ 6' N 4.? 360? 21 59' S 24~ 49' S 5. 236? 48.2 38 N? 6.? 289.2 W 20~ 48' N 17~ 13' N Ch. VI] NAVIGATION: DEAD RECKONING 63 7. A ship sails a course of 309~ for 74 miles from Michigan City. How far north and how far west is it from its port of departure? 8. Ludington is 59 miles east and 142 miles north of Chicago. Find a ship's course and distance from Chicago to Ludington. 46. Unfavorable case. When the departure p and difference of latitude I are given and the course C and distance D are desired, Traverse Table VI is not as convenient as in all other such problems. While logarithms could be used, it is customary in the practice of navigation to employ the traverse table, as in the following solutions. EXAMPLE 1. Given p = 82.3, 1 = 34.3, find C and D. Solution. Since p is a little more than the double of 1, angle C is a little more than 60~. After inspecting Table VI for several such angles, we extract the numbers in the first and third lines below: C dis. D dep. p lat. I 67~ 88 81.0 34.38 67~18' 89.2 82.3 34.3 68~ 92 85.3 34.46 We ignore the variation of the given I from the entries 1, since an error less than 0.2 of a mile is negligible for the purposes of Plane Sailing. In view of the values of p, the interpolation ratio is 13/43, so that the corrections to C and D are 13 13 43 X 60' = 18', 1X 4 = 1.2. The method used in Ex. 1 applies for all distances < 100 and all courses between 60~ and 90~, an error of 0.2 in I being allowed. For courses between 0~ and 30~, we employ entries from the table having the given p, to within 0.2, and interpolate for C, D, 1. But for distances <100 and courses between 30~ and 60~ we may commit an error of 0.3 even though we use the better of the two methods (the first if the course is >45~, and the second if <45~). When greater accuracy is required, proceed as in Ex. 2. EXAMPLE 2.1 Given p = 63.1, 1 = 68.7, find C and D. 'The instructor may elect to omit this example. 64 TRIGONOMETRY [Art. 47 Solution. Evidently C is only a few degrees less than 45~. From the traverse table we read off the first and third lines of the following tablettes: C = 42~ C = 43~ p 1 D 62.90 69.86 94 63.10 70.08 94.30 63.57 70.60 95 p I D 62.74 67.28 92 63.10 67.67 92.52 63.43 68.02 93 In the middle line of each tablette, we insert the given value of p, and (starting with p instead of I since C <45~) interpolate the values of I and D in the usual manner. We now have the first and third lines of the new tablette: C I D 42~ 70.08 94.30 42034' 68.70 93.28 43~ 67.67 92.52 Insert in the middle line the given value of I and interpolate the values of C and D. We have carried the work to two decimal places (although one is enough for navigation) to show the accuracy of the method. In fact, computation by five-place logarithms gives C = 42~34'0", D = 93.282. EXERCISES ON INTERPOLATION Given the departure p and difference of latitude I, find the course C and distance D by Table VI, allowing errors of 0.2. 1. p = 14, 1 = 85. 2. p = 25, 1 = 90.;i,\' B3. p = 90, 1 = 25. s S 4. p = 85, 1 = 14. E,;\ 5. p = 14.7, 1= 65.7. IT I \66. p = 64.8, 1= 53. \i ~ C '\ t D 47. Traverse Sailing. Let a ship sail a \ '/, short distance AB on one course and then a \' second short distance BC on a new course, r< etc. (Fig. 34), so that the method of Plane FIG. 34 Sailing is applicable to each of the sailings. Ch. VI] NAVIGATION: DEAD RECKONING 65 In four columns we enter the north and south differences of latitude and the east and west departures. By footing up the columns and subtracting the totals, we get the difference of latitude and departure of the final point F reached with respect to the point A of starting, and hence have the combined effect of the successive sailings. We thus speak of the course and distance made good, just as if the ship had sailed in a straight line AF from its initial point A to its final point F. The letters S and E of the course S 22~ E tell that the difference of latitude is south and the departure is east. EXAMPLE. A ship sails the five successive courses and distances in the following table. Find the course and distance made good. Solution. The courses and distances are shown to scale in Fig. 34. Courses Dist. Diff. lat. Departure N S E W S 220 E 15 13.9 5.6 S 45~ E 34 24.0 24.0 S 79~ W 16 3.1 15.7 N 68~ W 39 14.6 36.2 S 110 E 40 39.3 7.6 14.6 80.3 37.2 51.9 14.6 37.2 S 12~ 36' W 67.3 65..7 Hence the effect of the successive sailings is to carry the ship 67.3 miles S 12036' W (from A to F in Fig. 34). EXERCISES ON TRAVERSE SAILING Given courses C and distances D, find the course and distance made good: Ex. 1 C D 80~ 25 130~ 38 210~ 50 Ex. 2 C D 172~ 18 219~ 37 205~ 56 Ex. 3 C D S 25~ W 43 S 28~ W 39 S 17~ W 27 Ex. 4 C D N 70~ W 21 N 31 E 9 N 20~ E 9 S 25~ W 30 66 TRIGONOMETRY [Art. 48 5. A ship in lat. 30~20' S sails 54 miles S 39~ E, 90 miles S 56~ W, 96 miles S 34~ E, 64 miles E, and 36 miles S 28~ E. Find the latitude in and course and distance back to the starting point. 6. A ship in lat. 34040' N sails 124 miles on course 307~, 32 miles on course 349~, 80 miles on course 273~, 58 miles on course 307~, 60 miles on course 11~, and 28 miles on course 8~. Find the latitude in and the course and distance back to the starting point. 7. The distance PA across an island is 44 miles and A is due north of P. A ship sailed N 25~ W from P to a point B and then N 65~ E to A. The return trip was made on the opposite side of the island, sailing S 29~ E from A to a point F and then S 61~W to P. Find the shorter of the distances sailed going and returning. 48. Parallel Sailing. This term is used when a ship sails on a parallel of latitude. While this is a very special case, the method used solves the important question to find the difference of longip tude GH of two places A and B on the same parallel of latitude, given the departure 0 _ _ _,._\ -\ \ AB (Fig. 35). Let P be the adjacent pole, C the center of the earth, and draw AO and C \ / \ BO perpendicular to PC. - - - - - - - Then 0 is the center of the " '. small circle which gives the \ I G H / parallel of latitude through A I / / and B. Since BO and HC \/ / / are in the plane of the meridian PBH and are perpendicular to PC, they are parallel FIG. 35 lines. Similarly, AO and GC are parallel. Hence the angles AOB and GCH are equal, so that they subtend arcs AB and GH proportional to the radii AO and GC of their circles. Also GC and AC are equal, being radii of the earth. Denote by L the latitude AG of A; it subtends angle ACG, which equals the alternate angle CAO. Hence in Ch. VI] NAVIGATION: DEAD RECKONING 67 triangle CAO, angle 0 is a right angle, and angle CAO equals L. Thus AO AO AB dep. Cos CAOs or cos L= AC CAO= GC GH diff. long. (1) dep. = (diff. long.) cos L. EXAMPLE 1. A ship in latitude 49~ N and longitude 100~ W sails due west to a place in longitude 101035' W. Find the distance sailed. Solution. The difference of longitude is 1~35' or 95 nautical miles. The distance sailed is here the departure and, by (1), equals 95 cos 49~. By Table VI, this product equals 62.33. EXAMPLE 2. A ship in latitude 38~ N sails 71.8 miles due west. Required the difference of longitude D. Solution. By (1), 71.8 =D cos 38~. By Table VI, 71.7 = 91 cos 38~, 72.5 = 92 cos 38~. Hence the interpolation ratio is.1/.8. Adding 1/8 of 1 to 91, we get D = 91.1 miles or 1~31.1'. EXERCISES ON PARALLEL SAILING Find the quantities indicated by question marks. Ex. Lat. Dep. Long. out Long. in 1. 52~? 0~59' W 2~24' E 2. 61025'? 179~20' W 176~52' E 3. 60~ 204 E 160~ 2'E? 4. 51~28' 70.9 E 32~ 7' W? 5. 34057' 981 E? 53~20' E 6. _? 156 W 25040'W 30054 W 7. A ship in lat. 40~ N, long. 165~ W, sails due east until her longitude is 155030' W. Find the distance sailed. 8. How far must a ship sail due east in lat. 60~ N to change her longitude by 5~? 9. From lat. 30~ N, long. 45~ W, a ship sails due west 240 miles, then due north 240 miles, and finally due east 240 miles. Find lat. in and long. in. 10. Two ships in lat. 35~ N, 150.7 miles apart, sail 300 miles due north. How much closer are they at the end of the run? 68 TRIGONOMETRY [Art. 49 11. Two ships A and B steam due west at the same speed, while B changes longitude twice as fast as A. If A is in lat. 20~ N and to the southward of B, find B's latitude. 12. In what latitude is the length of a degree of longitude 47 miles? 49. Middle Latitude Sailing. We now abandon the assumption made in Plane Sailing and take into account the curvature of the earth. Let the ship's track be a rhumb line (AB in Fig. 36), p. making the same angle C with all the meridians crossed. By thus keeping the ship on a constant true course, the navigating officer not only simplifies his F \ B computations but also is spared lo/ / /d \ the trouble of sending new orders as to the course to the helms4/3 vj|~7 \ man. The distance between two P/^ / 3 / places A and B is the number of nautical miles in the rhumb,, _ _i X^ line joining them. In Fig. 36, P represents the AV ' / north pole, AB the ship's rhumb track, AG and FB the correFIG. 36 sponding parallels of latitude, each above the equator, and FAB the constant course C. Divide the distance D = AB into parts dl,...,d4, each so short that the resulting right triangles having them as hypotenuses may be regarded as plane triangles. Let pl,...,,4 be the corresponding intercepts on the parallels of latitude, and l1,....14 the corresponding intercepts on the meridians and hence the differences of latitude. Apply to each small triangle the formulas derived in Plane Sailing (Art. 45). First, we get 1 = di cos C, 12 = d2 cos C, 13 = d3 cos C, 14 = d4 cos C. Adding, we have Ch. VI] NAVIGATION: DEAD RECKONING 69 AF= (di + d2 + d3 + d4) cos C = D cos C, diff. lat. = D cos C. Second, pi = di sin C, P2 = d2 sin C, p3 = d3 sin C, p4 = d4 sin C. Hence, on adding, we get P1 + p2 - P3 + p4 = D sin C. Since the meridians converge toward the pole P, we see that pi is less than the corresponding intercept on the parallel AG and greater than that on FB. Hence pi + P2 + PS + p4 is less than AG and greater than FB and thus equals the intercept UV made by the meridians PA and PBG on a certain intermediate parallel of latitude. It is assumed that the latter is the parallel half way between FB and AG. Then UV is called the departure in middle latitude, i.e., the east and west arc intercepted by the meridians through A and B on the parallel whose latitude is midway between the latitudes of A and B. Hence our second result becomes dep. in middle lat. = D sin C. For purposes of computation we may therefore replace our spherical triangle AFB by the plane right triangle AFB (Fig. 33) of Plane Sailing, provided departure in Plane Sailing be replaced by departure in middle latitude. The last replacement is to be made also in applying formula (1) of Parallel Sailing to compute the difference of longitude. The assumption made concerning UV will not introduce inadmissible errors1 if the distance sailed is not greater than a day's run, and the ship is not in a large north or south latitude. EXAMPLE 1. A ship in lat. 42~30' N, long. 58~51' W, sails 300 miles S 34~ E. Required the latitude and longitude arrived at, using Middle Latitude Sailing. Solution. By Table VI with course S 34~E and distance 300, we find diff. lat. = 248.7 miles or 408.7' S and dep. in mid. lat. = 167.8 E. lat. out = 42~30' N lat. out = 42~30' diff. lat. = 4~ 8.7'S ~ diff. lat. = 2~ 4.4' lat. in = 38~21.3' N mid. lat. = 40~25.6' 1 Corrections to be used when the distance is large are given in a table in Bowditch's American Practical Navigator, p. 77. 70 TRIGONOMETRY [Art. 49 We now apply formula (1) of Parallel Sailing to find diff. long., given dep. = 167.8 and L = 40~25.6'. Dividing by 3 to get numbers directly within Table VI, we have dep. = 55.9. From 55.9 = d cos 40~, Table VI gives diff. long. d = 73. From 55.9 = d cos 41~, d = 74.1. Adding 25.6/60 X 1.1 = 0.5 to the former, we get 73.5. Its product by 3 gives diff. long. = 220.5 miles, or 3~ 40.5' E. This must be subtracted from the given longitude. Hence long. in = 55011' W. EXAMPLE 2. A ship in lat. 49~57' N, long. 15~16' W, is bound for a port in lat. 47018' N, long. 20010' W. Find the course and distance to be sailed. Solution. diff. lat. = 2039' S = 159 miles, diff. long. = 4~54' W = 294 miles, mid. lat. = 49057' - (2039') = 48~37.5' N = L, dep. = 294 cos L. Taking L to be 48~ and 49~ in turn, we obtain by Table VI the departures 196.7 and 192.9 respectively. By interpolation, dep. = 194.3. Dividing the numbers by 3, we have diff. lat. = 53, dep. = 64.8. The course is thus > 45~. By mental interpolations on lat. in Table VI, we get C lat. dep. dist. 50~ 53 63.2 82.5 510 53 65.5 84.2 Hence, by interpolation-on dep., C = S 50042'W, dist. = 83.7 X 3 = 251.1 EXERCISES ON MIDDLE LATITUDE SAILING Find the quantities indicated by question marks. Ex. Lat. out Lat. in Long. out Long. in True course Dist. 1. 25035' N 27028' N 60~ W 54055' W?? 2. 32030' N 34010' N 25024' W 29~ 8'W?? 3. 46024' S? 178028' E? S 53026' E 278 4. 20029' N? 179010' W? 2530 333 5. 41038' N 41026' N 59016' W? 101015? 6. 4r19' N 41011' N 57047' W?? 167 7. 46028' N 45~17' N 22018' W 19039'W?? 8. 36052'N? 75051' W? N66 E 175 9. 36052' N 38042.2' N 75051' W 71051.6' W?? Ch. VI] NAVIGATION: DEAD RECKONING 71 50. The Mercator chart. A clear understanding of the leading method of making a map of the earth's surface is of great importance not only in navigation but also in geography and other earth sciences. The student will not fail to appreciate the practical nature of this topic. The earth's surface is mapped on the interior of a rectangle in such a way that the meridians are represented by parallel straight lines perpendicular to the straight line representing the equator, while the parallels of latitude are represented by straight lines parallel to the line representing the equator. Since the earth's meridians converge at the poles and yet have been plotted as parallel lines, there has been an opening out of these meridians, i.e., a stretching of east and west lengths. But we desire that any small figure on the map shall be of the same shape as the corresponding figure on the earth. Hence there must be simultaneously a stretching of north and south lengths. For a very short such vertical length in latitude L, the stretching factor is sec L, when the earth is regarded as a sphere. For, by Art. 48, diff. long. = dep. X see L. On a Mercator chart we agreed that the east and west length called departure should be stretched until it becomes equal to the corresponding diff. long. mapped unstretched on the line representing the equator. Hence dep. has been stretched in the ratio sec L, and we agreed to use the same stretching factor for small vertical lines. This argument is valid only for a very short arc, say one minute of arc. Given a longer arc extending from the equator vertically to lat. 5~ N, we divide it into 300 arcs each equal to a minute and hence obtain the stretched arc containing sec 1' + sec 2' +... + sec 300' minutes. It is too laborious to compute such sums without the aid of integral calculus, which leads to a formula convenient for computation.1 1 For latitude L the number of nautical miles in the stretched latitude is log tan (45~+ 'L) - r(e2 sin L+ Ie4 sin' L +...).4343 where r is the equatorial radius and e is the eccentricity of the ellipse whose rotation produces the earth's surface. 72 TRIGONOMETRY [Art. 50 The resulting stretched latitudes are called meridional parts and are given by Table IX. For example, the meridional parts corresponding to latitude 45~ (or 2700') are 3013.4. But the meridional parts for lat. 1~ are 59.6, showing a shrinking and not a stretching, which is explained by the fact that the earth is not a sphere. By use of Table IX we readily construct a Mercator chart to scale, for example for the region extending from 40~ to 55~ north 75 a 65 60 latitudes and from 60~ to 5.... 55 75~ west longitudes (Fig. 37). On a horizontal line representing the parallel of latitude 40~ take any B convenient segment to represent 5~ difference of 50. — — / ---. 50 longitude, and at one of its ends construct an equal A.- -/ M r segment, and another one at the remote end of the latter. Mark the equally ____ __45 / ^45 spaced points with the A A' labels 60, 65, 70, 75. Then the vertical lines through these points represent the meridians whose longitudes are 60~, 65~, 70~, 75~ W. 4 - -1I 40 ' 5 5.75 70 65 60 It remains to locate the FIG. 37 horizontal lines which represent the parallels of latitudes 45~, 50~, 55~. This is done by means of Table IX as follows: Meridional parts for lat. 40~ = 2607.6 Meridional parts for lat. 45~ = 3013.4 Meridional diff. of lat. = 405.8 Ch. VI] NAVIGATION: DEAD RECKONING 73 Divide the last number by the number 300 of minutes in 5~, The quotient 1.35 is the number of units of length in the distance of parallel 45~ above parallel 40~. Similarly, the meridional parts for lat. 50~ are 3456.5, and (443.1)/300 = 1.48 is the distance of parallel 50~ above parallel 45~. Likewise, parallel 55~ is 1.64 units above parallel 50~. By employing similarly the meridional parts for each intermediate degree of latitude 41~, 42~,..., we locate the points of division shown on the scale at the right of Fig. 37. For greater distinctness, alternate intervals of one degree are marked by double lines. 51. Angle and distance on a Mercator chart. Any angle on the earth's surface is represented by an equal angle on a Mercator chart. Since the rhumb line on which a ship sails makes the same angle C with all the meridians crossed, it is mapped as a straight line. For this reason Mercator charts are of special importance in navigation. To find from the chart the length of the rhumb line from place A, in latitude 45~ N and longitude 70~ W, to place B in latitude 51~ N and longitude 65~ W, plot A and B on the chart, as in Fig. 37; the straight line AB represents the rhumb line on the earth. Project the middle point M of AB on to the vertical scale at the right of the chart, and measure up and down from the projection M' a distance equal to one-half of AB. The number 6.9 of degrees between the extreme points A' and B', when reduced to miles by multiplication by 60, is the approximate length 414 of the rhumb line. 52. Mercator's Sailing. This method has the advantage that the computations can be conveniently checked graphically on the Mercator chart which shows the ship's position at all times and hence its relation to possible danger places. Further, it involves no assumption restricting its accuracy, such as was made in Middle Latitude Sailing. The latter method is not sufficiently accu 74 TRIGONOMETRY [Art. 52 rate when the distance sailed is more than 500 miles, especially when the course is less than 45~ or the ship reaches a high latitude. In these cases Mercator's Sailing should be used. In Mercator's Sailing we use two plane right triangles each having an angle equal to the ship's course C. One triangle (Fig. 38) is drawn on a Mercator chart and has as legs the difference of difft long. C FIG. 39 FIG. 38 longitude, and the meridional difference of latitude m corresponding to the actual latitudes of the point sailed from and the point reached. The hypotenuse is not used. The other triangle (Fig. 39) has as hypotenuse the distance D sailed and as vertical leg the difference of actual latitudes; its horizontal leg is not used. This plane triangle is the one which we saw (Art. 49) can be substituted for the spherical triangle on the earth's surface. For computation by logarithms, we use the formulas diff. long. = m tan C, diff. lat. = D cos C, which follow from Fig. 38 and Fig. 39 respectively. To use Table VI, we have only to note that, in Fig. 38, m is the leg adjacent to C and diff. long. is the leg opposite. EXAMPLE 1. A ship in lat. 42~30'N and long. 58~51'W sails 300 miles S 34~ E. Find the latitude and longitude arrived at, using Mercator's Sailing. Solution. By Table VI, diff. lat. = 248.7 miles = 4~8.7' S. lat. out = 42030' N, whose merid. parts = 2806.4 diff. lat.= 4~ 8.7' S lat. in = 38021.3' N, whose merid. parts = 2480.8 merid. diff. lat. m = 325.6. Ch. VI] NAVIGATION: DEAD RECKONING 75 Since m exceeds the entries in Table VI, divide it by 4. Then by Table VI with course S 34~ E, 4 m = 81.4 in adj. leg column, read off 4 diff. long. 54.9 in opp. leg column. Hence diff. long. = 219.6 miles or 3~40' E. Thus long. in = 55011' W. Cf. Ex. 1, Art. 49. EXAMPLE 2. A ship sails on a rhumb line from lat. 4203' N, long. 70~4' W to lat. 36059' N, long. 25~10' W. Find the course and distance sailed. lat. out = 42~ 3' N merid. parts = 2770.1 long. out = 70~ 4' W lat. in = 36059' N merid. parts = 2377.3 long. in = 25010' W diff. lat. = 5~ 4' S m = 392.8 diff. long. = 44~54' E = 304'S = 2694' E Since the numbers are so large, solution by logarithms is preferable: tan C = - diff. long. D = diff. lat. / cos C m log 2694 = 3.4304 log 304 = 2.4829 log 392.8 = 2.5942 log cos C = 9.1592 - 10 log tan 81042.2' = 10.8362 - 10 log D = 3.3237 C = S 81042.2' E dist. = 2107. EXERCISES ON MERCATOR'S SAILING 1. Construct a Mercator chart of the region from 30~ to 50~ north latitudes and from 20~ to 40~ east longitudes. If a ship sails from lat. 30~ N, long. 25~ E, to lat. 40~ N, long. 35~ E, find from the chart the course and distance. In the following exercises, find the quantities indicated by question marks. Ex. Lat. out Long. out Lat. in Long. in True course Dist. 2. 40~ N 70~ W 20 S 15~ W?? 3. 30022' N 68015' W 43~17' N 15~18' W?? 4. 10~15' S 170~10' E 5~10' N 81012' W? 5. 45015' N 35026' W?? 49~ 175 6. 50~48' N 9~10' W?? S 41~ W 275 7. 37~ N 48020' W 51~18' N?? 1027 8. 51015' N 9050' W 3705' N? 214~? 9. A ship sails from lat. 34022' S, long. 18024' E, to lat. 52021' S, long. 59018' W. Find the true course and distance by Middle Latitude and Mercator's Sailings. Which method is more accurate? 76 TRIGONOMETRY [Art. 53 10. A ship sails 1022 miles 214~ from latitude 51026/ N, longitude 9~29' W. Find lat. in and long. in by Middle Latitude and Mercator's Sailings. Which method is more accurate? PART II. FINDING THE TRUE COURSE; COMPASS CORRECTIONS 53. The mariner's compass. The compass consists of the compass card (Art. 44), one or more magnetic needles firmly attached to the lower surface of the card in a position parallel to the north and south line, and a bowl with a pivot at its center upon which the card turns - the whole being enclosed in the compass box. The angle between the ship's fore-and-aft line and the magnetic needle equals the arc between the north or south point on the circumference of the card and the point of the latter which is adjacent to a certain mark on the inner surface of the bowl (such that this mark and the pivot of the card are in the same plane with the ship's keel line). 54. Variation and deviation of the compass. The variation of the compass at a place is the angle through which the needle is deflected from true north by terrestrial magnetism alone (for details, see Art. 71). Thus, in a ship without any magnetic substance or motors, the needle lies in the magnetic meridian, whose angle with the true meridian is the variation of the compass. The true meridian of a place is the great circle passing through it and the earth's poles. As a modern ship contains considerable iron or other magnetic metal, and carries motors, there is an additional source of error in the ship's compass. The angle which the direction of the needle makes with the magnetic meridian is called the deviation of the compass. When the effect of the deviation is to draw the north end of the needle eastward (i.e., to the right) of the magnetic meridian, the deviation is marked E or +; in the contrary case, W or -. The marking of variation is similar. 55. Leeway. The angle which the ship's heading (fore-and-aft line) makes with her track through the water (as indicated by the Ch. VI] NAVIGATION: DEAD RECKONING 77 ship's wake) is called the leeway. It is estimated from various factors including the direction and velocity of the wind. Leeway is marked E or + when the wind turns the ship's head to the right or clockwise, but W or - when the wind turns the ship to the left or counter-clockwise. 56. Courses. The compass course of a ship is the angle which the direction of the ship's head makes with the needle as it actually points (as deflected from true north by both variation and deviation), and is measured clockwise from compass north to the direction of the ship's head. The true course of a ship is the angle which the ship's track (not its head) makes with the true meridian, and is measured clockwise from true north to the ship's track. To find the total correction which must be added algebraically (i.e., with attention to sign) to the compass course to obtain the true course, we have only to combine algebraically the separate corrections for variation, deviation, and leeway. This rule may be proved as in the following Ex. 1. EXAMPLE 1. The compass course is 45~, variation 8~ E, deviation 5~ E, and leeway 11~ W (due to a wind blowing from southeast). Find the true course. Solution. From the observer's position 0 on ship (Fig. 40), let ON represent true north, OM magnetic C north (direction of needle deflected by variation only), OC compass north (direction of needle deflected by both variation and deviation), OH the direction of the ship's head, and OT the direction of the ship's track through the water. Here the compass course is Z COH = 45~, and the leeway is Z TOH = 11~, whence 0 the track's course COT by compass is 34~. Adding the FIG. 40 deviation Z MOC=5~ and the variation ZNOM =8~ to COT, we get the true course NOT =47~. Or we may find the total correction +8~+5'-110 = +2~, add it to the compass course 45~, and obtain the true course' 47~. 78 TRIGONOMETRY [Art. 56 EXAMPLE 2. The true course is 124~, variation 10~ W, deviation 4~ E, and leeway 17~ W. Find the compass course. Solution. The total correction is -10~ + 4~ - 17~ = -23~. This must be subtracted from the true course to obtain the compass course 147~. EXERCISES ON COMPASS CORRECTIONS In each exercise find the unknown course. Ex. Compass Variation Deviation Leeway True course course 1. 11~ 10~ W 50 E 6~ E? 2. S 67~ E 5~ E 3~W 11~ W? 3. 28~ 21~ E 6~ E 11~W? 4.? 150 W 6~ E 6~E 20~ 5.? 10~ E 18~30'W 8~ E 259~ 6.? 15~ E 18~ E 6~ E S 85~ E 7. At a certain place the rate of the Gulf Stream is 4 miles per hour. If a ship heads directly across it at 20 miles per hour, what is the leeway due to the current? MISCELLANEOUS EXERCISES ON THE SAILINGS 1. From latitude 36~ N, a ship sails 81 miles on compass course 230~, the variation being 15~ W, deviation 5~ E, and leeway 11~ W. Find the lat. in and the departure. 2. A ship sails 20~ west in lat. 60~ N. The variation at the initial point I is 26~ W, and deviation 7040' W. Find the distance sailed and the compass course at I. 3. From lat. 49~ 28.5' N, long. 13030' E, a ship sailed 36 miles on compass course 312~, the variation being - 15~, and deviation- 7~. Find lat. in and long. in by Middle Latitude Sailing. 4. From lat. 15~15' N, long. 45~ W, a ship sailed 110.8 miles S 44~E E by compass, the variation being 13~ W, and deviation 6~35' E. Find lat. in and long. in by Middle Latitude Sailing. 5. Find by Mercator's Sailing the compass course and distance from Cape East, New Zealand (lat. 37050' S, long. 178~36' E), to San Francisco Ch. VI] NAVIGATION: DEAD RECKONING 79 (lat. 37048' N, long. 122024' W), the variation being 14020' E, and deviation 5040' E. 57. Dead reckoning. Given several successive short sailings each with a constant course, we find the true courses, then compute the net difference of latitude and net departure as in Traverse Sailing, and finally find the course and distance made good and the position at sea usually by Middle Latitude Sailing (not by Plane Sailing as in the final step in Traverse Sailing), but by Mercator's Sailing if still greater accuracy is needed. EXAMPLE. A ship sails from a point near Cape Henry lighthouse (lat. 36~ 55.6' N, long. 76~ 0.5'W) on the following series of courses, with variations and deviations of compass and leeway shown. Find the course and distance made good and the position by dead reckoning. Compass Varia- Devia- Total True Dis- W I N S E w course tion tion eewy error course tance 28~ 6~W 30E 0 30 25~ 1.4 1.3 0.6 530 50 W 30 E 3 W - 5~ 48~ 27.6 18.5 20.5 191~ 6~ W 0 3~ E -3~ 188~ 31.5 31.2 4.4 67~ 6~W 3 E 5 W — 8~ 59~ 14.2 7.3 12.2 1770 6~ W 0 6~ E 0 177~ 11.0 11.0 0.6 42~ 8~W 3 E 3 W — 8~ 340 87.0 72.1 48.7 99.2 42.2 82.6 4.4 Made good 53054' 96.82 82.6 4.4 57.0 78.2 Diff. lat. 57.0' N Dep. 78.2 = D cos L, Lat. out = 36055.6' N D = diff. long. = 98 for L = 37~ Lat. in = 37052.6' N 99.2 for L = 38~ Mid. lat. L = 37024.1' 98.5 for L = 37~24, Long. in = 7600.5' - 138.5' = 74022' W. EXERCISES ON DEAD RECKONING 1. From lat. 48~20.5'N, long. 10~10' W, a ship sailed 22.2 miles on compass course 257~, variation - 20~, deviation - 3~; thence 216.5 miles on compass course 232~, variation - 20~, deviation - 1~. Find lat. in and long. in. 2. From lat. 49040.51 N, long. 15~ W, a ship sailed the courses 80 TRIGONOMETRY [Art. 57 Compass C. N 76~ W N 70~E N 341~ E S67 WV Find lat. in and long. in. 3. At noon a ship was in lat. 49015' N, long. 24o15' W. Time Comp. C, Dev. Var. Speed noon N 69~ W 4020' W 16040' W 20 3 P.M. N 68~ W 4020' W 17040' W 20 6 P.M. N 23~ E 4020' W 18~40' V 21 1 A.M. S 66~ E 5020' E 18~40' W 20 3:30 A.M. S 67~ E 5020' E 17040' V 20 The speed is in knots, a knot being 1 nautical mile per hour. Find the ship's position at 7:00 A.M. 4. At 8:00 P.M. a ship was in lat. 5008' N, long. 16~ W, and was steaming 35~ at 20 knots; at midnight, true course 300~; at 5:00 A.M., true course 204~; at 8:00 A.M., true course 132~, new speed 22 knots. Find the position at noon. 5. At noon, a ship in lat. 48023.5' N, long. 10032.8' W, was steaming 354040' at 25 knots, deviation 5~10' W. At 7:00 P.M. it altered its (compass) course to 96020', deviation 3040' E. Constant variation 17~ W. Find its position at midnight. 6. Took departure, Cape Henry lighthouse bearing 293~ (see Art. 61) by ship's compass, distant 10 miles, deviation + 3~, variation - 6~. Thence ran as follows: Compass Compass Dist. Dev. Var. Leeway courses 73~ 60 +3~ -6~ +3~ 118~ 20 +6 -6~ +3~ 160~ 10 +3~ -6~ +3~ 319~ 10 -6~ -6 -3~ 26~ 28 +3~ -6~0 3~ Ch. VI] NAVIGATION: DEAD RECKONING 81 Find the course and distance made good from the lighthouse, and the position by dead reckoning. Hint: Since the true bearing of the lighthouse is 290~, the true departure course is 110~ and distance 10. 7. At noon a ship was in lat. 48o22' N, long. 70050' E, steaming S 63~ E by compass, speed 20 knots, deviation 5030' E. At midnight, the course was N 81~ E, speed 20 knots, deviation 2~ E. At 6:00 A.M., course N 6~ E, speed 21 knots, deviation 5~ W. Constant variation 1~ W. A current was setting E (by magnetic compass) with velocity 1 knot from noon until 6:00 A.M. Find the position at noon. (Treat current as an additional course.) CHAPTER VII LAND SURVEYING 58. Branch of surveying treated. Land surveying treats of the determination of the lengths and directions of the boundary lines and the area of a tract of land, as well as the accurate representation of the boundary lines on a map. In plane surveying, which alone is considered here, the curvature of the earth is ignored and the part surveyed is regarded as a plane. No discussion will be given here of the following further branches of surveying: geodetic surveying, in which the curvature of the earth is taken into account; leveling, in which the relative elevations of points are determined; topographic surveying, which combines the methods of leveling and horizontal location, and treats of the construction of contour maps; mine, marine, and railroad surveying. The object of the present chapter is merely to define and illustrate the terms used in the elements of land surveying, to describe the instruments employed, without discussing their adjustments, and to treat in detail the application of plane trigonometry to the balancing of land surveys and the computing of areas. 59. Chains, tapes, area. A Gunter's chain is 66 feet long and is composed of 100 links connected by small rings. Since there are 5280 feet in a (statute) mile, there are 80 chains in a mile. Since a square mile contains 640 acres, an acre is equal to 10 square chains. Hence an acre contains 10 X 662 = 43560 square feet. Instead of a Gunter's chain, it is now customary to use the more accurate steel tapes. When the number of acres in a field is desired, it is convenient to use a steel tape 66 (or 33) feet in length, graduated to feet and tenths of a foot on one side and to links on the reverse side. In city surveying and often in farm 82 Ch. VII] LAND SURVEYING 83 surveying, use is made of steel tapes 50 or 100 feet in length, graduated to feet and tenths of a foot and often also to hundredths of a foot. Since the distances measured are assumed to be horizontal (cf. Art. 60), it is necessary in the case of sloping ground to elevate one end of the chain or tape. By the area of a tract of land is meant the area of its projection on a horizontal plane. Hence if a piece of hilly ground were completely leveled, its area would remain the same. 60. Course. By a course IT is meant any one of the horizontal straight lines AB extending from the vertical line through the initial point I to the vertical line through the terminal point T. The vertical plane through i — I and T intersects the ground in an irregular A B curve ICT (Fig. 41). The course from I to T FIG. 41 is not this curve, but is a horizontal straight line AB which is a horizontal projection of the curve. The distance between I and T, in the sense used in surveying, is AB. In brief, we do not employ the actual lines, distances, and areas on the ground, but rather their horizontal projections. 61. Trite bearing. A true north and south line is called a true meridian. The true bearing of a course IT is the angle which it makes with the true meridian through the B initial point I of the course, and is measured from the north or south point toward \ A the east or west point to give an angle not exceeding 90~. For example, if (Fig. 42) we i —W E start from the initial point I and measure C 70~ D in the direction IA proceeding between the north and east points and making an S angle of 55~ with the meridian IN, the FIG. 42 bearing of IA is N 55~ E and is read "north 55~ east." For the course IB, which proceeds between the north and west points 84 TRIGONOMETRY [Art. 62 and makes an angle of 20~ with the meridian, the bearing is N 20~ W. Similarly, the bearings of the courses IC and ID are S 70~ W and S 75~ E respectively. We postpone to Part II the description of the surveyor's compass and transit and the measurement of certain angles by them from which the true bearings of courses are easily found. PART I. BALANCING A SURVEY, AREA (TRUE BEARINGS ASSUMED) 62. Latitude and departure. Given a course IT of length D and true bearing B, as N 55~ E in Fig. 43 or S 55~ W in Fig. 44, draw the perpendicular TF F dep. T to the true meridian through D I. Then IF is called the Us/D /.latitude and FT the departure BT/ - F of the course IT. We have dep. lat. = D cos B, FIG. 43 FIG. 44 dep. = D sin B. According as T is north (Fig. 43) or south (Fig. 44) of I, the latitude of the course IT is called north or south. According as T is east (Fig. 43) or west (Fig. 44) of I, the departure of the course IT is called east or west. The letters N and E of a bearing N 55~ E tell that the latitude is north and the departure is east. Given two of the four quantities B, D, lat., dep., we can find the remaining two by Traverse Table VI with sufficient accuracy for ordinary farm surveying. But for city surveying the work should be done by logarithms and checked against gross errors by the traverse table, perhaps omitting interpolations. 63. Balancing a survey. All measurements involve errors. Hence before attempting to find the area of a tract of land, we must adjust the measurements recorded in the field notes of a survey by making such corrections as will properly distribute Ch. VII] LAND SURVEYING 85 the errors of measurement. This adjustment of the errors is known as balancing the survey. Consider the following survey of a triangular field (Fig. 45), in which the data in the first three columns are taken from the field notes of the survey, while the entries in the next four columns have been read off from the traverse table: Lat. Dep. Balance Course Bearing Dist. -N-. S p. Balance _________ N S E W Lat. Dep. AB N 60010'E 26 12.94 22.56 +13.02 +22.56 BC S 18~ 5' W 31.2 29.66 9.68 -29.56 - 9.69 CA N 38~ W 20.9 16.47 ____12.87 +16.54 -12.87 Sum of Dist. = 78.1 29.41 29.66 22.56 22.55 29.41 22.55 Error in Lat. = 0.25 0.01 = error in Dep. We have added the north latitudes, then the south latitudes, etc. If the survey had been exact (which is not to be expected in view of errors of measurement in the field), the sum of the north latitudes would have been equal to the sum of the south latitudes. In our example, the latter sum exceeds the former by 0.25, which is called the error in latitude. This error is distributed among the individual latitudes in proportion to the lengths of the courses, the proportionate parts of the error 0.25 being 26 31.2 20.9 8X.25=.08, 781 X.25=.10, 78 X.25=.07. The partial errors in the north latitudes are here to be added to them, while the partial error in the south latitude is to be subtracted from it. The corrected latitudes are entered in the first of the columns headed "Balance"; to the north latitudes prefix the sign +, and to the south latitudes the sign -, thus enabling us to put all of them in a single column. Moreover, these signs are needed in the further problem to find the area of the field. Similarly we obtain the corrected departures and prefix the sign + to the east departures and the sign - to the west departures. 86 TRIGONOMETRY [Art. 64 The bearings and distances are now inconsistent with the balanced latitudes and departures and should be corrected. Thus for the first course we have the legs 13.02 and 22.56 of a right triangle and require the hypotenuse and an acute angle, which may be most readily found by logarithms (but also from the traverse table by double interpolation, Art. 46). If a map of the field is made by plotting the courses in turn, the final point, which should coincide with the initial point, lies at a distance from it, called the error of closure. It is equal to the hypotenuse of a right triangle whose legs are the errors in latitude and departure. These were.25 and.01 in our example, so that the error of closure is.25. The ratio of this to the perimeter 78.1 is approximately 1/312, which is too large a relative error, even N for farm surveying for which the relative error M- --- B should be < 1/500 and preferably < 1/2000. / — / 64. Double meridian distances. Select the meridian which passes through that A / one of the stations which is farthest west. H - - - In our example this is the initial station A (Fig. 45). The meridian distance (called K -— X / I -also longitude) of a course AB is the \. / distance FD of its mid-point D from the P -- - — j meridian NAS. Likewise HG and XJ s c Q are the meridian distances of BC and CA. FIG. 45 We have 2FD = MB, 2HG = MB + PC = MB + MB + (-QC), 2KJ = PC = MB- QC = MB + PC+ (- QC) + (-PC). Hence, the double meridian distance of the first course is equal to its departure; the double meridian distance of any new course is equal to the algebraic sum 1 of the double meridian distance and departure of the preceding course together with the departure of the course itself. 65. Area of a field. The area of triangle ABC in Fig. 45 is evidently obtained by subtracting triangles ABM and ACP from the trapezoid MBCP. Hence the algebraic sum of the By the algebraic sum of 10, 2, and -5 is meant 7. Ch. VII] LAND SURVEYING 87 latter two triangles and the negative of the trapezoid is equal to the negative of ABC. As the sign before the area obtained for the field is of no interest and is changed by passing around the field in the reverse sense, it is ignored. The double areas of the three auxiliary figures are found by the following multiplications: Course Lat. Double Double area mer. dist. AB +AM 2FD 2ABM BC -PM 2HIG -2MBCP Alg. sum CA +PA 2KJ 2PAC J =-2ABC Hence, if we multiply the latitude of each course by its double meridian distance, paying attention to their signs, and divide the algebraic sum of these products by 2, we obtain the area of the closed field. We shall now illustrate the rules given in the last two sections by performing the computations necessary to find the area of the triangle in Arts. 63, 64 obtained after balancing its survey. Double Course Lat. Dep. mer.dis Double area AB +13.02 +22.56 +22.56 13.02X22.56 = +293.73 +22.56 - 9.69 BC -29.56 - 9.69 +35.43 -29.56X35.43 = -1047.30 - 9.69 -12.87 CA +16.54 -12.87 +12.87 16.54X12.87 = +212.87 d-506.60 +506.60 2)540.70 270.35 As a check, the double meridian distance +12.87 for CA is the negative of its departure. If the given distances are in chains, the area is 27.035 acres. 88 TRIGONOMETRY [Art. 66 66. Plotting. To plot our triangular field on square-ruled paper, choose a vertical ruled line to represent the meridian NS through the most westerly station A. To represent A take the intersection of NS with any horizontal line of the ruling. We locate stations B and C by means of their balanced latitudes and departures. On AN lay off AM (Fig. 45) to contain 13.02 units and on the perpendicular at M lay off MB to contain 22.56 units, thus plotting B. By adding algebraically the latitude - 29.56 and departure - 9.69 of C with respect to B to the latitude 13.02 and departure 22.56 respectively of B with respect to A, we obtain the latitude - 16.54 and departure + 12.87 of C with respect to A. Hence we lay off AP downwards of length 16.54 and on the perpendicular at P lay off PC to the right to contain 12.87 units, thus plotting C. EXERCISES ON BALANCING SURVEYS AND FINDING AREA Balance each of the following surveys, plot, and find the area in acres. Use logarithms for Exs. 5, 6. 1. Course Bearing Chains AB N10~E 6.50 BC S72015'E 7.35 CA S63~W 9.15 3. Course Bearing Chains. AB S45~30'E 8.61 BC S29015'W 12.48 CD N58010'W 13.89 DA N51010'E 15.15 5. Course Bearing Chains AB S21015'E 24.68 BC N72~15'W 25.84 CD N9030'W 13.36 DA N84~E 18.08 2. Course Bearing Chains AB N15~E 9.6 BC S10040'E 7.2 CA S60~5'W 4.4 4. Course Bearing Chains AB S60~10'E 13.0 BC S38~W 10.45 CA N18~5'W 15.6 6. Course Bearing Chains AB S64~E 5.40 BC S21~30'W 9.60 CD N74~45'W 4.43 DA N16030'E 10.52 Ch. VII] LAND SURVEYING.89 PART II. SURVEYING INSTRUMENTS; FINDING TRUE BEARINGS 67. Verniers. There is a simple device, called a vernier, which increases the accuracy of measurements made by instruments used in surveying, astronomy, physics, etc. The vernier (invented in 1631 by Pierre Vernier) is a short attachment which slides along the side of a scale, enabling us to measure portions of the scale smaller than the least divisions on the scale. Consider a scale S graduated to feet and tenths of a foot. Let 9 small divisions (each.1 of a foot) of the scale have together the V 01 3 4 5 6 78910 8 9 3 4 5 6 7 8 9 3 x,? 6 FIG. 46 same length as 10 divisions on the vernier V. Hence each division on the vernier is equal to.9/10 of a foot. Thus a small division of the scale exceeds a division on the vernier by.1 -.09 =.01 of a foot. The reading in Fig. 46 is 5 feet. We always seek the reading on the scale S of the point exactly opposite to the sV 1 2 3 4 5 6 7 8 9 10I S \8 9 1 3 4 56 7 89 1 x 5 6 / FIG. 47 zero of the vernier. Let the vernier move to the right until 1 on the vernier becomes exactly opposite to 1 on the scale, so that the vernier has moved.01 of a foot; then the reading of the point opposite to 0 is 5.01 feet. In Fig. 47, 5 on the vernier is exactly 90 TRIGONOMETRY [Art. 67 opposite to a division point on the scale, while 0 is opposite to a point P between divisions 2 and 3; whence the reading at P is 5.25. Such a vernier is attached to a leveling-rod. The present form of the vernier is called a direct vernier since the markings 1, 2, 3,... on it proceed in the same direction as the markings on the scale. S I FIG. 48 We next describe a type of double vernier for reading angles. Let S (Fig. 48) represent a portion of a circular scale graduated to half degrees from 0~ to 360~ in each direction; let 29 of these small divisions of the scale have together the same length as 30 small divisions on the vernier V. FIG. 49 Each small division of the scale exceeds a division on the vernier 29 by 30' -2 X 30' = 1', so that readings may be made to minutes. Next, let the vernier take the new position shown in Fig. 49. Ch. VII] LAND SURVEYING 91 We read from the zero of the scale S to the zero of the vernier V, then in the same direction along the vernier until we reach coincident lines (marked dotted); add the reading of the vernier to the reading of the scale. Thus, if we use the lower markings on the scale S, which increase to the left, we add the reading 16' of the left half of the vernier to the reading 68~30' of the scale to obtain the angle 68~46'. But when using the upper markings on the scale, we add the reading 14' of the right half of the vernier to the reading 291~ of the scale to obtain the angle 291~14'. EXERCISES ON VERNIERS 1. The least count (i.e., smallest reading) by any direct single vernier is found by dividing the value of the smallest division of the scale by the number n of divisions on the vernier, if these n divisions cover n - smallest divisions of the scale. 2. On a scale whose smallest division is equal to 20 minutes, 59 divisions cover 60 divisions on the vernier. Show that the least count is 20 seconds. 3. Construct a retrograde vernier whose markings 0, 1, 2,..., proceed in the opposite direction from the markings on the scale, and such that 11 small divisions (each 0.1 of a foot) of the scale have together the same length as 10 divisions on the vernier. What least length can be read by its use? 68. Surveyor's compass. A surveyor's vernier compass (Fig. 50) consists of a horizontal brass plate which supports at its center a compass circle and a magnetic needle, a vertical sight vane at each end, two spirit levels (Art. 4) placed at right angles to each other, a vernier (the role of which is explained at the end of Art. 71), and a tripod or Jacob's staff on which the instrument is supported. Each sight vane has long narrow slits terminated by larger circular holes through which the object to be sighted is more readily found. The observer always looks through the sight vane adjacent to the south point of the compass circle toward the sight vane 92 TRIGONOMETRY [Art. 68 adjacent to the north point. The latter vane has graduations for the rough measurement of vertical angles. The brass head, by which the compass is attached to the tripod, has a ball-and-socket joint to give a universal motion, thus en I FIG. 50. THE SURVEYOR'S COMPASS. abling the surveyor to level the instrument, i.e., to bring the main plate into a horizontal plane. This will be the case when the bubble in each of the two spirit levels is at the middle, provided the levels have been properly adjusted. The compass circle is graduated to half degrees from 0~ to 90~ each way from the north and south points N and S. When the face of an ordinary pocket compass or a mariner's compass is viewed from above, the point 90~ to the right of N is marked E, and the point 90~ to the left of N is marked W. But on a surveyor's compass, these markings E and W are interchanged Ch. VII] LAND SURVEYING 93 (Fig. 51), with the result that, if the plane of the sight vanes is directed toward true east, the north end of the needle points to the mark E on the compass circle. ~ T FIG. 51 The needle is unreliable and often, as in city surveying, local attraction renders it practically useless. In any case the compass cannot be read closer than 5 or 10 minutes. Hence, except for rough work for which speed rather than accuracy is the main requirement, the compass has been displaced by the transit (Art. 72). But since the compass was the instrument in general use before the introduction of the transit, it is especially adapted to retracing old surveys. Also, owing to the relations of a new survey of farm lands to old official surveys with a compass, and to the fact that the transit usually carries a compass box for use in checking, the surveyor must understand the simple principles of the compass. 69. Bearing with respect to any course. We may select any course AB as the reference course. The bearing of a course IT with respect to AB is the angle between them which is measured from B or A toward the right (clockwise) or left to give an angle not exceeding 90~. In the case of true bearing T - - T (Art. 61), the reference line was a true meridian. Also here a bearing is written N 60~ E if IT lies 60~ to the right of IB (Fig. 52). FIG. 52 94 TRIGONOMETRY [Art. 70 The bearing of the reverse course TI (or IT') is S 600 W and is called the reverse bearing of the given course IT. 70. Magnetic bearing. When the reference line is the local magnetic meridian, determined by the magnetic needle at rest, a bearing is called the magnetic bearing. Thus the magnetic bearing of a course IT is the angle which it makes with the magnetic meridian through I, and is measured from the north or south point of the magnetic needle toward east or west to give an angle not exceeding 90~. To find the magnetic bearing of a course IT (Fig. 51), place the compass so that its center is vertically above I, level it, and sight toward T (or a point vertically above or below T); read the number of degrees between the north end of the needle and the nearest zero mark. Thus, in Fig. 51, the bearing is N 60~ E. As a check, take a back sight toward I upon arriving at T; the bearing of the reverse course1 TI should be S 60~ W. 71. Magnetic declination. The earth acts like a large, but irregular, magnet whose two magnetic poles2 are somewhat remote from the north and south geographical poles N and S of the earth. Hence the north point of a compass needle will not in general point true north, viz., toward N. The angle which the needle makes with the true meridian (north and south line of the place) is called the magnetic declination or variation of the compass at the given place and the given time. If the needle makes an angle of 8~ with the true meridian, the declination is 8~ E or 8~ W according as the north end of the needle points east or west of the true meridian. Besides the small change in the declination during the day, there is the more important secular variation, which is a change in the 1 When the point N of the compass card, with its center at T, is directed toward I, the new figure is like Fig. 51, but with interchanges of N with S, E with W, I with T. The beginner should draw the new figure. 2 The north magnetic pole is in latitude 70~ N, longitude 96|~ W, approximately; the south magnetic pole is in latitude 73~~ S, longitude 147~~ E, approximately. Ch. VII] LAND SURVEYING 95 same direction for a period of years, followed by a change in the opposite direction, with a period of about two and a half centuries before the declination returns to its initial value. Thus the declination was 830' W at Philadelphia in 1700; it diminished until it reached the minimum 1~30' W in 1800, and then increased to 7~45'W in 1915. Variation charts for different regions of the earth are published about every fifth year and show curves passing through the places where the variation is the same; the charts give also the yearly change at any place. The only use made of the vernier on a compass is to set it so that the compass readings will give the true bearings of lines (without requiring subsequent correction for the declination). To do this let the observer stand at the south end of the instrument and turn the vernier to the right or left, according as the variation is west or east, and through an angle equal to the variation. EXAMPLE. Given the magnetic bearing N 40~ E and the magnetic declination 10~ E, find the true bearing. NT! Solution. In Fig. 53, IN represents the true meridian, IM the magnetic meridian (containing the north point of the needle IM), and IT the course. Hence the true bearing is N 50~ E. FIG. 53 EXERCISES ON BEARINGS AND SURVEYS BY COMPASS 1. Given the magnetic bearing N 78~ E and the magnetic declination 6~ W, find the true bearing. 2. Given the true bearing N 80~15' W and the variation of the needle 13~ E, find the magnetic bearing. 3. Given the true bearing S 20~ W and magnetic bearing S 2~ E, find the variation of the needle. 4. The magnetic declination is 10~20' W. Find the magnetic bearings of the cardinal points, viz., true N, true E, S, W. 5. A course of an old survey had the magnetic bearing N 20~ E. By a recent survey it has the magnetic bearing N 17040' E. Given that the present magnetic declination is 6~30' W, find the declination at the time of the old survey. 96 TRIGONOMETRY [Art. 72 6. Given the following magnetic bearings of two courses from the same point, find the angle between them: (a) N35 E andS 42~ W. (b) S 80~ E and N 65~ E. (c) N 35~ E and N 87~ W. 7. Using the bearings in the table in Art. 63, find the bearings of BC and CA with respect to AB as the reference course. Balance the following surveys and find the area in acres. 8. 9. Course Magnetic Chains bearing AB N35025'E 5.29 BC S 25025'E 7.02 CA N71030'W 6.41 Course Magnetic Chains bearing AB S54~E 10.80 BC S31~30'W 19.20 CD N64~45'W 8.86 DA N26030'E 21.04 Magnetic declination 10~ E Magnetic declination 10~ W 72. Surveyor's transit. This instrument (Fig. 54) is the most important one used in land surveying. It enables us to measure directly the angle between two courses to minutes, often to 30 or 20 seconds, while with a compass it is necessary to measure the magnetic bearing of each course and the compass reads only to within 5 or 10 minutes. The principle of the transit is the same as that of the telescopes used in astronomy. The transit has a telescope which revolves in a vertical plane perpendicular to the horizontal supporting axis. The axis rests in bearings at the top of two standards which are rigidly attached to a horizontal circular plate, so that if the telescope is turned sidewise (horizontally), the plate must turn with it. This plate is called the upper or vernier plate and carries a vernier,1 two spirit levels placed at right angles to each other, and a graduated compass circle. Just below the vernier plate is another horizontal 1 Often two verniers placed diametrically opposite to each other, both being read to eliminate the error due to the imperfect centering of the graduated circle. Ch. VII] LAND SURVEYING 97 FIG. 54. THE SURVEYOR'S TRANSIT 98 TRIGONOMETRY [Art. 72 circular plate, called the lower plate or the limb of the instrument; it has on its outer edge the graduated circle. Each plate has its own separate motion controlled by a set screw and a tangent screw, as will be described in detail. The two plates may be rotated independently of one another, or may be clamped together by means of a set screw, which is not attached to the lower plate, although it appears to be so in Fig. 54. In fact, when the set screw is loose, the upper plate can revolve, taking the screw with it. When the screw is set, the upper plate is virtually clamped to the lower, but in such a manner that the upper plate may be rotated slowly relatively to the lower plate by turning the tangent screw, which operates at right angles to the set screw. For example, the zero of the vernier may be brought approximately opposite to the zero of the limb and, after the set screw is tightened, the zero of the vernier may be brought exactly opposite to the zero of the limb by slowly turning the tangent screw. In the same manner, without disturbing the limb, we may point the telescope approximately and then exactly on an object. At the bottom of Fig. 54 is shown the leveling head, composed of a central cylinder and a small circular plate separated by four vertical leveling screws which are used to level the transit. The limb may be clamped to the leveling head by means of a set screw and then rotated slowly by use of the lower tangent screw. Similarly, the movement of the telescope in a vertical plane is controlled by a set screw (working upon the supporting axis of the telescope) and a tangent screw. The axis of the telescope usually carries a large vertical circle or arc with a vernier attached to read vertical angles to minutes. Just underneath the telescope is attached a long spirit level used to level the telescope when finding the difference of elevation of two stations or when running horizontal courses. Within the telescope are two fine perpendicular cross wires whose intersection and the optical center of the object glass deter Ch. VII] LAND SURVEYING 99 mine the line of collimation. This line indicates the point toward which the telescope is pointed. The telescope may possess a pair of horizontal stadia wires fastened to movable slides so adjusted that the wires subtend say one foot on a graduated stadia rod standing 100 feet away, and hence subtend three feet on a rod 300 feet away, etc. With the stadia wires we can therefore rapidly measure distances approximately, which is especially useful over rough ground.' 73. Measuring angles with a transit. To measure a horizontal angle ABC, place the transit over the vertex B of the angle, level the transit, and set the zero of the vernier exactly opposite to the zero of the limb. Bring the line of collimation to bear approximately on A, clamp the lower plate to the leveling head and make the line of collimation bear exactly on A by means of the lower tangent screw. Unclamp the upper plate and turn it until the line of collimation bears approximately on C, clamp, and make the line of collimation bear exactly on C by means of the upper tangent screw. Then angle ABC is the arc over which the zero of the vernier has passed and can be read by observing the point on the limb at which the zero of the vernier stops. If it stops exactly opposite to a graduation mark of the limb, the angle is read without using the vernier. But usually it stops between graduation marks and then the vernier reading tells how far it has passed beyond the last mark (cf. Art. 67). To measure the angle of elevation or depression of an object relative to the horizontal plane (Art. 5), make the line of collimation horizontal (as shown by the level bulb attached to the telescope), whence the vertical circle or arc reads zero if in proper adjustment. Then sight approximately at the object, tighten the set screw, and, by turning the tangent screw, bring the line of collimation exactly on the object. Then read the vertical circle. 74. Traverse. A series of connected courses AB, B C, CD,... is called a traverse. It is called closed if it re- B turns to the starting point. Thus a survey of a field is a closed traverse.,' / In a traverse with a transit four main types A 10~o' 5 of angles are used: direct angle, deflection angle, / azimuth, and bearing. The first three types | / will be defined in turn. The fourth type has C been defined above. FIG. 55 1For details, see Tracy's Plane Surveying, 1914, pp. 300-317. 100 TRIGONOMETRY [Art. 75 75. Direct angle. The direct angle between two consecutive courses AB and BC is the angle ABC measured in a specified direction (right or left) and may have any value up to 360~. In a closed traverse we may employ only interior angles as in Fig. 55, or only exterior angles (that at B in Fig. 55 being 317~55'). 76. Deflection angle. It is often convenient to use the deflection angle which a course makes with the preceding course reversed, the angle P-^~ C' being measured,- _-D from the latter to xA:,~ Oo.~ J..~I2 '_ give a result less (R. A5 2 iS 1 A 4 the right or left, to C- ''E g than 180~. Thus, ru, in Fig. 56, the deFI. 56 flection angle at B FiG. 56 is A'BC and is read to the right from BA'; that at C is B'CD and is read to the left from CB'. 77. Azimuth. The angle which a course makes with a chosen reference line (such as the magnetic meridian) is the azimuth of the course; it may have any value up to 360~ and is read clockwise from a chosen end of the reference line (such as north N / on the magnetic meridian). In Fig. 57, B 05 the azimuth of BC with respect to the meridian BN is 205~, and that of AB / is 47~. C S EXERCISES ON THE FOUR TYPES OF ANGLES 1. Given the bearing N 60~10'E of the first F 57 course AB in Fig. 55, calculate the bearings of BC and CA. Check that the sum of the three angles at the right of the meridian through A is 180~. Ch. VIII LAND SURVEYING 101 2. Given the bearing N 70~25'E of the first course AB in Fig. 56, calculate the bearings of the remaining three courses. 3. In a triangular field ABC, the interior angles are A = 42~5', B = 81~50', C = 56~5/, and the bearing of AB is S 60010' E. Find the bearings of BC and CA. 4. Given the true bearings N 60010' E, S 18~5' W, N 38~ W of courses AB, BC, CA, find the azimuths of these courses with respect to the reference lines (i) true meridian, and (ii) AB. 5. In the second column of the following table are given the azimuths of the courses with respect to course AB. Given also the magnetic bearing N 30~ W of AB, verify the entries in the last two columns: Course Azimuth with AB Bearing with AB Magnetic Bearing AB 0O N N 30~ W BC 94~ S 86~ E N 64~ E CD 174~ S 6~ E S 36~ E DA 256~ S 76~ W S 46~ W 6. Given the last column in Ex. 5, derive the azimuths with AB. 7. Given the bearings with AB in Ex. 5, derive the azimuths with AB. 78. Balancing a transit survey. If we have measured the interior angles of a field of n sides, their sum should be equal to 2(n-2) right angles; any error is distributed equally among the n angles. When deflection angles are used, the difference between the sum of those measured to the right (R) and the sum of those measured to the left (L) should always be equal to 360~, as the reader may prove by means of the sum of the interior angles and the relation between any interior angle and the corresponding deflection angle. If, for example, the difference is 360~2'30" and if there are three R's and two L's, subtract 30" from each R and add 30" to each L; or we may compute the interior angles and distribute any error in their sum equally among the angles. From the angles so adjusted we compute the bearings and then the latitudes and departures by logarithms, and balance the survey as in Art. 63, and compute the area. 102 TRIGONOMETRY [Art. 78 EXERCISES ON BALANCING SURVEYS AND FINDING AREA Balance and find the area of the following surveys. In Exs. 3, 4, 5 assume that AB is a true meridian. In Ex. 6, use as the meridian a line not a course. 1. Course Azimuth Chains AB 252045' 8.20 BC 163~45' 12.05 CD 113~55' 9.80 DA 346005' 18.50 3. Course Deflection angle Feet AB 86~31' R 335.05 BC 10013' L 464.98 CD 124~53' R 483.72 DE 76~03' R 616.53 EA 82046' R 242.84 5. Course Direct angle Feet AB ABC = 132012' 678.53 BC BCD = 89032'53" 137.47 CD CDA = 90025'27" 502.98 DA DAB = 47050' 589.35 2. Course Azimuth Chains AB 00 9.60 BC 154~20' 7.20 CA 225005' 4.40 4. Course Direct angle Feet AB ABC= 44~24.5' 265.8 BC BCD = 107055' 391 CD CDA= 83052.5' 141.7 DA DAB= 123048' 250 6. Course Direct angle Feet AB ABC = 89052'40" 399.91 BC BCD=90015'47" 432.28 CD CDA =89041'33" 400.30 DA DAB = 9010' 433.26 CHAPTER VIII TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 79. Rectangular coordinates. The reader became familiar in Chapter VI with the idea of locating a point on the earth's surface by means of its latitude and longitude, the former being its distance north or south of the equator, and the latter being the arc on the equator which is intercepted between its meridian and a fixed meridian, as that through Greenwich. Similarly, in defining the two rectangular coordinates of a point P in a plane, we employ (Fig. 58) a horizontal line X'X (which plays the rl6e of the equator) and a vertical line Y'Y (which plays the role of the P meridian of Greenwich). If to the distance of a point P X from X'X we prefix the sign + or,X.. X -, according as P is above or p,, below X'X, we obtain a positive or negative number y which is called the ordinate of P (and plays the role of latitude, north or south). Likewise, if to the distance of P from Y'Y we prefix the sign + or -, according as P is to the right or left of Y'Y, we obtain a positive or negative number x which is called the abscissa of P (and plays the role of longitude, east or west). The abscissa x and ordinate y are together called the (rectangular) coordinates of P. The point P is designated by (x, y). In Fig. 58. the coordinates of P are x = + 3, y = + 2, whence P is (3,2); those of P' are x = -3, y = + 2, whence P' is (- 3, 2). Similarly, P" is (- 3, - 2), and P"' is (3, - 2). As a further illustration of the signs of coordinates, we recall from Art. 63 that it is convenient in surveying to mark north latitudes +, south latitudes -, east departures +, and west departures -. 103 104 TRIGONOMETRY [Art. 80 The line X'X is called the x-axis (less often, the axis of abscissas), and Y'Y the y-axis. Their point of intersection 0 is called the origin (or origin of coordinates). To plot a point is to locate it with reference to the axes by means of its coordinates. For example, if we plot the point (-3, 2), we obtain P' in Fig. 58. The work of plotting points is simplified by the use of square-ruled paper (called also rectangular coordinate paper), which has been used in the earlier chapters in drawing right triangles to scale. 80. Radius vector. The distance OP is called the radius vector of the point P, and denoted by r. By its definition, r is a positive number. Since OP is the hypotenuse of a right triangle whose base is of length x or -x, and vertical side is of length y or -y, we see that x2 + y2 = r2, so that r = + lx2 +y2 is the radius vector of the point (x, y). EXERCISES ON RECTANGULAR COORDINATES On square-ruled paper choose as the origin a convenient intersection of two rulings. After inspecting the size of the coordinates of the points to be plotted, select the number of units which shall be represented by a small (or a large) division of the ruled paper, and mark the number along one such division. 1. Plot the points (3, 4), (3, - 4), (- 3, 4), (-3, - 4), (4, 3), (4, -3), (-4,3), (-4,- 3), (5,0), (-5,0), (0,5), (0,-5), (1, -24), (2, 4-21). What is the radius vector of each point? On what circle do all the points lie? 2. Plot the points (- 33, - 56), ( — 56, =t 33), (- 16, ~ 63), (~ 63, 16), (=- 25, - 60), (- 60, - 25), ( — 39, ~ 52), ( 52, ~ 39), (, 65, 0), (0, - 65), using all combinations of signs. What is the radius vector of each point? On what circle do all the points lie? 3. Plot the points (- 2, -3), (- 1, -1), (0, 1), (1, 3), (2, 5). Do they appear to lie on a straight line? 4. Find y so that (0, - 200), (1, y), (2,400) shall lie on a straight line. Ch. VIII] FUNCTIONS OF ANY ANGLE 105 81. Generalized notion of angle. We have been concerned mainly with right triangles, no one of whose angles exceeds 90~. We shall soon consider oblique triangles, no one of whose angles exceeds 180~. But we learned in navigation to employ angles of any size up to 360~. Likewise in surveying, where we distinguished between the clockwise and counter-clockwise directions in which angles may be measured, starting for example from the north point. These remarks and especially the rotation of the telescope of a transit in the act of measuring an angle prepare us for the following general notion of angle: An angle may be considered as generated by the rotation of a straight line which first coincides with the initial side of the angle and then rotates about the vertex of the angle until finally it coincides with the terminal side of the angle. An angle is called positive if the rotation is counter-clockwise, negative if the rotation is clockwise. For example, in Figs. 59-61 are shown three angles IOT whose directions of rotation are indicated by curved arrows. In Fig. 59, the positive angle 320~ 3200 initial side initial side\ initial side 1 -40" i '400 FIG. 59 FIG. 60 FIG. 61 is generated by counter-clockwise rotation from the initial side OI to the terminal side OT. In Fig. 60, the negative angle - 40~ is generated by clockwise rotation. In Fig. 61, there has been a complete revolution clockwise through 360~ followed by a further rotation clockwise through 40~, the combined result being the negative angle - 400~. +3 106 TRIGONOMETRY [Art. 82 82. Trigonometric position of an angle; the four quadrants. When, as in Figs. 59-61, an angle IOT is placed so that its initial side 01 is horizontal and is drawn toward the right from 0, the angle is said to be in its trigonometric position. The horizontal line X'X and the vertical line Y'Y, which intersect at 0, divide the plane into four parts, called quadrants, which are numbered as in Fig. 62. If a line whose initial position is OX rotates counter-clockwise Y about 0, it passes in turn over the first, second, third, and Secand First fourth quadrants. In any one Quadrant Quadrant of its positions, the rotating X o ' Foth line is the terminal side of an Quadrant h urt angle having OX as initial Quadrant Quadrant side; this angle is said to be y' in the quadrant which conFIG. 62 tains the terminal side. For example, the angles in Figs. 59-61 are all in the fourth quadrant, while angle 120~ in its trigonometric position (Fig. 65) is in the second quadrant. EXERCISES ON THE QUADRANTS AND TRIGONOMETRIC POSITION 1. Name the quadrants of the following angles when placed in their trigonometric positions: 230~, -120~, - 300~, 950~, -1000~. 2. Give two positive and two negative angles in their trigonometric positions whose terminal sides bisect zX'OY'. 3. Name the quadrants in which lie the plots of the points (3, - 4), (-3, 4), (-3, -4). What least positive angles in their trigonometric positions have these points on their terminal sides? 4. What angle does the minute hand of a clock describe in 2 hours and 20 minutes? The hour hand? 5. What angle is described by a spoke of a bicycle wheel, 3 feet in diameter, when the bicycle travels 100 feet? Ch. VIII] FUNCTIONS OF ANY ANGLE 107 83. Trigonometric functions of any angle. The six trigonometric functions of any acute angle were defined in Art. 7. We shall now define the six trigonometric functions of any angle A, with the exception of certain of the functions of 0~, 90~, - 180~, or any multiple of 90~. Place the angle A in its trigonometric position XOT (so that its initial side is the horizontal line OX extending to the right from Y Y A TV 0iX AxT FIG. 63 FIG. 64 the origin 0 of a system of rectangular coordinates). Select any point T, other than 0, on the terminal side of angle A. Let x be the abscissa of T, y its ordinate, and r its radius vector. Not only for the cases of the angles A shown in Figs. 63, 64, but for an angle A of any size and any sign, we make the following definitions: sinA=, cosA=-, tan A=- (if x 0), r r x csc A=r (if y.0), sec A = (if xo0), cot A= x (if yO). y x y Or in words, the sine is the ratio of the ordinate to the radius vector,..., the cotangent is the ratio of the abscissa to the ordinate provided the ordinate is not zero. These ratios are the same, no matter what point T is selected. The values of the six functions depend only on the angle A. For the case of an acute angle (Fig. 63), these definitions agree with those given in Art. 7, where r was called the hypotenuse, x the adjacent side, and y the opposite side. 108 TRIGONOMETRY [Art. 83 When A is equal to 0~, 90~, 180~, or any multiple of 90~, one of the numbers x and y is zero and cannot be used as a divisor. It was therefore necessary in our definition of tan A as y/x to exclude the cases in which x = 0. No definition is given of the tangent of 90~, 270~, or of any odd multiple of 90~; the tangent is said to be undefined for these angles. Accordingly we avoid the symbols tan 90~, tan 270~, and similarly csc 0~, cot 0~, sec 90~, csc 180~, cot 180~, sec 270~, together with the symbols obtained by increasing or decreasing these angles by multiples of 360~. The fact that there exists no entirely satisfactory definition of these excluded symbols will become evident when we have studied the graphs of the functions (Arts. 107-8). Angles - 340~, 20~, 380~ and 740~ have the same terminal side when placed in their trigonometric positions, and hence by definition have the same sines, the same cosines, etc. So always, the trigonometric functions of angles which differ by any multiple of 360~ have the same values. EXAMPLE 1. Find the trigonometric functions of angle 120~. Solution. Place angle 120~ in its trigonometric position XOT (Fig. 65). For convenT ience take OT = r = 2. Since ZBOT = 60~, i' 1\ ~ we complete the equilateral triangle OTB, and see that OC and TC are of lengths 1 and / i.F\ i /3 respectively. Hence x = - 1, y = + 3, -/__t"1^^.-l so that C O FoC l f i>sin 120 ~=- cos 120 ~= FIG. 65 2 2 tan 120~= -=-3. EXAMPIE 2. Given tan A = 4/3, find sin A and cos A. Solution. The ratio of the ordinate y to the abscissa x is 4/3. Since it is a question of a ratio and not of actual lengths, we may take x = = 3 to avoid fractions. First, let x = + 3. Then y = + 4, r2 = 32 + 42, r = 5, and angle A is in the first quadrant (as in Fig. 63). Thus sin A = 4/5, Ch. VIII] FUNCTIONS OF ANY ANGLE 109 cos A = 3/5. Second, let x = - 3. Then y = -4, r = 5, and angle A is in the third quadrant (Fig. 64). Hence sin A = -4/5, cos A = -3/5. Thus there are two sets of answers. EXERCISES ON THE DEFINITIONS OF THE TRIGONOMETRIC FUNCTIONS 1. Prove that, for an angle in the second quadrant, the sine and cosecant are positive and the remaining four functions are negative. For an angle in the third quadrant, the tangent and cotangent alone are positive. For an angle in the fourth quadrant, the cosine and secant alone are positive. 2. By noting their relations to 45~, 30~, 60~ (Figs. 10, 11), find in terms of radicals the sine, cosine, and tangent of 135~, 150~, 210~, 225~, 240~, 300~, 315~, 330~. 3. Using the results in Ex. 2, find the sines and cosines of - 210~, 495~, -495~, - 60~, - 390~. 4. By using the present letters y, x, r in place of the letters a, b, c of Art. 10, extend the proofs given there of sin A cos A (1) sin2A+cos2A=1, (2) tanA= A, (3) cotA= siA, cos A sin A (4) sec2A=l+tan2 A, (5) csc2A=1+cot2A to any angle A, with exception in case of (2) and (4) of angles A which are odd multiples of 90~ (whence x = 0), and in case of (3) and (5) of angles which are even multiples of 90~ (whence y = 0), since in the excepted cases the formula involves either a function which is undefined or a fraction whose denominator is zero. 5. Which of the formulas in Ex. 4 determines the value of a new function when tan A - 4/3? How do we then get cos A? Why is it better to use (2) rather than (1) when finding sin A? Compare your answers with those found geometrically in Ex. 2 of the text above these exercises. Find both geometrically and by means of the formulas in Ex. 4 the remaining functions of angle A, when it is given that 1 5 3 6. tanA= —. 7. sin A =1-. 8. cosA=-. 6. ta 3' 13' 5'os 2 9. tan A =- with A in the third quadrant. 3 110 TRIGONOMETRY [Art. 84 2 10. sin A= — with A in the fourth quadrant. 3 11. sec A = 2, with tan A negative. By considering angles which approach 0~ or 90~, show that 12. sin 0~ = 0, cos 0 = 1, tan 0~ = 0, sec 0~ = 1. 13. sin 90~ = 1, cos 90~ = 0, cot 90~ = 0, csc 90~ = 1. 84. Trigonometric identities. An equality like sin2 A + cos2 A = 1 which is true of all angles A is called an identity. The same term is applied also to an equality (like tan A =sin A/cos A in Ex. 4 above) which is true for all those angles A for which both members have a meaning, i.e., for which the functions involved are defined and the denominators are not zero. Apart from the additional task of listing such exceptional angles, if any, the work of proving an identity is the same as in Art. 11. EXERCISES ON IDENTITIES Prove the following identities, stating1 the exceptional values, if any, for which either member is undefined. 1. (sin A + cos A)2 + (sin A - cos A)2 = 2. 2. sin2 A (csc2 A - 1) = cos 2 A. 3. sin2A + tan2 A sin2A = tan2 A. 4. sin2 A sec2 A = sec2A -1. 5. (1 + tan B)2 + (1 -tan B)2 = 2 sec2 B. 6. sin B + cos B cot B = csc B. 7. cos4 C sin4C = 2cos2C-1. 8. sin C tan2C cot3C = osC. sin a 1 - cos a -+cos a sin a sin a cos a 10. +1ta = sin a+ cos a. 1 - cot a 1 - tan a sin y+tan y 11. s= sin y tan y. csc y + cot y 1 The instructor may omit this requirement in some of the exercises assigned. Ch. VIII] FUNCTIONS OF ANY ANGLE 1ll 12. (cot y +cse y)2= + cos y. 1 - cos y (seoY+ cscY+)2 13. (se + csc)2 + 2 sin cos 0. sec2 0 + csc2 0 cos 6 14. sec0 —tan 0= + I q- sin 0 15. (1 + tan A) (1 + cot A) sin A cos A = (sin A + cos A)2. 16. (sin A + sec A)2 + (cos A + csc A)2 = (1 + sec A csc A)2. 17. (sin B + cos B) (tan B + cot B) = sec B + csc B. 18. (1 + sin C + cos C)2 = 2 (1 + sin C) (1 + cos C). 19. (1- sec x - tan x) (1 - csc x - cot x) = 2. 20. (1 tan 0 +sec0) (1 + tan - sec0) = 2tan0. 85. Reduction of the trigonometric functions of any angle to functions of an acute angle. Since our tables of the values of the trigonometric functions give directly only functions of acute angles, we seek formulas, like cos 130 = - cos 50~, which express the functions of any angle in terms of functions of acute angles. First, if A is any acute angle or any angle between 90~ and 180~, we shall prove the formulas sin A = sin (180~-A), csc A= csc (180 -A), (1) cos A= -cos (180 -A), see A= -sec (180 -A), tan A = -tan (180~-A), cot A= -cot (180~-A), which enable us to express the functions of obtuse angles in terms of functions of acute angles. If A is acute, write B for 180~ - A. But if A is between 90~ and 180~, write B for A. In either case, B is an angle between 90~ and 180~. Place B in its trigonometric position XOT (Fig. 66) and for convenience take the radius vector OT T to be of unit length. Construct 1 XOT' equal to ZX'OT= 180~ - B, andtakeOT'=OT. Draw the perpen- o diculars T'X and TX'. The right FIG. 66 112 TRIGONOMETRY [Art. 85 triangles T'OX and TOX' have equal hypotenuses and equal angles at 0, and hence are equal. Thus T'X = TX', OX = OX'. By the definitions of the trigonometric functions, sin B = TX' = T'X = sin (180~-B), cos B = -OX'= - OX = - cos (180 - B). Also, sin B sin (1800-B) tan B — = —tan (180-B). cos B -cos (180- B) Whether B is equal to A or to 180~ - A, the three formulas just proved are the same as the three formulas in the first column of (1). Taking reciprocals, we get those in the second column; for example, 1 csc A= ==csc (180-A). sin A sin (180 -A)csc (180A). When A = 90~, its tangent and secant are undefined (Art. 83), while the four formulas (1) which do not involve these two functions are true, since cos 90~ = cot 90~ = 0. 'Hence formulas (1) hold for every angle between 0~ and 180~ for which the functions involved are defined. Second, we shall prove the identities sin A= -sin (360~-A), csc A = -csc (360~- A), (2) cos A= cos (360~-A), sec A= sec (360~-A), tan A = -tan (360~- A), cot A = -cot (360~-A), which enable us to express the trigonometric functions of any angle A between 180~ and 360~ in terms of those of 360~ - A, which is an angle between 0~ and 180~. If the latter angle is between 90~ and 180~, we saw that, by means of formulas (1) we can pass to functions of an acute angle, which are given by the tables. If A is between 0~ and 180~, write B for 3600 - A. But if A is between 180~ and 360~, write B for A. In either case, B is an angle between 180~ and 360~. Place B in its trigonometric position XOT (Fig. 67 if B<270~, Fig. 68 if B> 270~). Ch. VIII] FUNCTIONS OF ANY ANGLE 113 x, F X _ F ---X T T FIG. 67 FIG. 68 Take the radius vector OT to be of unit length. Draw the perpendicular TF to OX and produce it to T' where T'F = TF. Since the right triangles OTF and OT'F are equal, L TOF = L T'OF. In Fig. 68, Z TOF + B = 360~, whence Z T'OF = 360~ - B. In Fig. 67, B = 180~ + FOT, Z XOT' = 180~ - T'OF; adding, we get B + Z XOT' = 360~. Hence, in either figure, Z XOT' = 360~ -B. By the definitions of the trigonometric functions, sin B= -FT= -FT'= -sin (360~-B) (Fig. 67 or Fig. 68), cos B= -OF = + cos (360~-B) (Fig. 67), cos B= +OF = + cos (360~-B) (Fig. 68). Then, if B ~ 270~, B sin B -sin (360~ ) -n ( ) tan B - = -tan (360 — B). cos B cos (3600 -B) The proof of these formulas holds true also in the limiting cases B = 180~, B = 360~. We have now proved the three formulas in the first column of (2), with exception of the tangent formula when A = 90~ or 2700. From these formulas, by taking reciprocals, we obtain the three in the last column of (2), with exception only of angles for which the functions involved are undefined. Hence formulas (2) are identities. Since angles 360~ - A and - A have the same terminal side and hence the same trigonometric functions, formulas (2) imply (3) sin(-A)=-sin A, cos (-A) =cos A, tan(-A) =-tanA. 114 TRIGONOMETRY [Art. 85 EXAMPLE. Express the functions of 580~ in terms of functions of an acute angle. Solution. Since 580~ = 220~ + 360~, the functions of 580~ are the same as the functions of 220~. By (2), for A = 220~, we have sin 220~ = - sin 140~, cos 220~ = cos 140~, tan 220~ = - tan 140~. By (1), for A = 140~, we get sin 140~ = sin 40~, cos 140~ = - cos 40~, tan 140~ = - tan 40~. Hence sin 580~ = - sin 40~, cos 580~ = - cos 40~, tan 580~ = tan 40~. EXERCISES ON REDUCTION TO ACUTE ANGLES Express as a function of an acute angle and find from the table of natural functions: 1. cos 510~. 2. tan 525~. 3. sin (- 165~). 4. tan 975~. 5. tan (- 30~). 6. sec (- 60~). 7. cot 800~. 8. csc 525~. Using the table of logarithms of the trigonometric functions, find 9. log cos 285~20'. 10. log tan (-120~15'). 11. log sin 484~10'. 12. Why is there no meaning for log cos 150~, log sin 225~, log tan 300~7 13.* Show that (1) are identities by proving them also for angles A between 180~ and 360~. Hint: Let A = 360~ - a, so that (1) and (2) are true when A is replaced by a, and (3) when A is replaced by 180~ - a. 14.* By replacing A by 180~ + A in (2) and then using (1), prove that sin (180~ + A) = -sin A, cos (180~ + A) = - cos A, tan (180~ + A) = tan A. Then by use of a table of natural functions, find sin 200~, cos 200~, tan 200~. CHAPTER IX SOLUTION OF OBLIQUE TRIANGLES 86. Altitude and area of any triangle. Select any angle A of any triangle ABC and place the triangle so that angle A is in its trigonometric position (Fig. 69 if A is acute, Fig. 70 if A is obtuse, Fig. 71 if A is a right angle). C ~CC c a h b A~ ~__I\B "B B....I1 C,j — 'C C FIG. 69 FIG. 70 FIG. 71 Denote the altitude on AB by h, the side A C by b, and the base AB by c. In each figure, sin A = h/b. Hence h = b sin A. Since the area of a triangle is equal to one-half the product of its base c by its altitude h, it is equal to ~bc sin A. The area of any triangle is equal to one-half the product of any two sides multiplied by the sine of the included angle. 87. Law of sines. The sides of any triangle are proportional to the sines of the opposite angles. If the sides are a, b, c, and the opposite angles are A, B, C, respectively, then a sin A a sin A b sinB (1)_, _ b sin B c sin C c sin C It will be sufficient to prove only the first formula (1), provided we agree that a and b denote any two sides of any triangle. By Art. 86, h = b sin A and, by applying the same result to angle B, we have h = a sin B. Thus b sin A = a sin B, and hence we obtain the desired first formula (1). 115 116 TRIGONOMETRY [Art. 88 We shall see that formulas (1) enable us to solve a triangle if we are given a side and two angles, or two sides and the angle opposite to one of them. A suitable check formula is furnished by any one of the formulas (5), (6), (7) of Art. 88. 88. Law of tangents. Given two sides a, b and the included angle C, we cannot solve the triangle by use of the law of sines, since each formula (1) contains two unknowns. We proceed to prove a formula suitable for this purpose. Let ABC be a triangle in which the ~ B side BC = a is greater than the side AC = b (Fig. 72). With center C and radius CB describe a semicircle meeting A C produced in D and E. Join B with D E C bA D and E. Then FIG. 72 EA = EC + b = a + b, AD = CD-b = a-b. Since Z CBD = Z CDB and Z CBD + Z CDB =180- C = A + B, we have ZADB = Z CDB= (A + B), ABD = Z CBD- Z B = (A +B)-B = (A- B). Since Z EBD is inscribed in a semicircle, it is a right angle. Hence, by Art. 12, ZAEB = 90~- ZADB=90~- ~(A+B), sin AEB=cos (A+B), ZEBA = 90~- Z ABD = 90 — (A - ), sin EBA = cos1 (A-B). Applying the law of sines to triangle ABD, we get a-b AD sin ABD sin '(A-B) c - AB sin ADB sinI (A+B) Similarly, applying the law of sines to triangle EAB, we get a+b EA sin EBA cos -(A-B) (3) c AB sinAEB cos c - AB - sin AEB - cos '(A +B) Hence by division of (2) by (3), Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 117 a-b tan 2(A-B) a+b -tan (A+ B) which is called the law of tangents. Since 2 (A + B) = 90~ - C, we may write (2), (3), (4) in the convenient form ~(5) a-b sin ~(A-B) (5) 2 c cos IC (6).a+b cos (A-B) (6) 2 c sin C a- b _tan ~(A-B) (7) a+b cot ~C Formulas (5) and (6) are known as Mollweide's equations. EXAMPLE 1. Given a = 789, B = 70~00', C = 63~53', solve the triangle. Solution. A = 180~ - B C = 110~ - 63053' = 46~7'. Solving the first two formulas (1) for b and c, we get a sin B a sin C sin A sin A log b = log a + log sin B - log sin A log c = log a + log sin C- log sin A log a = 2.8971 log a-log sin A = 3.0393 log sin A = 9.8578-10 log sin C = 9.9532- 10 3.0393 log c 2.9925 log sin B = 9.9730-10 c= 982.8 log b = 3.0123 b = 1028.7 Check by the formula derived from (6) by interchanging a and c: cb= a cos I(C-B) a cos ~(B-C) cb= - - sin AA sin 'A B-C = 607' log a = 2.8971 (B- C) = 3 3.5' log cos (B- C) = 9.9994- 10 2A = 23~3.5' 2.8965 c + b = 2011.5 log sin AA = 9.5929 - 10 log (c + b) = 3.3036 c + b = 2011.8 The values of b and c + b should be abridged to four significant digits, and then the check is exact to four digits. 118 TRIGONOMETRY [Art. 88 EXAMPLE 2. Given two sides a = 38.56, b = 25.69, and the included angle C = 59~55', solve the triangle. Solution. 4C = 29057.5', '(A + B) = 90~ - C = 60~2.5'. If we can find 4 (A - B), we can get A and B by addition and subtraction. From (7), we get tan (A- B) a- cot 4C a-b=12.87 log (a-b) = 1.1096 a + b = 64.25 log (a +b) = 1.8079 9.3017- 10 log cot 4C = 10.2393 -10 log tan 4 (A-B) = 9.5410 -10 4(A-B) = 19~ 9.8' 4 (A + B) =60~ 2.5' Adding, A = 79~12.3' Subtracting, B = 40~52.7' To find c, use the last formula (1), which gives b sin C = sin B ' log c = log b + log sin C - log sin B logb = 1.4097 log sin C = 9.9372-10 1.3469 log sin B = 9.8159- 10 log c = 1.5310 c = 33.96 Check by the second formula (1), which gives log a - log c = log sin A - log sin C log a = 1.5862 log sin A = 9.9922 - 10 log c = 1.5310 log sin C = 9.9372 -10.0552.0550 EXERCISES ON OBLIQUE TRIANGLES Solve and check the following triangles, given: 1. a = 111.4, A = 65~48', B = 37~24'. 2. b = 890, B = 26~00', A = 52~20'. 3. a = 364.2, A = 5421', C = 68~15'. 4. c = 43.09, B = 43025', A = 104~32'. 5. a = 148.0, b = 255.5, C = 62~32'. Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 119 6. a = 193.8, b = 287.3, C= 52~21'. 7. a = 156.0, c = 190.5, B = 56030/. 8. b = 1048, c = 909.9, A = 56028'. 9. From a ship, a lighthouse was observed to bear N 56~ W; after the ship had sailed 6 miles due east, the lighthouse bore N 67~ W. Find the distance from the lighthouse to the ship in each position. 10. Two observers 7 miles apart on a plain, and facing each other, find that the angles of elevation of a balloon in the same vertical plane with themselves are 60~ and 75~. Find the distance of the balloon from each observer and its height. 11. At one place on a road running due north an observer sees that a house is northeast. After walking north 2 miles, he sees that the house is S 30~ E. How far is the house from the road? 12. Find the distance of a fort F from a battery B if a point C is located such that BC is 800 yards and angles FBC and BCF are 60~ and 65~32;. 13. Find the height of a hill above a horizontal plane from one point of which the angle of elevation of the top of the hill is 11028', while, after walking 1000 yards toward the top up an incline of 3~25', a man finds the angle of elevation of the top to be 21030'. 14. To find the distance between two objects A and B, separated by a swamp, a station C was chosen, and the distances AC = 4604 ft. and BC = 6422 ft., and angle ACB = 78~44', were measured. Find the distance AB. 15. It is planned to tunnel through a mountain from A to B, points both visible from C. If AC = 179.1 yards, BC = 360.1 yards, and LACB = 36024', how long will the tunnel be? 16. Two ships start at the same time from the same place; one steams due west at the rate of 20.88 miles per hour, and the other steams southwest at the rate of 15.42 miles per hour. How far apart are they at the end of two hours? 17. The area of a parallelogram is 147684 square feet and the diagonals are 544 and 668 feet. Find the angles and sides. 18. Two adjacent sides of a parallelogram are 34.49 and 20.26 and their included angle is 118~44'. Find the two diagonals. 19. The two diagonals of a parallelogram are 100 and 120 and form the angle 130~42'. Find the sides. 120 TRIGONOMETRY [Art. 88 20. Two ships have wireless apparatus with a range of 1000 miles. One is 446 miles S 40~ E, and the other is 804 miles S 61~40' W from a certain port. Can the ships communicate directly with each other? 21. Find the distance between two pumping stations P and Q in Lake Michigan, given the distance 536 yards between two points A and B on shore and the angles BAQ = 40~16', QAP = 57~40', ABP = 42~22', PBQ = 71~7'. 22. Find the distance between two objects C andD not visible from each other, given the segments AD = 756 and DB = 562 of a line AB through D such that ZDAC = 47029' and ZDBC = 57045'. 23.* A steamer 20 miles south of a harbor sees a ship sail from it N 55~ E at the rate of 13.5 miles per hour. In what direction and at what rate must the steamer travel in order to overtake the ship in 2 hours? 24.* A ship is 4.31 miles due west of a tug and is moving 17 miles per hour in the direction S 59040' E. Find the direction and time the tug must travel at 11 miles per hour to overtake the ship as soon as possible. 25.* From a captive balloon 4000 feet high the angles of depression of two ships, one due north and the other northwest, are 11~40' and 14030'. Find the distance between the ships and the direction from the first ship to the second. 26.* From the top T of a hill, let TC be the perpendicular to the horizontal plane containing the points A, B, C. If the angle TBC of elevation is 10~43', and the horizontal angles ABC and CAB are 41024' and 96~28' respectively, while the length of AB is 1560 yards, find the height TC of the hill. 27.* At a point in a horizontal plane a tower subtends the angle A, and a flagstaff of height f standing on top of it subtends the angle B. Show that the height of the tower is f sin A cos (A + B) sin B and find the height when A = 20030/, B = 6~10', f = 50 feet. 28.* By means of a transit at the top T of a hill the angles of depression of two successive milestones A and B in the horizontal valley are found to be 38~ and 28~, and the horizontal projection AFB of angle ATB is 106~. Find the height of the hill above the valley. 29.* To find the distance d from the center E of the earth to the center M of the moon, a station P is chosen in north latitude 45~ and a Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 121 station P' in south latitude 55~ such that P' is in the plane EPM, which is perpendicular to the plane of the equator. The radius EP produced makes an angle of 55~20' with PM, while EP' produced makes an angle of 46~10' with P'M. Find d, given the earth's radius to be 3957 miles. Hint: Use the latitudes only to find Z PEP' = 45~ + 55~. 89. Solution of a triangle, given two sides a, b, and the angle A opposite to one of them. Geometric discussion.l Construct angle DAE equal to A, and on its arm AD lay off A C equal to b. Draw the perpendicular CF from C to the other arm A E. Describe an arc with C as center and a as radius. Case 1. If A is an acute angle and a< CF, the arc (1 in Fig. 73) does not intersect A E, and the triangle is impossible. D D C/ C ^A A F2 F E B - FIG. 73 FI2G. 74 FIG. 73 FIG. 74 Case 2. If A is an acute angle and a = CF, the arc (2 in Fig. 73) is tangent to AE at F, and the required triangle is the right triangle A CF. Case 3. If A is an acute angle and b>a> CF, the arc cuts AE at two points B, and B2, both to the right of A (Fig. 74), and there are two triangles AB1C and AB2C which have the given parts a, b, A. Case 4. If a>b, the arc cuts AE at a point B to the right of A and at a point B' to the left of A (Fig. 75 or Fig. 76). The 1 The instructor may omit this and assign only the trigonometric solution. If time must be saved, he may omit both and assign Ex. 3. 122 TRIGONOMETRY [Art. 89 triangle AB'C is excluded since it does not contain angle A, but contains its supplement. The only solution is triangle ABC. Case 5. If A is an acute angle and a = b, the single solution is the isosceles triangle ABC in Fig. 75 with B' and A coincident. D C / C.. < \ b B1\~ F i/ E Ag B" E FIG. 75 FIG. 76 Case 6. If A is an obtuse angle and a = b, there is no proper triangle (Fig. 76 with B and A coincident). Case 7. If A is an obtuse angle and a<b, the arc cuts AE at two points each to the left of A or fails to cut AE, so that there is no solution. This is also evident since a<b implies A < B, whence B and A are both obtuse, contrary to A + B + C = 180~. These results may be summarized as follows: There is no solution if angle A is acute and a < CF, or if A is obtuse and a b. There are two solutions if A is acute and b> a> CF. There is one and only one solution in the remaining cases. Trigonometric solution. The law of sines enables us to detect easily the cases in which the triangle is impossible and to solve the triangle or two triangles if the problem is possible, as well as to obtain anew the criteria found by the above geometric discussion. Write h for the altitude CF = b sin A (Art. 86). By the law of sines, b sin A h sin B = =a a Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 123 Case 1. If a<h, h/a exceeds unity and hence cannot be equal to sin B. Thus the triangle is impossible. Case 2. If a = h, sin B = 1 and B = 90~. Then if A is acute, a single triangle exists and it is a right triangle. Cases 3-7. Let a>h. Then h/a is a positive number <1 and we can find from a table of natural or logarithmic sines a single acute angle B1 for which sin B1 = h/a. Then (Art. 85) B2 = 180~ - B1 is the obtuse angle for which sin B2 = sin B1 = h/a. Hence the possible values of our angle B are B1 and B2. First, let A be an obtuse angle. The triangle cannot contain a second obtuse angle B2. If a<b (Cases 6, 7), then A <B, whereas the obtuse angle A exceeds the acute angle B = B1. Hence must a>b (Case 4, Fig. 76), and then a single triangle exists. For it, C and c are determined by a sin C (8) C = 180- A - B, c= sin Y sin A Second, let A be an acute angle. If a> b, then A >B, so that B is an acute angle, and the oblique angle B2 is not a value of B. For the single triangle (Cases 4, 5, Fig. 75) with B = B1, we may compute C and c by means of (8). Finally, if a<b (Case 3), then A <B1 by Fig. 74 or by b sin A sin A < = sin B1. Since A and B1 are each <90~, C1 = 180~ - A - B1 is positive, and A, B1 and C1 are angles of one triangle. There is here another solution, viz., the triangle with the angles A, B2 = 180 - B1, C2 = B1 - A. For each triangle, the third side cl or c2 is found by the law of sines (8). Hence only in Case 3 are there two solutions. EXAMPLE 1. Find the number of triangles having A = 30~, a = 2, b = 6. There is no solution since b sin A = 6 X ~ = 3 and a<3. 124 TRIGONOMETRY [Art. 89 EXAMPLE 2. Given a = 46.73, b = 79.80, A = 23020', solve the triangle or triangles, and check. Solution. h = b sin A. log h = log b + log sin A log b = 1.9020 log sin A = 9.5978-10 log h = 1.4998 log a = 1.6696 b> a> h, two solutions sin B = h a log sin B = log h - log a = 9.8302- 10 B1 = 42~33.6' B2 = 180~-B1 = 137~26.4' B1 +A = 65~53.6' B2 + A = 160~46.4' C = 180~ - (B + A) C1 = 11406.4' C2 = 19~13.6' c cos b (B-A) Check by (6): a+b- s2~ sin IC a c = sin A log c = D + log sin C D = log a- log sin A log a = 1.6696 log sin A = 9.5978- 10 D = 2.0718 log sin C1 = 9.9604-10 log cl = 2.0322 cl = 107.70 D = 2.0718 log sin C2 = 9.5176-10 log C2 = 1.5894 C2 = 38.85 C2 = B1 - A = 19~13.6' Ci = B2 - A = 114~6.4' 2C2 = (Bi - A) = 9036.8' Ci = (B2 - A) = 5703.2 log cl = 2.0322 log C2 = 1.5894 log cos 2 (Bi- A) =9.9939-10 log cos (B2- A) =9.7355 —10 2.0261 1.3249 log sin C1 = 9.9239-10 log sin C2 = 9.2227-10 2.1022 2.1022 log (a + b) = log 126.53 = 2.1022 Note. This method of solving and checking the triangles is not only a long one, but is liable to introduce too large an error when angle B1 exceeds 70~. For, when finding B1 by our table from its log sine, we may make an error of more than 1', and the final part of our solution may involve an accumulation of appreciable errors. For example, given a = 13.54, b = 16.08, A = 52~24', the above method yields B1 = 70~14', C1 = 57~22', C2 = 17~50', cl = 14.39, C2 = 5.234, whereas the values obtained by use of six-place tables are Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 125 B1 = 70~12'21.4", C1 = 57o23'38.6", C2 = 17~48'21.4", ci = 14.3963, C2 = 5.22594. Thus the former value of c2 presents too large an error, due to an accumulation of several small errors. It is preferable to proceed as in Ex. 3. Moreover, it is desirable that the student bear in mind that every oblique triangle can be solved by solving component right triangles. EXAMPLE 3. Given a = 13.54, b = 16.08, A = 52~24', solve the triangle or triangles. Solution. We solve the two component right triangles ACF and CFB1 in Fig. 74 by the method given in Art. 35. h = CF = b sin A FB = (a + h) (a -h) log b = 1.2063 a +h = 26.28 log sin A = 9.8989- 10 a-h = 0.80 log h = 1052 log (a + h) = 1.4197 h = 12.74 log (a-h) = 1.9031 2) 1.3228 AF = b cos A log FB1 = 0.6614 log b =1.2063 FB1=4.586 log cos A = 9.7854- 10 log AF = 0.9917 cos B1 = FB / a AF = 9.810 log a = 1.1316 log cos B1 = 9.5298- 10 cl = AF + FB1 = 14.396 B1 = 70~12.2' C2 = AF- FB1 = 5.224. C1 = 57~23.8' B2 = 180~ - B1 = 109047.8' C2 = 17~48.2'. Only two significant figures are reliable in view of a - h = 0.80. To solve this Ex. 3 by the traverse table, we employ triangles I and II obtained by mental interpolations between triangles with hypotenuses 16 and 17, and triangles III and IV obtained by interpolations between triangles with the hypotenuses 13 and 14: hyp. = 16.08 hyp. = 13.54 Triangle A opp adj. Triangle B. adj. I 52~ 12.67 9.90 III 70~ 12.73 4.63 ACF 52024' 12.74 9.81 CFB1 70~9' 12.74 4.60 II 530 12.84 9.68 IV 71~ 12.80 4.41 Hence AF = 9.81, FB1= 4.60, B1 = 70~9', ci = 14.41, c2 = 5.21. 126 TRIGONOMETRY [Art. 90 EXERCISES ON SOLVING TRIANGLES, GIVEN TWO SIDES AND THE ANGLE OPPOSITE TO ONE SIDE Find the number of triangles having 1. A =30~, b = 10, a = 15. 2. A = 30~, b = 10, a = 8. 3. A = 30~, b = 10, a = 5. 4. A = 30~,b = 10, a = 4. 5. B = 3723', a = 9.1, b = 7.5. 6. B = 61~16', a = 12.75, b = 9.512. Using Table VI, solve the triangles having 7. a = 9, b = 10, A= 55~. 8. a = 120, b = 100, A = 50~. 9. a = 42, b = 67, A = 52~. 10. a = 36, b = 38, A = 16~. Solve by logarithms and check the triangles having 11. a = 311, b = 374, A = 27018'. 12. a = 75.64, b = 82.66, A = 50"16'. 13. a = 7082, b = 8034, A = 6127'. 14. a = 342, b 214, A = 31053/. 15. A ship is 1190 feet from the nearer of two buoys which are 970 feet apart. The angle at the ship made by the lines to the buoys is 38036'. How far is the ship from the farther buoy? 16. The two faces of an embankment measure 145.5 ft. and 252.0 ft., and the angle of inclination of the second face is 12015'. Find that of the first face and the width of the embankment at its base. 17. Solve Exs. 2 and 7 of Art. 88 by using only right triangles. 90. Law of cosines. In any triangle the square of any side is equal to the sum of the squares of the remaining two sides diminished by double the product of those two sides multiplied by the cosine of their included angle. Or, if the initial side be called a, (9) a2 = b2 + c2- 2bc cos A. Drop a perpendicular CF from the verC tex C to the base AB or the base produced. If A and B are each < 90~, F falls within b/ I a AB (Fig. 77) or at A or B, so that c = AB = BF + AF. But cos B =BF/a, cos A = / F \B AF/b. HenceBF = a cos B, AF = b cos A, and FIG 77 (10) c = a os B + b cos A. Next, let A, for example, be an obtuse angle. Then F falls on AB produced (Fig. 78), so that c = AB = BF-AF. But Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 127 cos B =BF/a, cos A = -AFb. Hence BF = a cos B, AF= -b cos A, and c = a cosB-(- b cos A), C so that (10) again holds true. We have now i proved that any chosen side of any triangle is a equal to the sum of the products obtained by I multiplying each remaining side by the cosine of L _ the angle which it makes with the chosen side. F A c B FIG. 78 Hence, when the chosen side is a or b, we have (11) a = bcos C + ccosB, b = acos C + c cos A. Multiply equation (10) by c and the two equations (11) by -a and b, respectively, and add. We get (12) c2 -a2 + b2= 2bccosA. Transposing terms, we have (9). The law of cosines (9) is merely a combination of two theorems in geometry: The square of the side opposite an acute [or obtuse] angle is equal to the sum of the squares of the remaining two sides diminished [or increased] by double the product of one of those sides by the projection of the other upon it. Thus a2 = b2 + c2 - 2cAF (A acute, Fig. 77), a2 = b2 -+ 2 + 2c AF (A obtuse, Fig. 78). We saw that, in the first case, AF = b cos A, while, in the second case, AF = - b cos A. Thus each of the two different geometric formulas leads to the same trigonometric formula (9). Formula (9) may be used to compute a when b, c, A are given, or to compute A when a, b, c are given, since it may be written in the form (12) and hence in the form b2+ c2 - a2 (13) cos A 2bc 2be But, as these formulas are not adapted to logarithms, computation by them is unnecessarily laborious unless b and c in (9) and a, b, c in (13) are numbers with only one or two significant digits. Formulas adapted to logarithmic computation are given in Arts. 88, 93. The importance of (9) is due mainly to its being employed frequently in proofs of theorems in pure and applied mathematics. 128 TRIGONOMETRY [Art. 9i EXERCISES ON THE LAW OF COSINES 1. Using the law of cosines, write the two further equations of type (9), and from them deduce two of type (13). Hence solve the triangles in which 2. a = 2, b = 3,c = 4. 3. a = 7,b = 8, c = 9. 4. a = 2, b = 3, C= 30~. 5. b = 1, c = 10, A = 35024'. 6. The two diagonals of a parallelogram are 6 and 8, and their included angle is 60~. Find the sides. 7. Find the inclination of the face of an embankment if a ladder 26 feet long rests 10 feet from the foot of the embankment and reaches 23 feet up its face. 8. From the crossing, at an angle of 54~, of two railroads the distances to two bridges over the same river are 4 and 6 miles. How far apart are the bridges? 9. Two towns A and B are 15 and 20 miles distant from C, while angle ACB is 27~30'. Two men start at the same instant from C and walk at the same rate. One walks to A and then toward B. The other walks to B and then toward A. How far from A do they meet? 10. Divide each member of (10) by b sin A and replace a lb by its value given by the law of sines. Hence show that C cot B = - cot A. b sin A By this formula compute B, given b = 8, c = 12, A = 30~. 11.* The plane ABC of a side of a hill is inclined 30~ to the horizontal plane ABD and intersects the latter in an east and west line AB. A tree standing at A had its upper part broken at a point 12 feet above the ground by a WSW wind (i.e., blowing from S 67 ~ W), and the top of the broken tree now rests on the ground 40 feet from A. Find the height of the tree. 91. Area (A) of a triangle in terms of its sides.1 By Art. 86, A = ~bc sin A, A2 = b2c2 sin2 A = b2c2 (1-cos 2 A). We shall evaluate the two factors 1 + cos A, 1 - cos A. By (13), - c (b+c)2-a2 a2 - (b-c)2 1 + cos A - a, 1- cos A= 2be 2bc 1The instructor may assume this result from geometry. Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 129 Hence A2 = [(b + c)2-a2] [a2 -(b- c)2] = i6 (b + c + a) (b + c - a) (a + b - c) (a - b + c). This result becomes much simpler if we employ the abbreviation s for the semi-perimeter (half of the sum of the sides). Then a + c 2s, b + c - a = b c + a - 2a = 2 (s - a), a + b - c = 2 (s - c), a -b + c = 2 (s - b). Thus we obtain Heron of Alexandria's formula (14) A= Vs(s-a) (s-b) (s-c). For example, if a = 13, b = 14, c = 15, then a +b +c = 42, s = 21, s-a = 8, s-b = 7, s-c = 6, A = -21-8 -7.6 = V24 32.72 = 22.3.7=84. 92. Radius of the inscribed circle. Let 0 be the center and r the radius of the circle inscribed in the triangle ABC (Fig. 79). Then the three triangles AOB, AOC, BOC, which together make up triangle ABC, have equal altitudes OF = OE = OD = r. Hence the area A of triangle ABC is equal to C E D FI B A F FIG. 79 FIG. 79 A = ~cr + ~br + ~ar = ~ (a+b+c) r = sr, where s is the semi-perimeter of the triangle ABC, as in Art. 91. Employing the value (14) of A, we get (15) r = -a) (s-b) (s-c), 8 which determines the radius of the inscribed circle in terms of the sides of the triangle. For the example at the end of Art. 91, we have r = 4. 93. To compute the angles of a triangle, given the sides. In Fig. 79, the lengths AF and AE of the tangents from A to the 130 TRIGONOMETRY [Art. 93 circle are equal by geometry. Similarly, BF = BD, CE = CD. Hence the perimeter of the triangle is 2s = 2AF + 2BF + 2CD. But AF + BF = c. Thus s = c + CD. In this manner we get CD = s-c, BF = s-b, AF = s-a. Since the center 0 of the inscribed circle is the point of intersection of the bisectors of the angles of the triangle, ZOAF = ~A, ZOBF = ~B, ZOCD = C. In the right triangle OAF, side OF is the radius r of the inscribed circle. Hence (16) tan A= — tani B -- _ tanC= _ s-a as-b s — These three equations are usually called the half-angle formulas; in conjunction with (15), they furnish the best method of computing by logarithms the angles of a triangle whose sides are given. EXAMPLE. Given a = 104, b = 114, c = 141, find angles A, B, C. Solution. Use formulas (15), (16), and 2s = a + b + c. 2s = 359 log (s- a) = 1.8779 s = 179.5 log (s- b) = 1.8162 s-a= 75.5 log (s- c) =1.5855 s-b= 65.5 5.2796 s-c= 38.5 log s = 2.2541 4s - 2s = 359.0 log r2 = 3.0255 (check) log r = 1.5128 log (s- c) = 1.5855 log tan ~C = 9.9273- 10 log r = 1.5128 = 40~13.6' log (s- a) = 1.8779 C = 80027.2 log tan 'A = 9.6349- 10 log r = 1.5128 1A = 23020' log (s - b) = 1.8162 A = 46040' log tan ~B = 9.6966- 10 ~B = 26026.5' B = 52~53' Check: A +B + C = 180~0.2'. Ch. IX] SOLUTION OF OBLIQUE TRIANGLES 131 EXERCISES ON TRIANGLES WITH GIVEN SIDES Find the areas and angles of the triangles whose sides are 1. a = 2, b = 3, c = 4. 2. a = 51.38, b = 68.56, c = 60.35. 3. a = 146, b = 164, c = 182. 4. a = 12.66, b = 17.00, c = 15.70. Without using logarithms, find the areas of the triangles whose sides are 5. 25, 52, 63. 6. 35, 84, 91. 7. A pole 26 feet long rests 12 feet from the base of an embankment and reaches 16 feet up its face. Find the slope of the embankment. 8. Two islands, which are 6 miles apart, are 9 and 12 miles distant respectively from a boat. What angle is subtended by the islands at the boat? 9. Two circles are drawn with radii 14 and 16 inches and with centers 20 inches apart. Find the length of their common chord. 10. What is the radius of the largest circular track that can be made within a triangular field whose sides are 253.1, 344.3 and 462.1 yards long? 11. Given AB = 18.64, BC = 21.30, CA = 34.30, take a point P in the prolongation through A of AB such that zAPC = 15o35'. Find PC. 12. Three circles of radii 9, 11, and 13 are tangent externally. Find the angles of the triangle formed by joining their centers. 13.* Show by Fig. 79 that AO2 = be (s - a) /s and hence that (s-b) (s-c) Is(s-a) sin A = \ be -c, cos 4A= \ bc By means of one of these formulas find the least angle of the triangle whose sides are 13, 14, 15. 14.* Find the sides of a triangle which contains in its interior three circles each of diameter 20 feet and has each side tangent to two of the circles, given the distances between the centers of the circles to be 91, 82, and 73 feet. EXERCISES ON RESULTANTS AND COMPONENTS OF FORCES INVOLVING OBLIQUE TRIANGLES (For definitions and principles, see Arts. 18-21.) 1. Two forces of 3 and 5 pounds have a resultant force of 6 pounds. Find the angles made by the resultant with each component. 132 TRIGONOMETRY [Art. 93 2. The angle between the directions of two forces of 466.8 and 687.4 pounds acting on a point is 30~45'. Find their resultant force and the angle it makes with the direction of the smaller given force. 3. Two forces of 120 and 150 pounds have a resultant of 225 pounds. Find the angle between the directions of the two forces. 4. A force of 513 pounds is resolved into two components which make angles of 34~38' and 9208/ respectively with the direction of the given force. Find the component forces. 5. A force of 50 pounds is resolved into two components, one of which is 24.4 pounds and makes an angle of 25~ with the resolved force. Find the other component and its angle with the resolved force. 6. A train is moving N 54~ W at 36 miles per hour and the wind is blowing from northeast at 40 miles per hour. What direction and velocity does the wind appear to have to a passenger? 7. A ship is steaming 20 miles per hour S 10~ W and leaves a trail of smoke in the direction N 35~ E. If the wind is from the north, what is its velocity? 8. A passenger walks approximately southeast 4 miles an hour across the deck at right angles to the length of the ship, which is steaming N 35~ E at 20 miles per hour in a current moving S 50~ E at 5 miles per hour. Find the velocity and direction of the motion of the passenger. 9. An aeroplane is flying S 67~ W at a speed which would carry it 75 miles per hour in still air. A wind blows from northwest at 28 miles per hour. Find the ground-speed and the ground-course. 10.* Two aeroplanes fly in opposite directions at 60 miles per hour around a triangular field which is 20 miles on each side. A wind of 40 miles per hour blows down one side. For each aeroplane find the time of flight along each side, and compare the total times. 11.* A man can row his boat 6 miles per hour in still water. After rowing 2 hours in a straight line in a river, running due south, he reached a place 15 miles southeast from his starting point. Find the velocity of the current (a) if his course was south of east and (b) if his course was north of east. CHAPTER X RELATIONS BETWEEN FUNCTIONS OF SEVERAL ANGLES 94. The addition theorem for sine.' If A and B are any two angles, (1) sin (A + B) = sin A cos B + cos A sin B. Case 1. Let A and B be any positive angles whose sum is less than 180~. At the ends of any convenient segment AB, lay off angles equal to A and B, thus determining a triangle ABC. Denote its sides by a, b, c. By the law of sines (Art. 87), if we divide each side by the sine of its opposite angle, we obtain the same quotient q. Hence a = qsinA, b = qsinB, c = qsin C. By formula (1) of Art. 85, sin C = sin (180~- C) = sin (A + B). Using formula (10) of Art. 90, we have q sin (A+ B) = c = a cos B + b cos A = qsin A cos B + qsin B cos A. Dividing each member by q, we obtain (1). Also, (1) is true if A = 0 or B = 0 since sin 0~ = 0, cos 0~ = 1. Hence (1) is proved for A + B<180~, A 0, B>0. Case 2. Let A and B be angles each >0 and <180~, whose sum is greater than 180~. Then a = 180 - A and A = 180~ - B are angles >0 whose sum 360~ - (A + B) is less than 180~. Hence, by Case 1, (1') sin (a + 3) = sin a cos / + cos a sin d. Substitute for a and / their values and apply (1) and (2) of Art. 85. Thus 1 To economize time, the instructor may prefer to assign only Case 1. However the other cases involve only straightforward applications of earlier formulas and afford excellent practice with them. 133 134 TRIGONOMETRY [Art. 94 sin a =sin (180~-A) =sin A, cos a = cos (180~-A) = -cos A, sin 0 = sin (180~- B) = sin B, cos j = cos (180~- B) = - cos B, sin (a+-f) =sin [360~- (A +B)] = -sin (A +B). Hence (1') becomes - sin (A + B) = sin A (- cos B) - cos A sin B, which proves (1) for the present case. Case 3. Let A and B be angles >0 whose sum is equal to 180~. Since cos B = cos (180~- A) = -cos A, sin B = sin (180~- A) = sin A, the right member of (1) is zero, while the left member is sin 180~ = 0. We have now proved (1) when A and B are any angles each 0 and 180~. Case 4. Let 180~0<A360~, O<B5180~. Then A = 180~ + a, where 0<a<180~. Thus (1) holds for angles a and B: (1") sin (a + B) = sin a cos B + cos a sin B. For B = 180~, this reduces to sin (a + 180~) = - sin a, or sin A = sin a, since cos 180 = - 1, sin 180~ = 0. Thus to prove that (2) sin (x + 180~) = - sin x (for every angle x), it remains to prove it when 180~ <x<360~. Then x = a + 180~, where 0<a < 180~, so that sin x = - sin a, as just proved. But sin (x + 180~) = sin (a + 360~) = sin a. This completes the proof of (2). Taking x = a + B, we have sin (A + B) = - sin (a + B). Finally, by (2) and (1) of Art. 85, we have cos A = cos (360~ - A) = cos (180~ - a) = - cos a. Hence (1") becomes - sin (A + B) = - sin A cos B - cos A sin B, which proves (1) for the present case. Ch. X] FUNCTIONS OF SEVERAL ANGLES 135 Case 5. Let 180~ A <360~, 180~ B <360~. Let A = 180~+ a, B = 180~ + t. Then (1) holds for angles a and /3 which are each >0 and <180~: sin (a + /3) = sin a cos + cos a sin 3. By the results in Case 4, the right member is equal to (- sin A) (- cos B) + (- cos A) (- sin B). Also, sin (a + 3) = sin (A + B - 360~) = sin (A + B). Hence (1) is proved. We have now proved (1) for all angles 0 which do not exceed 360~. If to such an angle we add a suitably chosen multiple of 360~, we obtain any assigned angle. But any function of the latter is equal to the same function of the former. Hence (1) is true for arbitrary angles A and B. 95. Functions of A + 90~. In formula (1) take B = 90~. Since sin 90 = 1, cos 90~ = 0, we obtain (3) sin (A + 90~) = cos A (for every angle A). Let x be any angle, and take A = x + 90~ in (3). Thus cos (x + 90~) = sin (x + 180~) = - sin x, by (2). Writing A for x for uniformity, we have (4) cos (A + 90~) = - sin A (for every angle A). From (3) and (4) we get, by division when sin A 5 0, (5) tan (A + 90~) = - cot A (for A not a multiple of 180~). Passing to reciprocals, we have (6) csc (A + 90~) = sec A, cot (A + 90~) = -tan A (A not an odd multiple of 90~), (7) sec (A + 90~) = - csc A (A not a multiple of 180~). It is seen that formulas (3) - (7) are true for all angles except those for which the functions entering are undefined, and hence are identities. 136 TRIGONOMETRY [Art. 96 96. The addition theorem for cosine. If A and B are any angles, (8) cos (A + B) = cos A cos B - sin A sin B. For proof, apply the addition theorem for sine to the angles A + 90~, B: sin (A + 90~ + B) = sin (A + 90~) cos B + cos (A + 90~) sin B. In view of (3) and (4), this reduces to (8). 97. The subtraction theorems for sine and cosine. If A and B are any angles, (9) sin (A - B) = sin A cos B- cos A sin B, (10) cos (A - B) = cos A cos B + sin A sin B. For proof, apply the addition theorems for sine and cosine to the angles A and - B: sin [A + ( — B)] = sin A cos (- B) + cos A sin (- B), cos [A + (- B)] = cos A cos (- B) - sin A sin (- B). These reduce to (9) and (10) since sin (- B) = - sin B, cos (- B) = cos B by formulas (3) of Art. 85. EXERCISES ON THE ADDITION AND SUBTRACTION THEOREMS FOR SINE AND COSINE Express in terms of radicals the sines and cosines of 1. 15~ = 45~- 30~. 2. 75~. 3. 105~. 4. 120~. 5 3 5. Given sinA = 5, sin B= 3, where A and B are acute angles, find 13' 5' cos (A - B) and sin (A + B). Prove that the following relations are identities, stating the exceptional values, if any, for which either member is undefined: 6. sin (60~ + B) - sin (60~ - B) = sin B. 7. sin (A + B) sin (A -B) = sin2 A - sin2 B = cos2 B-cos2 A. 8. cos (A + B) cos (A -B) = cos2 A- sin B = cos2 B - sin2 A. 9. sin (x + y + z) = sin x cos y cos z + cos x sin y cos z + cos x cos y sin z - sin x sin y sin z. 1The instructor may omit this requirement. Ch. X] FUNCTIONS OF SEVERAL ANGLES 137 10. sin (A + B) cos B- cos (A + B) sin B = sin A. 3 11. cos2 C + cos2 (C + 60~) + cos2 (C -60~) = - 12. sin (90~ - x) = cos x, cos (90~ - x) = sin x, tan (90~ - x) = cot x, csc (90~ -x) = sec x, sec (90~ - x) = csc x, cot (90~ - X) = tan x. 13. sin (180~ + x) = - sin x, cos (180~ + x) = - cos x, tan (180~ + x) = tan x, csc (180~ + x) = - csc x, sec (180~ + x) = - sec x, cot (180~ + x) = cot x. 14. sin (270~ x) = -cos x, cos (270~ - x) = sin x, tan (270~ = x) = F cot x, csc (270~ = x) = - sec x, sec (270~ x) = i csc x, cot (270~ - x) = F tan x. 15. By Exs. 12 -14 and Arts. 85, 95, prove that any function of 180~ - x, x - 180~, or 360~ = x is equal to - the same function of x; while any function of 90~ - x, x - 90~, 270~ - x, or x - 270~ is equal to - the cofunction of x (defined in Art. 12). Each such formula holds for every angle x for which its members are defined, so that the proper sign may be determined by selecting any convenient angle x, for example a positive acute angle. Hence this summary furnishes a useful aid to the memory. 98. Heights and distances. Several earlier formulas may be converted into formulas more convenient for logarithmic computation by replacing tan x by sin X/cos x, or cot x by cos x/sin x, and then applying the addition or subtraction theorem for sine. EXERCISES ON HEIGHTS AND DISTANCES 1. Show that formula (3) of Art. 39 is equivalent to h sin B cos T X sin (T-B) Give another proof by applying the law of sines to the triangle containing angle T-B (Fig. 30). Use this formula to solve Exs. 1-6 of Art. 39. 2. Show that the formula in Ex. 7, Art. 39, is equivalent to d sin A sin B hsin (B-A) Use this formula to solve Exs. 8-12 of Art. 39. 138 TRIGONOMETRY [Art. 99 3. Show that the formula in Ex. 14 of Art. 8 is equivalent to sin a sin 3 h /sin (a+3) sin (a-3) Hence find h, given a = 60~2', A = 45~3', 1 = 2.01. 4.* To find the height h of an object above a horizontal plane, measure its angle A of elevation at a point of that plane and its angle B of elevation at a point directly above the first point and at a distance d from it. Prove that d cot B d sin A cos B (h —d) cotB=hcot A,h= cotB - cot A sin (A-B) Find h, given A = 25~40', B = 23~10', d = 60 ft. 5.* From a point which is f feet above the surface of a pond, the angle of elevation of the top of a tree standing at an edge of the pond is A, while the angle of depression of the reflection of the top in the water is B. If h is the height of the tree and d its horizontal distance from the point of observation, show that (h -f) cot A = d = (h + f) cot B, f (cot A + cot B) f sin (B+A) cot A- cot B sin (B-A) Given = 40, A = 41~12', B = 58~43', find h and d. 99. The addition and subtraction theorems for tangent and cotangent. If A and B are any angles such that neither A nor B nor A + B is an odd multiple of 90~, tan A+tan B (11) tan (A +B)= -tan A tan B The exceptional values are those for which one of the three tangents is undefined. We have A + sin (A+B) sin A cos B+cos A sin B tan A B) cos (A+B) - cos A cos B- sin A sin B Divide both numerator and denominator by cos A cos B, which is not zero in view of the restrictions on A and B. Thus Ch. X] FUNCTIONS OF SEVERAL ANGLES 139 sin A cos B cos A sin B sin A sin B cos A cos B cos A cos B cos A cos B tan(A +-B) = sin A sin B 1sin A sin B cosA cos B cos A cosB which proves formula (11). Similarly, if sin (A + B), sin A, sin B are not zero, cosA cosB sin A sin B cos (A+B) sin A sin B sin A sin B sin (A+B) sin A cos B+ cos A sin B sin A sin B sin A sin B gives cot A cot B-l (12) cot (A +B)= cot A cot B cot A + cot B if neither A nor B nor A + B is a multiple of 180~. Applying formulas (11) and (12) to the angles A and -B, we have tan A -tan (- B) tan [A +(-B)] = -tan A tn (- B) - -tan A tan (-B) cot [A+(-B)]= cot A cot (-B)cot A+ cot (-B) By (3) of Art. 85, tan (- B) = - tan B, whence cot (- B) = -cot B. Thus tan A -tan B tan (A-B) 1+tan A tan B' (13) ~(13) / )co A cot B+1 cot (A - B) cot B- cot A EXERCISES ON THE ADDITION THEOREM FOR TANGENTS 1. Prove formula (13) by the method used to prove (11) and (12). 2. Find tan 75~ and tan 15~ in terms of square roots, using formulas (11) and (13) and the functions of 30~ and 45~. 140 TRIGONOMETRY [Art. 100 3. Find cot 75~ and cot 15~, using (12) and (13). Check by Ex. 2. 4. Given tan A = I, tan B = 1, find tan (A + B) and tan (A - B). 5. The slope (Art. 6) of a roof of a house is 2/5. Find its (smaller) angle of inclination with the ground if the slope of the ground is 0.1. 6. The tangent of the angle of inclination of a roof with the ground is 2/5. If the slope of the roof is 1, what is the slope of the ground? Prove the following identities: 7. tan(45 + x) =- tanx 1 + cot x 9. cot (x - 45~)= 1 + cot x i - cot x cos (A - B) 1+ tan A tan B cos (A+B) 1 - tanA tan B 13 sin (A+B) 13sin (A +B)= tan A + tan B. cos A cos B cot x-1 8. cot (45 + x) = - cot X+l sin(A+B) tanA+tanB sin (A -B) - tan A - tan B 12. tan (A-B) + tan B _ tan A 1 - tan (A-B) tan B 14.cos (A-B tan A cot B. cos A sin B 15. Prove formula (12) by taking the reciprocals of the members of (11) and afterwards verifying (12) when A or A + B is 90~ or 270~. If A, B, C are angles of a triangle, prove that 16.* tan A + tanB + tanC = tan A tan B tan C. 17.* tan 'A tan 'B + tan 'A tan IC + tan 'B tan IC = 1. 18.* cot '4 + cot 4B + cot IC = cot 4A cot 'B cot IC. 100. Functions of double angles. By the addition theorems for sine and cosine, we have sin (A + A) = sin A cos A + cos A sin A = 2 sin A cos A, cos (A + A) = cos A cos A - sin A sin A = cos 2A - sin 2A. Hence (14) sin 2A = 2 sin A cos A, cos 2A = cos 2A - sin 2A. We may replace cos 2A by 1 - sin 2A, or sin 2A by 1 - cos 2A. Hence (15) cos 2A = 1 - 2 sin 2A, cos 2A = 2 cos 2A - 1. Formulas (14) and (15) should be memorized. Taking B = A in (11) and (12), we get Ch. X] FUNCTIONS OF SEVERAL ANGLES 141 2 tan A cot2 A-1 (16) tan2A= - ta A- cot2A= 2 A1- tan2 A 2 cot A 101. Functions of multiple angles. We may express sin3A and cos 4A in terms of sin A without introducing square roots. sin (A + 2A) = sin A cos 2A + cos A sin 2A = sin A (cos 2A-sin 2A) + cosA (2 sinA cosA) = 3 sin A cos 2A - sin 3A = 3 sin A (1 - sin 2A) - sin 3A, (17) sin 3A = 3 sin A - 4 sin 3A. In the first formula (15) we may replace A by 2A and obtain cos 4A = 1 - 2 sin 22A = 1 - 2 (2 sin A cos A)2 = 1 - 8sin2A cos2A = 1-8 sin2A (1 -sin2A), (18) cos 4A = 1 - 8 sin 2A + 8 sin 4A. EXERCISES ON FUNCTIONS OF MULTIPLE ANGLES 1. Prove formulas (16) from (14) by division. 2. Given sin A = 3 /5, with A acute, find sin 2A, cos 2A, tan 2A. 3. Given tan A = r/s, find sin 2A, cos 2A, tan 2A. 4. Express cos 3A in terms of cos A. 5. Express tan 3A and tan 4A in terms of tan A. 6. Express cos 4A in terms of cos A. Prove the identities 2 tan A 1 - tan2 7. sin 2A= — tanA 8. cos 2A= I-tan2A 7l+tan2A l+tan2A 1- sin 2A cot x -1 sin 2x-1 9. tan2(450 —A)= -s2A 10. s2 1 l+sin 2A cot x-1 cos 2x, 11. tan (45~ + A) - tan (45~ - A) = 2 tan 2 A. 12. tan x + cot x = 2 csc 2 x. 13. (sin x + cos x)2 = 1 + sin 2 x. 14. cot x - tan x = 2 cot 2 x. sin 2B sin 2(7 1 - -cosn 2B 1 — scos 2C * - sin z + tan -2 cos3 z 17. 18. =1 — sin2z. l+sin 2y \tan y+1/ sin z + cos z 142 TRIGONOMETRY [Art. 102 19. 1 + tan a tan 2a = sec 2a. 20. cos 4 -sin4 3 = cos 2 3. 21. tan + cot 2 = csc 2 0. 22. tan + tan 6 sec 2 6 = tan 2 6. cos x + sin x 23. = tan 2 x + sec 2 x. cos x - sin x 1- cos 2A + sin 2A 24. = tan A. 1 + cos 2A + sin 2A 4 tan2 x sin 2 x-sin x 25. sin 2x tan 2x =1 --- 26. t = tan x. 1 -tan4 x 1-cosx+cos 2 x 27. tan (45~ + x) + tan (45~ - x) = 2 sec 2 x. 28. 3sin2x-sin 6 x = 32 sin3 x cos 3 x. 29. tan 3 x - tan x = 2 sin x sec 3 x. 30. tan A + tan (A + 60~) + tan (A - 60~) = 3 tan 3A. 31. sin 3 x = 4 sin x sin (60~ + x) sin (60~ - x); deduce sin 20~ sin 40~ sin 60~ sin 80~ = 3 -16 By a mere change of notation of (14) and (15), prove that X X X X 32. sinx=2sin- cos ^33. cos = 1-2 sin2 2 2 3 34. sin6x=2sin3xcos3x. 35. cos8x = 2cos24x - 1. 36.* Express sin 5A in terms of sin A. 37.* Express cos 5A in terms of cos A. 38.* Why is cos 3-18~ = sin 2.18~? Use (14) and Ex. 4, cancel the factor cos 18~, and then replace cos 2 18~ by 1 - sin 2 18~. Solve the resulting quadratic equation for sin 18~. Hence obtain a construction of the regular pentagon. 39.* Express sin 3A + cos 3A in terms of d = cos A - sin A. 40.* The angle of elevation of the top of a tree from a point 30 yards from its foot is three times as great as the angle of elevation from a point 150 yards from its foot. Express the height in terms of a square root. 102. Trajectories. Practical applications of functions of the double angle are furnished by problems concerning the trajectory (or path of flight) of a projectile (or bullet) fired at a velocity of v feet per second from a gun having a as its angle of elevation. Ch. X] FUNCTIONS OF SEVERAL ANGLES 143 Fig. 80 shows the trajectory when v = 2000, a = 25~. In the plane of the trajectory, take the gun as the origin 0, the horizontal line through 0 as the x-axis, and the vertical line through 0 as the y-axis. The component of the velocity v in the direction of +V P 0 f O t h ~. 3 Xi, +3 FIG. 80 the x-axis (Art. 21) is v cos a. Hence if the projectile reaches the point P after t seconds, the abscissa of P is equal to t(v cos a) feet. The component of the velocity v in the direction of the y-axis is v sin a. But under the action of gravity, a body, starting at rest, falls in t seconds a distance of ~ gt 2 feet, where g is approximately equal to 32. Hence the coordinates of P are (19) x = vt cos a, y = vt sin a - gt 2. To find the horizontal range r =OA, we seek the value r of x for y = 0, when the projectile P has reached the point A in the x-axis. From y = 0, we get v sin a = gt. Inserting the resulting value of t into the first equation (19), we see that the value of x is V2 V2 r= -(2 sin a cos a) =-sin 2a. g g EXERCISES ON TRAJECTORIES 1. Draw the graph of the trajectory when a = 30~ and v = 2000 ft. per second, by assigning various convenient values to t. Measure the horizontal range. 2. For a and v as in Ex. 1, compute the horizontal range. 3. Given v = 1600, r = 61280, find the two angles a. 4. What is the maximum range for a given velocity? 144 TRIGONOMETRY [Art. 103 5. An athlete puts (throws) the shot from a position 6 ft. above the ground, at an angle of elevation of 45~. If the horizontal range is 48 ft., find the velocity. Hint: Use formulas (19) with y= -6. 6. Given v = 2000, x = 5000, y = 8260, find t and a. 103. Functions of half angles. We shall write formulas (15) in other forms, which the student need not memorize or make use of in the exercises below. In (15) take 2A = x; we get cos x = 1 - 2sin2 2 cos x = 2 cos2 1. 2' 2 Transposing terms, dividing by 2, and extracting square roots, we obtain x I - cos x x 1+cosx (20) sin=i- cos ^= -- - 2 2 2 2 When x is given, these equations enable us to find the sine and cosine of ~x. If 0<x <180~, 2x is an acute angle and its functions are positive; hence the + sign must be chosen in both equations (20). If 360~ <x <540~, 1x is in the third quadrant and its sine and cosine are negative; hence the - sign must be chosen in both equations (20). The signs always depend upon the quadrant of 1x. From (20) we get, by division, - 1-COS x x -+cos x (21) tan = - i cot = -=an2 1+cos X 2 1-cos X In each the sign is determined by the quadrant of ~x. When x is an acute angle, the first formula (21) was proved geometrically in Art. 35. EXAMPLE. Given cos 45~=0 2, find the sine and cosine of 22~. Solution. By (15), 2=-cos 45~=1-2 sin2 22 =2cos2 22 —1. Transposing terms and dividing by 2, we get *2 1 -i-2 _2-V2 2-2 2d+-V2 sin2 220 - --- cos2 2210 -2- 2 - 'cs2 2=.4 Ch. X] FUNCTIONS OF SEVERAL ANGLES 145 Since 22~~ is an acute angle, its functions are positive and sin 22~~ =2 /2 —2, cos 22~~=4\2+/ 2. EXERCISES ON FUNCTIONS OF HALF ANGLES 1. Given cos 30~= =-3, find sin 15~, cos 15~, tan 15~ in terms of square roots. 2. Given cos 150~= -= 33, find sin 75~, cos 75~, tan 75~ exactly. 3. Find the sine and cosine of 72~. Prove the identities x 1-cos x x 1+ cos x 4. tanX - os 5. cot-=+cs 2 sin x 2 sin x 6. 1 + tan x tan I x = sec x. 7. csc x - cot x = tan x. 8. sec A + tan A = tan (45~ + I A). 9. 1 + cot A cot l A = csc A cot ~ A. cos x l+tan x_ lq+sin x+-cos x 10.. cot I x.. 1-sin x 1-tanx 1l-sin x-cos x 12. tan A + 2sin2 A cotA = sin A. 13. A balloon rose vertically at a point whose horizontal distance is 2400 yards from an observer who found the angle of elevation to be 15~ when he first sighted the balloon, and 30~ at a later time. Find an exact expression for the distance the balloon rose between the two observations. 14.* From the values of 1 + cos A and 1 - cos A in Art. 91, prove that A- s(s-a) s - /A (s-b)(s -c) t (s-b)(s-c) cos~A== bc sin~A= - - C, tan~A =C)- be s be tbc s (s - a) where s = ~ (a + b + c). 15.* Express the six functions of x in terms of t = tan I x. 16.* Given the functions of 18~ (Ex. 38 of Art. 101), how can we find the functions of 12~, 6~, 3~? 104. Sum or difference of two sines or two cosines expressed as a product.' We employ formulas (1) and (9), viz., sin (A + B) = sin A cos B + cos A sin B, sin (A - B) = sin A cos B - cos A sin B. 1 The instructor who omits Art. 104 should omit Example 2 and Exercises 23-30 of Art. 105. 146 TRIGONOMETRY [Art. 104 By addition and subtraction, we get sin (A + B) + sin (A - B) = 2 sin A cos B, sin (A + B) - sin (A -B) = 2 cos A sin B. Write x for A + B and y for A - B. Then A = 2 (x +y), B (x-y), and the preceding two equations become (22) sin x + sin y = 2sin (x + y) cos (x - y), (23) sin x - sin y = 2 cos 1 (x + y) sin ( - y). If we start with the expansions (8) and (10) of cos (A + B) and cos (A - B), and proceed in the same manner, we will get (24) cos x + cos y = 2 cos (x + y) cos ~ (x - y), (25) cos x - cos y = - 2 sin I (x + y) sin (x — y). EXERCISES ON EXPRESSING SUMS AS PRODUCTS 1. Carry out the indicated steps leading to (24) and (25). Express as products the following sums or differences: 2. sin x + sin 3 x. 3. cos x - cos 3 x. 4. sin 7C + sin 3C. 5. cos 5 B + cos9B. 6. sin 5 A + cos 3 A = sin 5 A + sin (90 - 3 A). Prove the. identities sin 6A +sin 4A. sin 7A-sin 5A 7. tan 5A. 8. tan A. cos6A +cos4A cos7A +cos5A sinA+sinB tan 1(A+B) 10 sin A+sin B tan (A+B). sin A - sin B tan 2(A-B) cosA+cos B 11. sin sin s + 3 sin in 5A + sin 7A = 4 cos A cos 2A sin 4A. 12. sin2x + sin4x +sin 6x = 4cosxcos2xsin 3 x. 13. sin x - sin 2 x + sin 3 x = 4 sin 2 x cos x cos 2 x. 14. sin x+sin y+sin z-sin (x —y+-z)=4 sin sin sin -- 15. Given a/b = s/t we see on adding or subtracting unity from each fraction that a+b s+t a-b s-t a+b s+t b t 'b t ' a-b s-t' Ch. X] FUNCTIONS OF SEVERAL ANGLES 147 the third equation following by division. Applying this principle of "composition and division" to the law of sines a sin A b sin B' we get a+b sinA+sinB a-b sin A-sin B Replacing the second member by its value in Ex. 9, we have the law of tangents (Art. 88). If A, B, C are angles of a triangle, prove that 16.* sinA + sinB + sin C = 4cos I A cos I Bcos C. 17.* sinA + sinB - sin C = 4sin A sin Bcos C. 18.* cos A + cosB + cosC = + 4sin A sin Bsin C. 19.* cos A - cos B + cosC = 4cos i Asin B cos C. 20.* sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C. 21.* sin 4A + sin 4B + sin 4C = - 4 sin 2A sin 2B sin 2C. 105. Trigonometric equations. Unlike an identity, an equation is true only for special values. The exercises below include only equations which can be solved by methods wholly similar to those employed in the following four illustrative examples. More difficult equations are solved graphically in Arts. 109, 112. EXAMPLE 1. Find the positive angles <360~ for which 2 sin2x + 3 cos x = 2. Solution. Replacing sin2 x by 1 - cos2 x, we get 3 cos x - 2 cos2 x = 0. Hence cos x = 0 or 3/2. The second value must be discarded since cos x cannot exceed 1 numerically. Thus cos x = 0, x = 90~ or 270~. EXAMPLE 2. Find the positive angles <360~ for which cos x + cos 2x + cos 3x = 0. Solution. By formula (24), cos 3x + cos x = 2 cos 2x cos x. Hence the proposed equation becomes cos 2x (2 cos x + 1) = 0. Hence either cos 2x = 0 or 2 cos x = -1. The positive solutions <360~ of the latter are x = 120~, 240~. Next, cos 2x = 0 only when 2x is 90~ or 270~ or 148 TRIGONOMETRY [Art. 105 one of these angles increased by a multiple of 360~, whence x is 45~ or 135~ or one of these angles increased by a multiple of 180~. But if we add 180~ more than once, we obtain an angle > 360~. Hence the answers are 45~, 135~, 225~, 315~, together with the former angles 120~, 240~. EXAMPLE 3. Find the positive angles <360~ for which tan2x = 2 sin x. Solution. By formula (16), 2 tan x 2 sin x tan 2x =_ tan2 x - cos x (1-tan2 x) Thus the last fraction shall be equal to 2 sin x. This is evidently true if sin x = 0, whence x = 0~ or 180~. Next, if the common factor 2 sin x is not zero, it may be cancelled, and we obtain =1, sec x = 1 -tan2 x. cos x (1-tan2 x) We recall the identity sec 2 x = 1 + tan 2 x. Adding, we get sec 2 x + sec x = 2. Solving this as a quadratic equation for sec x, we obtain sec x = 1 or - 2. Hence cos x=l or -1/2, and the positive angles <360~ are 0~, 120~, 240~. These with the former angle 180~ give the four answers. EXAMPLE 4. Find the positive angles < 360~ for which 20 cos y - 15 sin y = 12. Solution. The most obvious method of solution is to transpose 15 sin y, square each of the new members, replace cos 2 y by 1 - sin 2 y, and solve the resulting quadratic equation for sin y. But only part of the angles found in this manner will satisfy the given equation. For, when we square each member of an equation x = a, we obtain an equation x 2 = a 2 having a solution x = - a which does not satisfy the former equation x = a. To give a solution which shall not introduce extraneous values, divide each member by -202+152 = 25. We get 4 3. 12 cos y - sin y=-2. 5 cos 5 sln 25 We can find an angle x such that sin x = 4/5, cos x = 3/5. By Table I, x 53~8', nearly. Hence our equation becomes sin (x - y) = 0.48. By the same table, 53~8' - y is equal to 28041' or 180~ - 28041', or one of these angles increased by a multiple of 360~. Hence the positive values <360~ of y are 24027' and 261049'. Ch. X] FUNCTIONS OF SEVERAL ANGLES 149 EXERCISES ON TRIGONOMETRIC EQUATIONS Find all the positive angles < 360~ which satisfy the following equations, excluding angles for which any of the functions are undefined: 1. 2 sin2 x + 3 cosx = 0. 2. 16 sin2 x + 24 cos x = 9. 3. 2 cos2 x + 5 sinx = 4. 4. 2 tanx sinx = 3. 5. cosx + cos 2 x =-1. 6. cosx + secx = 5. 7. tan A + cot A = 2. 8. sin 3A = cos 2A [= sin (90~-2A)]. 9. tan2y + cot2y = 2. 10. sec2 y + csc2y = 4. 11. sin 2 x = 2 sin x. 12. tan (45~ - x) + tan (45~ + x) = 4. 13. sin 4 x + cos 4 x = 1. 14. tan x + tan 2x = tan 3 x [ = tan (x + 2 x)]. 15. tan x sec x = - 2. 16. sin 2 x cos x = sin x. 17. 4 tan B- cot B = 3. 18. tan B - cot B = cot 2B. 19. tan 2 0 = tan 0. 20. sc 0 - cot 0 = sin I 0. 21. 6tan 2 0- 4sin 2 0 =1. 22. csc = 1 + cot0. By converting one or both members into a product (Art. 104), solve 23. sin 3 x + sin x = - sin 2 x. 24. sin 2 x-sinx = cos 2 x -cosx. 25. cos x - cos 3 x = sin 2 x. 26. sin x + sin 3 x = cos x - cos 3 x. 27. sin 5A + sin 3A = cos A. 28. cos 7A + cos A = - cos 5A - cos 3A. 29. sin 4A - sin 2A = cos 3A. 30. sin (A + 120~) + sin (A + 60~) = 1.5. By the second method of Example 4, solve 31. sin x + cos x = /2. 33. 2 cos x - sinx = 1. 32. 5 sin x + 12 cos x = 6.5. 34. 5sin x + 2 cosx = 5. CHAPTER XI GRAPHS OF THE TRIGONOMETRIC FUNCTIONS AND THEIR INVERSES, RADIANS 106. Line representations of the trigonometric functions. Consider any angle A in its trigonometric position and recall the definitions (Art. 83) that sin A = y/r, cos A = x/r, etc., where x and y are the coordinates of any point P of the terminal side of angle A, while r is the distance from P to the origin 0. Since r is the denominator of the fractions defining sin A and cos A, take r = 1 for convenience, i.e., take OP as the unit of length of our drawings. Then sin A is represented to scale by the ordinate y (PF in Figs. 81, 82; -PF in Figs. 83, 84) and cos A is repref cot T cot Y T A cos X cos 0 F~X F o T FIG. 81 FIG. 82 sented by the abscissa x (OF in Figs. 81, 84; -OF in Figs. 82, 83). Hence the sine of an angle in its trigonometric position is equal, in magnitude and sign, to the ordinate of the point P in which the terminal side of the angle intersects the unit circle. The cosine is equal to the abscissa of P. Next, consider tan A = y/x, where x 3 0. First, let A be in the first or fourth quadrant, so that the abscissa x is positive. 150 Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 151 Since it is convenient to have our present denominator x equal to unity, we choose the point T = (x, y) on the terminal side of angle A such that its abscissa x shall be OX, the unit of length in our figures. Then the corresponding ordinate y (TX in Fig. 81, -TX in Fig. 84) represents to scale tan A = y/1. Second, let A be in the second or third quadrant, so that the abscissa is negative. We write tan A = y/x in the form tan A = (-y)/ (-x) and choose the point T = (-x,-y) on the prolongation of the terminal side of A such that its positive abscissa - x shall be OX, the unit of length. Then the corresponding ordinate -y (-TX in Fig. 82, TX in Fig. 83) represents tan A = (- y)/1. Hence, the tangent of an angle in its trigonometric position is equal, in magnitude and sign, to the ordinate of the point T in which the terminal side of the angle, prolonged if necessary, Y cot _ -T o. T o X'<0co.F X T FIG. 83 FIG. 84 intersects the line tangent to the unit circle at the point X on the initial side of the angle. Similarly, the cotangent of an angle in its trigonometric position is equal, in magnitude and sign, to the abscissa of the point T' in which the terminal side of the angle, prolonged if necessary, intersects the line tangent to the unit circle at the point Y on the positive y-axis. 152 TRIGONOMETRY [Art. 107 Y 91200 ~ _ sine curve 150~/ '\30~ xtl — 0 ___ Pto o 0_ 0\ 210o240"270'000o330~3600 X 180~ '_360 00 300 600 90~10 10~ 10 210 1 o_ A 27 0 - FIG. 85. SINE AND COSINE CURVES (ART. 107) 00 300 If angle A is in the first or fourth quadrant (Figs. 81, 84), sec A =r/x is represented by the radius vector 60-7 OT to the point in the terminal side of A whose ab90o / scissa x is OX = 1, If A is in the second or third quadrant (Figs. 82, 83), sec A is equal to the negative 120, of the ratio of the length of OT to OX = 1; since T 1.5 1, is on the prolongation of the terminal side of A, it is _. customary to regard OT as negative, while lines in 180 the direction OP along the terminal side of A are l f D m ~ positive. Hence the secant of an angle in its trigonometric position is equal, in magnitude and sign, to the \ _ ) directed line OT from the origin to the point T in which \ 7 f the terminal side of the angle, prolonged if necessary, intersects the line tangent to the unit circle at the point X 3000 > on the initial side of the angle, prpvided OT is counted positive or negative according as T is on the terminal side 3300 or on its prolongation. 3600 Similarly, the cosecant of an angle in its trigonometric position is equal, in magnitude and sign, to the directed line OT' from the origin to the point T' in which the Z terminal side, prolonged if necessary (and then OT' is counted negative), intersects the line tangent to the unit circle at the point Y on the positive y-axis. 107. The sine and cosine curves. These curves may be drawn rapidly by means of the line representations just discussed. Draw a circle whose radius is any convenient length representing unity. Divide the circumference of the circle into 12 equal parts and label the points of division 0~, 30~, 60~,..., 330~ (Fig. 85). Through Ch. XIn GRAPHS, RADIANS, INVERSE FUNCTIONS 153 them draw lines parallel to the horizontal diameter X'X. Choose any convenient segment OA of X'X to represent 360~, subdivide it into 12 equal parts, label the points of division 30~, 60~,.., 330~, and draw vertical lines through each to intersect the horizontal line through the point on the circumference with the same label. The points of intersection are points on the sine curve. The abscissas of the points of the circumference marked 0~, 30~,.., 360~ are the line representations of cos 0~, cos 30~,.., cos 360~. In the lower part of Fig. 85, they have been transferred (by means of parallels to YZ) into the positions of the horizontal segments extending from equally spaced points on YZ, labeled 0~, 30~,.., 360~. If we now rotate the page of the book until YZ becomes horizontal, with Z to the right of Y, we have a part of the cosine curve in its conventional position (Fig. 86). g- 90 0~ 90 i180_ 2700~ 60~ 450 FIG. 86. THE COSINE CURVE Y=COS X Since sin (360~ + x) = sin x, the part of the sine curve y = sin x from 360~ to 720~ is an exact copy of the part from 0~ to 360~ which is shown at the right of Fig. 85. Similarly for the part from - 360~ to 0~ (Fig. 87, which is drawn on different scales from those used in Fig. 85). - 360\ -18018 soO 6o -- 20 FIG. 87. THE SINE CURVE Y=SIN X 108. The tangent curve. Using the line representation of the tangent function (Art. 106), we obtain the following construction of part of the tangent curve (Fig. 88): 154 TRIGONOMETRY [Art. 108 I I l l 1 tso OI I 1800.. 0 A\^y\> 0~ 300 60 ~ 90 80~210~ 27g \ '8 FIG. 88. THE TANGENT CURVE y=TAN X When the arc of the circumference increases from 0~ toward 900, the representation of the tangent function by the intercept on the line tangent to the circle increases without limit. The corresponding part I of the tangent curve extends upward indefinitely, approaching nearer and nearer the vertical line through the point marked 90~, but always remaining to the left of it. When the angle decreases from 180~ toward 90~, its tangent decreases from zero, through negative values, without limit. The corresponding part II of the tangent curve extends downward indefinitely, approaching nearer and nearer the vertical line through the point marked 90~, but always remaining to the right of it. We now see graphically why the tangent function is undefined at 90~, 270~, or any odd multiple of 90~. Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 155 If we elsewhere meet the statement that tan 90~ is positive or negative infinity, written symbolically tan 90~ = + oo, we should not understand this to define or give a value to tan 90~, but merely to mean that the tangent function increases or decreases without limit when the angle increases or decreases toward 90~, respectively. It is simpler in the case of the square of the tangent, for which we may appropriately write tan2 90~ = + co. EXERCISES ON GRAPHS OF TRIGONOMETRIC FUNCTIONS 1. By means of a circle and a line tangent to it at its highest point, construct the graphs of y = cot x and y = csc x. For the latter, transfer by means of a pair of compasses the lines OT' of Figs. 81-84, into vertical lines starting at equally spaced points marked 0~, 30~,..., 360~ on a horizontal line. While we may proceed similarly for y = cot x, we may also 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 i I I I I I II I i I I I I I.I 0~ 901\ ]o,0~ \ I 1 I I i I.! ~ 270 j\ 3b5' I I I I I I I \ I I 11 I0~ I I I I I I I Ono I J00 270~.3( I I I I I I I I I I I I III i I I I II FIG. 89. COTANGENT CURVE FIG. 90. COSECANT CURVE by the method used for y = cos x draw it rapidly in a rotated position, in Art. 107. 2. Draw the secant curve and note that it may be derived from Fig. 90 by taking 90~ from each angle, i.e., replacing the labels 90~, 180~,..., by 0~, 90,.... 156 TRIGONOMETRY [Art. 109 3. In view of the preceding graphs, state the angles for which (a) cotangent is undefined, (b) cosecant, (c) secant. 4. Explain by means of the identity sin (A + 90~) = cos A why the cosine curve may be obtained from the sine curve by replacing the labels 0~, 90~, 180~,... on the latter by - 90, 0~, 90~,.... 5. Since tan (A + 90~) = - cot A, how may we deduce the cotangent curve from the tangent curve? 6. State the identity which proves the second part of Ex. 2. 7. What property of the successive arches of the sine curve in Fig. 87 follows from the identity sin (180~ + A) = - sin A? 109o Graphical solution of trigonometric equations; harmonic curves. If a, b, c,... are constants, the graph of y=a sin cos sin cos sin 2 x+ d cos 2 x+ -* * is called a harmonic curve. Such curves occur frequently in physics. By means of the curve, we can solve the equation obtained by assigning to y any given value not too large numerically. This method of solving trigonometric equations is available when the special algebraic devices of Art. 105 fail. EXAMPLE. Plot the harmonic curve whose equation is (1) y = 2 sin x + sin 2 x, and find the positive angles <360~ for which sin x + si n 2 x = 0.64. Solution. First, we reproduce from Fig. 85 the graph of y = sin x. Then, by bisecting the ordinates of several of its points, we obtain points on the graph of yl = 4 sin x (Fig. 91). To find points on y2 = sin 2 x, take as the /f \ y=sin x y= sin 2 /00~ \= sin 2X / ' 360~ FIG. 91 Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 157 ordinate of x = 30~ the ordinate sin 60~ of x = 60~ in the graph of y = sin x, etc. Corresponding to each value of x, the ordinate y of (1) is equal to yl + Y2, or the algebraic sum of the ordinates of the two dotted curves. For example, when x = 135~, yl = AP1 is positive and y2 = - AP2 is negative and numerically larger than yl; hence we lay off PP2 equal to AP1, upward from P2, and obtain y = yl + Y2 = -AP, measured downward. The resulting graph of (1) is shown as a heavy curve in Fig. 91. Next, we draw a line parallel to the x-axis at a distance 0.64 above it. This line is seen to cross the heavy curve at four points for which x is approximately 15~, 86~, 213~, 225~. More exact values will be found by interpolation, using Tables I and II. For x = 15~ and x = 15~30', the values of (1) are 0.6294 and 0.6486, whose difference is 0.0192. The desired value 0.64 exceeds the lesser by 0.0106. Then 106 192 X30' 62', x =15~1 6'. When x is about 86~, set x = 90~ - A. Then y = 2 cos A + sin 2A. For A = 4, we get y = 0.6380. Since this is nearly the desired value 0.64, try A = 4~5', which gives y = 0.6409. Since 20/29 of 5' is 34', A = 403/', whence x = 85056~'. For the large angles x, set x = 180~ + B. Then y = sin 2B - 4 sin B. For B = 45~ and 46~, y equals 0.6464 and 0.6397. Since 64/67 of 60' is 57', B = 45~57', x = 225~57'. Similarly, the fourth answer is x = 212~49'. EXERCISES ON THE GRAPHICAL SOLUTION OF EQUATIONS Solve graphically and by subsequent interpolations 1. sin x + sin 2 x = 0.7. 2. sin x + sin 2 x = 1.2. 3. 2sinx + sin 3 x = 0.8. 4. 2sinx - sin x = 0.4. 110. The radian unit of angle. When we plotted the sine curve in Fig. 85, we were at liberty to choose any segment OA of the x-axis to represent the angle 360~, and any segment of the y-axis to represent sin 90~ or the radius of the circle which we employed to secure the line representations of the sines of angles. A different choice of these two arbitrary units of measurements would have led to a different appearance of the graph. For the sake of simplicity and uniformity, it is desirable to have the same unit of 158 TRIGONOMETRY [Art. 110 length for the x's as for the y's. In particular, any two sine curves would then have the same shape, although one might be a magnification of the other (as in Figs. 85, 87), just as is true of any two circles. How shall we secure this desirable state of affairs? We recall that the y's (or sines) were represented in Fig. 85 by vertical lines measured with respect to the radius of the circle as the unit of length. Since angles at the center are proportional to the arcs intercepted on the circumference, and since these arcs are measured naturally in terms of the radius as the unit of length, we shall obtain the desired goal of having the same unit of length for the x's (angles) as for the y's (sines), if we agree that an angle x has the same measure as its intercepted arc. The resulting unit of angle, called a radian, is therefore an angle at the center of a circle which subtends an arc equal to the radius. y ^ D /eL \ Thus, in Fig. 92, if arc AB E. r /; \ \R equals the radius r, angle E, i / \ \ AOB is a radian. Let DE be any second arc of the same s y circumference. Since angles at O' r A ~ -- - -> the center are proportional to FIG. 92 their subtended arcs, zDOE arc DE arc DE arc D'E' ZBOA arc AB r - R or length of subtended arc number of radians in angle at center=.length of su dd r length of radius and this ratio is independent of the particular radius chosen. If Z DOE contains n radians, the first fraction is equal to n, so that arc DE is of length nr. An angle of n radians at the center of a circle of radius r intercepts on the circumference an arc of length nr. Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 159 Since the length of the semicircumference is 7rr, where = 3.1415926 +, we have (2) t radians = 180~. Thus one radian is equal to 180/7 = 57.2958 - degrees, or approximately 57017'45". The student should memorize formula (2). REDUCTION OF DEGREES, MINUTES, AND SECONDS TO RADIANS Radians / Radians " Radians 1 0.0174533 1 0.0002909 1 0.0000048 2 0.0349066 2 0.0005818 2 0.0000097 3 0.0523599 3 0.0008727 3 0.0000145 4 0.0698132 4 0.0011636 4 0.0000194 5 0.0872665 5 0.0014544 5 0.0000242 6 0.1047198 6 0.0017453 6 0.0000291 7 0.1221731 7 0.0020362 7 0.0000339 8 0.1396263 8 0.0023271 8 0.0000388 9 0.1570796 9 0.0026180 9 0.0000436 EXAMPLE. Find the number of radians in 73~36'40". Solution. From the table we see that there are 1.221731 radians in 70~ (by shifting decimal point for 7~).052360 " " 3~.008727 " " 30' ( " " " " i 3').001745 " " 6'.000194 " " 40" ( " " " (" " 4") 1.284757 " " 73036'40". Although not so stated at the time, the graphs (Figs. 85-90) of the trigonometric functions were actually drawn so that the units of lengths on the two axes are the same, since the length on the x-axis which represents 360~ was chosen equal to 2r times the length of the radius of the auxiliary circle. EXERCISES ON RADIANS 1. Find the number of radians in 45~, 60~, -270~. 2. Express in degrees G/6, — r /4, 1.570796 radians. 160 TRIGONOMETRY [Art. 110 3. Copy Figs. 86 and 87, but label in terms of radians each angle which is a multiple of 90~. 4. Using Fig. 89, sketch the cotangent curve for angles from -27r to + 3w. 5. Find the number of radians in an angle at the center of a circle of radius 50 feet which intercepts an arc of 75 feet. 6. Find the length of an arc subtending an angle of 2.5 radians at the center of a circle of radius 50 feet. 7. Find the length of the radius of a circle at whose center an angle of 2.1 radians is subtended by an arc 42 feet long. 8. Find the length of an arc of 70~ on a circle of 9 ft. radius, using 7r = 22/7. 9. If the angle of a sector of a circle of radius r contains n radians, the area of the sector is equal to the product ~nr2 of n / (2 r) by the area r r2 of the circle. Subtracting the area 2r2 sin n of the triangle two of whose sides are radii and included angle is n (Art. 86), we obtain I (n - sin n) r2 as the area of the segment of the circle. Compute the area of a circular segment of radius 10 feet whose arc is 40~. 3w 5w w 7w 10. Prove that cos + cos ~ +2 cos -7 cos -= 0. 11 11 11 11 Express in terms of radians all solutions of the following equations: 11. cos x + tan x = sec x. 12. cot x - csc 2 x = 1. 13. tan 2 x tan 3 x =1. 14. In field artillery, a mil is the angle subtended by an arc equal to 1/6400 of the circumference. Show that a mil is approximately onethousandth of a radian. 15.* An endless band passes around two wheels the diameter of one of which equals the circumference of the other. When the larger wheel makes one complete revolution, what is the number of radians in the angle described by the radius of the smaller wheel? 16.* Two straight railroad tracks intersect at P making an angle of 55~. They are to be connected by a curved track, tangent to each, and forming a circular arc of radius 300 feet. Find the length of the curved track and the distance from P to the point of tangency. 17.* Three circles, whose radii are proportional to 1, 3, 5, are tangent externally. The points of tangency are the vertices of a curvilinear triangle whose area is 10 square inches. Find the radii. Ch. XI] GRAPHS. RADIANS, INVERSE FUNCTIONS 161 18.* A railroad curve forms a circular arc of 19~13.4', the radius to the center line of the track being 1680 feet. If the gauge is 5 feet, what is the difference in length of the two rails? 111. Approximate values of sines and tangents of small angles. Let n be the number of radians in an acute angle ACT (Fig. 93). With C as a center and any T radius r = CA, describe the arc AB. B Let the tangent at A meet CB produced at T. The area of A TAC is 2 TA CA = r2 tan n. The area of ACAB is ~r2 sinn, and the area of sector CAB is ~ r2n (Ex. C A 9, Art. 110). But FIG. 93 ACAB < sector CAB < A TAC. Hence r2 sin n < ~ r2n < r 2 tann, sin n < n < tan n. Dividing by the positive number sin n or tan n, we get n 1 n 1< n <-, cos n< <1. sin n cos n tan n But cos n approaches 1 when n approaches zero. Hence n/sin n and n/tan n each approaches 1 as n approaches zero, since each lies between 1 and a number which approaches 1. In other words, the sine and the tangent of a very small angle are each approximately equal to the number of radians in the angle. For angles <2~, the approximation is sufficiently exact to 5 decimal places, as is clear from the following values: A = 1~30' = 0.02618 radians, sin A = 0.02618, tan A = 0.02619, B = 2~ = 0.03491 radians, sin B = 0.03490, tan B = 0.03492. EXAMPLE 1. Find the inclination A of a railroad track to the horizontal if its grade is 1 per cent (i.e., rises 1 foot per 100 feet). Solution. Since tan A = 0.01, A contains approximately 0.01 radians. Since 1 radian is equal to 57.296 degrees, A = 0.573~ = 34.4'. 162 TRIGONOMETRY [Art. 112 EXAMPLE 2. A bolt 1 inch in diameter has 10 threads to the inch. Find the inclination A of the thread to a cross section of the bolt. Solution. If a small part of the cylindrical surface of the bolt were flattened into a plane (as when a roll of paper is unrolled), a thread would appear as a straight line. The length of one thread is approximately 7r inches. The length of the 10 threads is 107r inches. Hence we have a plane right triangle with the hypotenuse 10 7r inches, a leg 1 inch (height of ten threads) and opposite angle A. Thus sin A = 1/(10 7r), which is the approximate number of radians in A. Its product 18/7r2 by 180/7r is the number N of degrees. log 18 = 1.2553 log r = 0.49715 log r2 = 0.9943 log N = 0.2610 N = 1.824~ = 1~49'. EXERCISES ON SMALL ANGLES 1. How high is a tower which subtends 1~ at a distance of 2 miles? 2. What is the per cent grade of a railroad track which is inclined 1~ to the horizontal? 3. A bolt 2 inches in diameter has 6 threads to the inch. Find the inclination of the thread to a cross section of the bolt. 4. The diameter of the moon subtends the angle 31.5' at the earth, and the moon is 239200 miles from the earth. Find the moon's diameter. 5. How high is a tower which subtends 40' at a distance of 800 feet? 6. How far from the eye must a coin, one inch in diameter, be placed so as to just hide the moon, if its diameter subtends the angle 31'5"? 7. The sun is about 92 million miles from the earth and subtends the angle 32' at the earth. Find the sun's diameter. 112. Equations involving both an angle and its trigonometric functions. In any such equation it is understood that the angle is measured in radians. Thus in Fig. 94, the unit of length is the same on the x-axis as on the y-axis. EXAMPLE 1. Find the positive angles <27r satisfying 1 * 7r x - ~ sin x = -4 which is a case of Kepler's equation in astronomy. Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 163 Solution. The required angles x are the abscissas of the points of intersection of y = sin x and y = 2(x -. The latter represents a straight line, two of whose points are A-(, )0andB= (, j. B The abscissa xi of the point / of intersection is just less than 5w/12 or 75 ~./ A more exact value of xl can be found by Table II and a I table (Art. 110) of the radians in 1~,.., 9~. We shall compute 0 /A7 the values of 7r. TFIG. 94 F = x — sinx - for x = 72~ and x = 73~, and find that these values have opposite signs; hence F is zero for a value xl between 72~ and 73~. sin 72~ = 0.9511 sin 73~ = 0.9563 ~ sin 72~ = 0.4755 2 sin 73~ = 0.4781 = 0.7854 = 0.7854 _ 1.2609 1.2635 72~ = 1.2566 radians 73~ = 1.2741 radians F = - 0.0043 F = + 0.0106 The difference of the two F's is 0.0149. Hence xl exceeds 72~ by 43 49 X 60' = 17'; xl = 72~17', approximately. EXAMPLE 2. The area of a segment of a circle of radius 5 feet is 28.56 square feet. Find the length of the chord. Solution. Let x be the number of radians in the angle at the center subtended by the chord. By Ex. 9, Art. 110, 25 2- (x - sin x) = 28.56, x - sin x = 2.2848. Thus x is the abscissa of the point of intersection of y = sin x and y = x 2.2848, whose graphs are shown in Fig. 94, the straight line being determined by the points having x = 0 and x = r. As in Ex. 1, we get x = 15504', nearly. The length of the chord is 10 sin i x = 9.764 ft. 164 TRIGONOMETRY [Art. 113 EXERCISES ON TRIGONOMETRIC EQUATIONS 1. The area of a segment of a circle of radius 10 feet is 80 square feet. Find the length of the chord. Find the angle at the center of a circle subtended by a chord which cuts off a segment the ratio of whose area to the circle is 2. 1/8. 3. 1/4. 4. 3/8. Find the positive angles <27r for which 5. x - sin x= 6. x - sin x = 7. x - r sin x = 2. 2 3 113. The inverse trigonometric functions. With most operations in mathematics there are associated inverse operations. For example, the direct operation may be the finding of the square y = 2 of a given number x; the inverse operation then consists in finding a number x whose square is equal to a given number y, and there are two answers denoted by x = -= V-y. Again, we have often employed the direct operation of finding the cosine of a given angle; for example, finding the value - of the cosine of 60~. Now the inverse operation consists in deducing an angle 60~ (among the possible answers) when the value 2 of its cosine is given, and we shall employ either of the symbols cos-1 4 and arc cos 4 to denote any of the angles n * 360~ - 60~ whose cosine is i. The first symbol cos-1 is read inverse cosine of 4, and must be carefully distinguished from the reciprocal 1/(cos 4) of cos 4, which is properly written (cos ~)-1. Thus the part -1 of our symbol is not an exponent. This possible, but only apparent, confusion is avoided by the use of our second symbol arc cos 2 which is read arc whose cosine is 4. A simple illustration will make clear that we are not using the term inverse in the sense of reciprocal: let the direct operation be that of walking 4 miles north; then the inverse operation is that of walking 4 miles south, and not that of walking 4 of a mile north. Similarly, either of the symbols tan-l 1 and arc tan 1 denotes any of the angles nr + 7r/4 (in radians) whose tangent is 1. Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 165 Again, sin-1 ~ or arc sin ~ denotes any one of the angles n 360~ + 30~ and n 360~ + 150~ whose sine is 2. The nature of these inverse functions becomes clear from their graphs. To construct the graph of arc sin x, equate it to a variable y and plot the graph of the equation (3) y = arc sin x. Since this equation has the same meaning as x = sin y, we may plot the graph of the latter instead of (3). To do this, start with the graph (Fig. 87) of y = sin x, and rotate the plane of the paper 90~ counter-clockwise; then the former positive x-axis becomes the new positive y-axis, while the former positive y-axis becomes the new negative x-axis. It remains therefore to reverse the direction of measuring the new abscissas, which amounts to revolving the graph in its A I Y new position through 180~ about the ver- I tical axis. We therefore obtain the graph (Fig. 95) of x =sin y and hence of (3). P It lies between the vertical lines through \ the points with the abscissas -1 and + 1, and consists of an infinite number of repe- I titions both upward and downward of I B the portion between y= -r and y = + 7r. Hence for each value of x which is I I admissible (i.e., between - 1 and + 1), _ there is an infinite number of values of a P1 l X y, viz., the ordinates of the points (P, I P', P", etc.) of intersection of the curve I with the line parallel to OY and at a dis- tance x to the right of OY or — x to the left of OY, according as x is positive or negative. I _, Of the points, just mentioned, which FIG. 95 have any given abscissa x (between - 1 and + 1), a single such point lies in the portion BOC (drawn 166 TRIGONOMETRY [Art. 113 heavier) of the curve and that point evidently has an ordinate which is numerically less than the ordinates of all of the remaining points of the curve having the same abscissa. This numerically least angle or arc whose sine is x is called the principal value of arc sin x or sin-1 x and designated by Arc sin x or Sin-l x. By definition it lies between -7r/2 and + r/2. For example, Arc sin (- 1/2) = - 30~ = - 7/6. Similarly, the principal value of an inverse tangent, cotangent, or cosecant is the value between - /2 and + 7r/2, and the first letter in its name is written with a cap/A 7 ital. But the principal value of an inverse cosine 'i\ I or secant is the positive value ~ r. For ^ ~I \example, I S ": Arc cos (-1/2) = 120 = 2 7r/3. The reason for the latter definition becomes clear from the graph (Fig. 96) of y = arc C cos x. The portion BCD of the curve whose - 0 -1l ordinates lie between -7r/2 and + 7r/2 contains only points with positive abscissas. But any given number x, between — 1 and D - + 1, is the abscissa of a point of the portion FIG. 96 ABC (drawn heavier), whose ordinate lies between 0 and 7r. EXAMPLE 1. Prove that sec (2 cot-1 2) = 5/3. Solution. Let A be any angle whose cotangent is 2. We are to find sec 2A, given cot A = 2. Since csc2 A = 1 + cot2 A = 5, we get sin2 A = 1/5 and cos 2A = 1 - 2 sin2 A = 3/5. Hence see 2A = 5/3. EXAMPLE 2. If x and y are numerically <1, prove that sin (Arc sin x + Arc sin y) = x-ll-y2 + yJl-x2. Solution. Write A = Arc sin x, B = Arc sin y, so that A and B are positive or negative acute angles. Since sin A = x, sin B = y, we have Ch. XI] GRAPHS, RADIANS, INVERSE FUNCTIONS 167 cos A = + ll-cX2, cos B = -+ /- y2, sin (A + B) = sin A cos B + cos A sin B = x1_- y2 + y-1- x2. EXERCISES ON INVERSE FUNCTIONS 1. From the graphs (Figs. 88-90) of the tangent, cotangent, cosecant, and secant, derive the graphs of their inverse functions and justify the definition of their principal values. Find the values of 2. sin (Arc tan 1). 3. sec (Arc tan 2). 4. cos (Arc cos 2). 5. tan (2 Cos-1 ). 6. cos (2 tan-1 ~3). 7. tan (Sin1 5). Prove the following formulas (with x and y numerically <1 in Exs. 8, 10, 12, 14 and 15): 2c 8. cos (2 cos1 x) = 2 x2-1. 9. tan (2 tan c) = 2 10. Arc cos x= Arc cos (2 x2-1) if 0 < x < 1. 2n7 11. Arc sin i +Arc sin 1 = 2 -12. Sinl x + Cosl x = 7r. 13. Sin- 1 - Sin-1 2 = Cos-1. _ 14. cos (Sin1 x - Sin'1 y) = xy + 1-x2. V1 -y2. 15. sin (Sin' x - Sin-1 y) = xW/1-y2 - y/1l-x2. 16. Tan'1 x- Tan'1 y = Tan-l -y if x>0, y>0. 1+xy 17. 2 Tan-1 2 = Tan-1 2 18. 3 Sin-1 = Sin1 — i. 1 —C19. 'Sin-1 3 + Sin-l 8 = Sin-1 7 7 20. Cos- Cos-1 2 = Cos'1. 21. Arc tan + Arc tan = 22. Tan'-1 +Tan- = 23. 2 Tan' + Tan'l =- 24. Tan' + Tan- 2= 4~ —~a 2 25.* Tan'1 x + Tan'1 y =n7r + Tan- 1 -xy where n = 1 if x and y are l -xy positive and xy > 1; n -1 if x and y are negative and xy > 1; n = 0 in all remaining cases. 26.* Arc tan 5 + Arc tan (-3)= Arc tan i. Solve and discuss the two equations: 27.* cos-1 x + cos'- (1-x) = cos-' (-x). 28.* tan-L (x + 1) + tan'l (x — 1) = tan'1 s3. LIST AND INDEX OF FORMULAS 1 csc A = in A smnA 1 sec A= -, cos A 1 cot A= - tan A sin A cos A sin2A + cos2 A =1, tanA= si, cot A cosA cos A sin A sec2 A = 1 + tan2 A, csc2 A = 1 + cot2 A (pp. 13, log MN = log M + log N, log N = p log N, logM log M - log N, log N log N (pp. 3 N r In a right triangle with hypotenuse c and leg b adjacent to ZA, tan iA= c- (1 c+ b, 109). 6, 37). p. 51);?. 53). Area = 2ab = ~a2 cot A = ~c2 sin A cos A Navigation (D = distance, C = course, pp. 62, 67, 69, 74): diff. lat. = D cos C, dep. = D sin C, dep. = (diff. longitude) X (cos lat.), diff. long. = (merid. diff. lat.) X tan C. Surveying (D = length of course, B = true bearing, p. 84): lat. =D cos B, dep. = D sin B. (I sin( 90~ —x)= cosx, cos( 90~-x)= sinx, tan( 90-x) = cotx, sin ( 90~+x) = cos x, cos ( 90+x) = -sin x, tan ( 90~+x) =-cot x, sin (180~-x)= sinx, cos (180~-x)= -cosx, tan (180- x)= -tanx, sin (180~+x) = -sinx, cos (180~+x)=-cosx, tan (180~+x)= tanx, sin (360~- x) = sin (- x) = -sin x, sin (270~ = x)= - cos x, cos (360~- x) = cos (- x) = cos x, cos (270~ x)= = sin x, tan (360~- x) = tan (- x) = -tan x, tan (270~ == x) = =F cot x. Analogous formulas for secant, cosecant, cotangent, pp. 17, 111-4, 134-7. Law of sines: Law of cosines: Law of tangents: a b c sin A sin B sin C a2 = b2 + c2- 2bc cos A a - b tan 2 (A-B) _ tan (A-B) a+ b tan (A+B) cot 2 C 169 (p. 115). (p. 126). (p. 117). 170 LIST AND INDEX OF FORMULAS Half-angle formulas (pp. 130-1, 145): tan2A= -, tan2B= r-, tan2C=- r, s- a s-b s-c = T(a+ b + ) r (s-a) (s-b) (s-c) sin 'A= (s-b) (s-c) cos s (s-a tan -A= (s-b) (s-c) b cos= bcs (an-a) bc s(s —a) Area of triangle = 2bc sin A = /s (s-a) (s-b) (s-c) (pp. 115, 129). Addition and subtraction theorems (pp. 133, 136, 138-9): sin (A + B) = sin A cos B + cos A sinB, cos (A + B) = cos A cos B- sin A sin B, sin (A - B) = sin A cos B - cos A sin B, cos (A - B) = cos A cos B + sin A sin B, tan A +- tan B cot A cot B -1 tan (A +B)-=,-t + tn B- cot(A+B)=ctActB-1 1-tanA -tanB cotA+cotB tan (A - B)= ----—, cot (A-B) - cot A cot B-. 1+ tan Atan B cotB-cotA Functions of double angles (pp. 140-1): sin 2A = 2 sin A cos A, cos 2A = 1 - 2 sin2 A, 2 tan A tan 2A = tan A 1 - tan2 A Functions of half angles (p. 144): x 1-cos x sin - = 2 2 x 1= —cos x tan = 1cos 2 1+cos X cos 2A = cos2 A - sin2 A, cos 2A = 2 cos2A -1, cot2 A-1 cot2A = -- 2 cot A x 1 + cos x cos- = = 2 cot2 2 =+ J 2 =I-cos x Sums or differences expressed as products (p. 146): sin x +sin y= 2 sin (x + y) cos 2 (x - y), sin x - sin y= 2 cos 2 (x + y) sin 2 (x - y), cos x+ cos y = 2 cos (x +y) cos (x-y), cos x - cos y = -2 sin 2 (x + y) sin 2 ( - y). Miscellaneous formulas for heights and distances (pp. 55, 57, 120, 137 -8), trajectories (p. 143), multiple angles (p. 141), length of arc of circle (p. 158), vr radians= 180~ (p. 159). INDEX INCLUDING INDEX TO DEFINITIONS Abscissa, 103. Deviation, 76. Absolute error, 3. Direct angle, 100. Acre, 82. Directed line, 2. Addition theorem for cosine, 136. Double meridian dist., 86. for sine, 133. for tangent, cotangent, 138. Equator, 60. Altitude, 115. Error, 23, 51, 52. Angle, general, 105. of closure, 86. in trigonometric position, 106. Extraction of roots, 37, 41. measurement of, 99. negative, 105. Force, 29, 131. of depression, 5, 99. Function, 6. of elevation, 5, 99. of incidence, 31. Grade, 5. of refraction, 32. Graphs, 150. positive, 105. Gunter's chain, 82. quadrant of, 106. Area of field, 83, 86. Half-angle formulas, 130. oblique triangle, 115, 128-9. Harmonic curve, 156. right triangle, 53. Heights and distances, 9, 28, 55, 137. Axis, 104. Hexagon, 54. Azimuth, 100. Horizontal angle, 4, 99. line, 4. Balancing surveys, 84, 101. plane, 4. Bearing, 83, 93, 94. range, 143. reverse, 94. Identities, 16, 110. Casting out nines, 51. Index laws, 34. Chain, 82. of refraction, 32. Characteristic, 39. Interpolation, 19, 27, 63. Co-function, 17. Inverse functions, 164. Collimation, 99. Isosceles triangle, 53. Compass, 61, 76. circle, 92. Kepler's equation, 162. course, 77. Knot, 80. surveyor's, 91. Complementary angle, 16. Ltitu Component force, 30, 131. Latitude, 60, 84. Coordinates, 103. Law o cne, 12 Cosecant, 7, 107. L oosines, 126. curve, 155. tangents, 117. Cosine, 7, 107. tangens, 6 1. C ritca 1anl4. 3.Leeway, 76. arc, 164. curve, 9152-3. Level, 92, 98. Finvrse, 1564. Limb of transit, 98. law ofn, 16. Logarithm, 36. Cangent, 7, 107. Logarithmic scales, 43. curve,7 155. Longitude, 60, 66. Course, 61, 77, 83. made good, 65. Made good, 65. Critical angle, 33. Magnetic, 94. Cross wires, 98. Mantissa, 39. Maps, 71, 88. Dead reckoning, 59, 79. Mercator chart, 71. Decagon, 54. sailing, 73. Declination, 94. Meridian, 59. Deflection angle, 100. distance, 86. Departure, 62, 84. magnetic, 76. in mid lat., 69, prime, 60. 171 172 Meridional parts, 72. difference of lat., 74. Middle latitude sailing, 68. Mil, 160. Mollweide's equations, 117. Natural functions, 19. Nautical astronomy, 59. mile, 60. Navigation, 59. Oblique triangle, 115. Octagon, 54. Ordinate, 103. Origin, 104. Parallel of latitude, 60. sailing, 66. Parallelogram of forces, 29. Parts of triangle, 21. Patent log, 61. Pentagon, 53. Perimeter, 129. Piloting, 59. Plane Sailing, 62. Plotting, 88, 104. Plumb line, 4. Points of compass, 61. Poles, 59, 94. Polygon, 53. Protractor, 1. Quadrants, 106. Radian, 158. Radius vector, 104. of inscribed circle, 129. Rational, 35. Reciprocal, 7. Refraction of light, 31. Relative error, 3. Resultant of forces, 29, 131. Rhumb line, 68, 73. Right triangle, 19, 48, 125. Sailing, Mercator's, 73. middle latitude, 68. parallel, 66. plane, 62. traverse, 64. Secant, 7, 107. curve, 155. Sight vane, 91. Significant, 38. INDEX Sine, 6, 7, 107. arc, 165. curve, 152-3. inverse, 165. law of, 115. of small angle, 161. Slide rule, 44. Slope, 5. Spirit level, 4. Square-ruled paper, 2, 104. Stadia wires, 99. Statute mile, 60. Subtraction theorems for sine and cosine, 136. for tangent and cotangent, 138. Surveying, 82. instruments, 89. Tangent, 5, 7, 107. curve, 154. law of, 117. of 90~, 155. of small angle, 161. Tape, 82. Telescope, 96. Trajectory, 142. Transit, 96. Traverse, 99. sailing, 64. table, 26, 63. Trigonometric equations, 147, 156, 162. Trigonometric functions, 6, 107. inverse of, 164. line representation, 150. of double angle, 140. of half angle, 144. of multiple angle, 141. of several angles, 133. reduced to acute angles, 111. sum of, expressed as a product, 145. True bearing, 83. course, 61, 77. Undefined, 108, 154. Unit's place, 38. Variation, 76, 94. charts, 95. Vernier, 89, 95. double, 90. plate, 96. Vertical angle, 4. circle, 98. line, 4. plane, 4. ANSWERS TO CERTAIN OF THE FIRST FIVE EXERCISES OF EACH SET Pages 8, 9 1. sin 45~ =cos 45~= V2, tan 45= cot 45 = 1, sec 45 = csc 4 2. sin 60~ = ~ /V3, cos 60~ =, tan 60~ = /3, csc 60~ = -3, sec 60~ = 2, cot 60~ = 1 3. 3. sin 30~ =, cos 30~ = <3, tan 30~ = -3, csc 30~ = 2, sec 30~ = 3V3, cot 30~ = /3. 5. sin =, cos = 4, tan =, cot = 4, sec = C, cs Pages 10, 11 1. 40(3 +-3). 2. 6 + 11/3. 3. 250. Page 12 1. 53~, see ans. 1, page 14. 3. 67~, A, 1- 5. 53~, Page 14 1. sinA =, cosA=, tanA=, cotA =, secA =-, csc 3. sin A =-, cosA=l, tanA=-5, cotA= '=, secA==I, csc Pages 17, 18 1. cos 25~, sin 70~, cot 34~ 40', tan 7~ 48', csc 23~, sec 23~. 2. sin 70 =cos 20~, cos 70~ =sin 20~, tan 70 =cot 20~. 5. 45~. Page 21 1. (a) 0.3156, 0.9488, 0.3327, 3.006. 3. 2.0051. 4. x = 27~ 10', y = 40~ 3',z = 64~ 47', w = 82~ 58'. Pages 24, 25 50 = -/2. c = 20 ft. A= 4.;A = 7. 8. 1. a = 281.9, b = 102.6. 2. a = 10.72, c = 41.41. 5. A = 19~ 12', B = 70~ 48', a = 5.916. Pages 28, 29 1. 60. 2. 901.1. 5. 888.4. Page 31 1, 13 Ibs., 22~ 37'. 3. 85.02 lbs. 4. 6.12 miles. Page 33 1. 4.49 ft. 3. 56.28 ft. Pages 37, 38 1. 5,-1, -3, 2. 2. 2. 3. (b)1.38021, (c) 0.35218. Pages 41, 42 1. 141.8. 0.01668. 0.0002154. 2. 464.7. 3. 4.152. -.,. I. _ 173 11 74 ANSWERS Page 47 1. 9.7018-10, 9.8645-10, 9.9689-10. 2. x =42~ 57', y = 24~ 45', z = 87~ 51', w = 70~ 27'. Page 52 1. a = 536.6, b = 1006. 2. b = 130.1, c 3. A = 31~ 24', B = 58~ 36', b = 3666. Page 53 1. 297.4. 2. 857.7. 5. a = 23.2, c = 30.63, A = 49 14', B = 4 Pages 54, 55 1. A = 67~ 22.9', V = 45~ 14.2', h = 264.0, A = 2. r = 41.11, h = 33.72, A = 793.0. Pages 56-58 1. 69.95. 2. 0.298. 3. 1842 = 162.3. L0~ 46'. 29040. (approx.). Pages 62, 63 1. 16.4 E, 43~ 40.3' N. 3. 199.7, 58.4 W. Page 64 1. 9~0 21', 86.1 2. 15~ 31', 93.4. Pages 65, 66 1. S 24~ 21' E, 69.6. 5. 33~ 44' S, N 240 46' W, 224.4. Pages 67, 68 1. 125.0E. 3. 166~ 50'E. Page 70 1. N 67~ 29' E, 295.5. 3. 49~ 10'S, 176~ W. Pages 75, 76 2. S 40~ 47.2' E, 4754. 5. 47~ 10' N, 32~ 15' W. Page 78, top 1. 12~. 4. 23~. Page 78, bottom 1. 34~ 49' N, 39.3 W. 2. 600, N 56~ 20' W. Pages 79-81 1. 45~ 1.9'N, 13~ 18.7'W. 3. 51~ 19.5'N, 24~ 9'W. Page 88 1. 2.370. 3. 15.137. 1. N 72~ E. 1. S 18~ 5' W, N 38~ W. Pages 95, 96 2. S 86~ 45' W. 5. 8~ 50'W. Pages 100, 101 2. S 57~ 22' E, N 75~ 26' E, S 56~ 14' E. ANSWERS 175 Page 102 1. 12.101 acres. 3. 226330 sq. ft. = 5.1958 acres. Page 104 1. 5, center at origin and radius 5. 4. 100. Page 106 1. Third, third, first, third, first. 5. 3819~ 44'. Pages 109, 110 2. 135~ 150~ 210~ 225~ 240~ 300~ 315~ 330~ 1 1 1 -1 -- / 3 _-_/ 3 -1 1 sn 2 2 2 a2 2 2 2 - 2 2 os — 1 _ /3 3 - 1 1 1 _ 3 V2 2 2 12 2 2 72 2 tan -1 s$3 at3 1 3 - -1 Page 114 1. -0.8660. 2. -0.2679. 3. - 0.2588. Pages 118-120 1. C=76~48',b=74.17, c=118.9. 5. A=35~2.6',B=82~ 25.4', c = 228.7. Page 126 1. One. 2. Two. 4. None. 5. Two. Page 128 1. b2 = a2 + c2- 2 ac cos B, cos B = (a2 + c - b2) /(2 ac). 2. A = 28~ 57', B = 46~ 34', C = 104~ 29'. 5. a = 6.456, B = 80~ 46', C = 63~ 48'. Page 131, top 1. 2.905, see ans. 2, page 128. 5. 630. 2. 1499, A = 46~ 26.3', B = 75~ 13.5', C = 58~ 20.2'. Pages 131, bottom, 132 1. 56~ 15', 29~ 55'. 2. 1114, 18~ 23'. 4. 640, 364. Page 136 1. J2(V 3-1), iV]2(3 + 1). 5. 6, Page 138 3. 2.467. 4. 457.8. Pages 139, 140 2. 2+-/3, 2 —/3. 4. A, 2. 5. 16~ 6'. 176 A) Pag 47 24 2. 2 4)T-2-4T Pag, 2. 108250. 3 F I~~~~~~~1 VSWERS es 141, 142 4. 4 cos3A - 3 cos A. es 143, 144. 25~, 65~. 5. 36.95. Page 145 1. ~ 2-a/3, ~2+-i3, 2-63. Page 146 2. 2 sin 2x cos x. 3. 2 sin 2x sin Page 149 1. 120~, 240~. 4. 60~, 300~. 5. 90~, ] Page 156 3. (a), (b) 0, 180~, 360~; (c) 90~, 270~. Page 157 1. 16~ 51', 84~ 9' (maximum 0.6636 at x = 219~ 26'). Pages 159-161 1. 7r/4 = 0.7854, 7/3, -fr. 5. 1.5. Page 162 1. 184.3 ft. 3. 1~ 31'. Page 164 1. 18.36. 2. 101~ 12'. Page 167 2. 1/1V2. 3. 15. 120~, 240~, 270*. 4. 2192 miles. 5. 115~ 47.6'. 5. - 13. TABLES PAGE EXAMPLES SHOWING HOW THE TABLES ARE USED.. 2 TABLES I, II. NATURAL SINES AND COSINES. 3 TABLES III, IV, V. NATURAL TANGENTS AND COTANGENTS.. 5 TABLE VI. TRAVERSE TABLE.... 8 TABLE VII. LOGARITHMS OF NUMBERS.. 26 TABLE VIII. LOGARITHMS OF THE TRIGONOMETRIC FUNCTIONS. 30 TABLE IX. MERIDIONAL PARTS.... 35 1 EXAMPLES SHOWING HOW THE TABLES ARE USED EXAMPLES. Find by Table I sin 25~ 23' and cos 64~ 37'. Angle < 45~ sin 25~ 20' = 0.4279 degrees at left sin 25~ 30' = 0.4305 minutes at top 10 X.0026= To the upper number add.0008.. sin 25~ 23' = 0.4287 Angle > 45~ cos 64~ 30' = 0.4305 degrees at right cos 64~ 40' = 0.4279 minutes at bottom 10- X.0026= From the upper number subtract.0018.~. cos 64~ 37' = 0.4287 Or in P.P. tablette headed 26, read entry 18 opposite to 7. EXAMPLE. Given cot x = 2, find the acute Lx by Table V. cot 26~ 30' = 2.006 cot 26~ 40' = 1.991.015 2.006 —2 =.006 61 X 10' = 4', x = 26~ 34'. Traverse Table VI gives, for each ZA and each hypotenuse, distance, or number D, the lengths of the adjacent and opposite legs of the right triangle, or the latitude and departure, or the products D cos A and D sin A. ( 85, adj. = 61.14,opp. = 59.05 For A = 44~ and D = t 300, 215.80 208.40 385, 276.94 267.45 Or multiply by 5 the entires adj. = 55.39, opp. = 53.49 for D = 385/5 = 77. When seeking, by Table VII, the logarithm of a positive number N, use pages 26, 27 if the first digit of N is 1. When finding the mantissa (decimal part) of log N, ignore the decimal point in N. To the mantissa, prefix + p if the first significant digit of N lies p places to the left of unit's place in N, but prefix p if it lies p places to the right. For example, log 16.17 = 1.2087, log 0.01617 = 2.2087. In Table VIII employ the labels at the top (including 9. or 10.) when reading angles at the left, but the labels at the bottom when reading angles at the right. Always annex -10. For example, log sin 10~ 30' = 9.2606 - 10, log tan 75~ 40' = 10.5926 - 10. 2 TABLE I. NATURAL SINES o 0' 10' 20' 30' 40' 50' 60' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 0.0000 0.0175 0.0349 0.0523 0.0698 0.0872 0.1045 0.1219 0.1392 0.1564 0.1736 0.1908 0.2079 0.2250 0.2419 0.2588 0.2756 0.2924 0.3090 0.3256 0.3420 0.3584 0.3746 0.3907 0.4067 0.4226 0.4384 0.4540 0.4695 0.4848 0.5000 0.5150 0.5299 0029 0058 0087 0204 0233 0262 0378 0407 0436 0552 0581 0610 0727 0756 0785 0901 0929 0958 1074 1103 1132 1248 1276 1305 1421 1449 1478 1593 1622 1650 1765 1794 1822 1937 1965 1994 2108 2136 2164 2278 2306 2334 2447 2476 2504 2616 2644 2672 2784 2812 2840 2952 2979 3007 3118 3145 3173 3283 3311 3338 3448 3475 3502 3611 3638 3665 3773 3800 3827 3934 3961 3987 4094 4120 4147 4253 4279 4305 4410 4436 4462 4566 4592 4617 4720 4746 4772 4874 4899 4924 5025 5050 5075 5175 5200 5225 5324 5348 5373 5471 5495 5519 5616 5640 5664 5760 5783 5807 5901 5925 5948 6041 6065 6088 6180 6202 6225 6316 6338 6361 6450 6472 6494 6583 6604 6626 6713 6734 6756 6841 6862 6884 6967 6988 7009 0116 0291 0465 0640 0814 0987 1161 1334 1507 1679 1851 2022 2193 2363 2532 2700 2868 3035 3201 3365 3529 3692 3854 4014 4173 4331 4488 4643 4797 4950 5100 5250 5398 5544 5688 5831 5972 6111 6248 6383 6517 6648 6777 6905 7030 0145 0175 0320 0349 0494 0523 0669 0698 0843 0872 1016 1045 1190 1219 1363 1392 1536 1564 1708 1736 1880 1908 2051 2079 2221 2250 2391 2419 2560 2588 2728 2756 2896 2924 3062 3090 3228 3256 3393 3420 3557 3584 3719 3746 3881 3907 4041 4067 4200 4226 4358 4384 4514 4540 4669 4695 4823 4848 4975 5000 5125 5150 5275 5299 5422 5446 5568 5592 5712 5736 5854 5878 5995 6018 6134 6157 6271 6293 6406 6428 6539 6561 6670 6691 6799 6820 6926 6947 7050 7071 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 P. P. 29 28 1 3 3 2 6 6 3 9 8 4 12 11 5 15 14 6 17 17 7 20 20 8 23 22 9 26 25 27 26 1 3 3 2 5 5 3 8 8 4 11 10 5 14 13 6 16 16 7 19 18 8 22 21 9 24 23 25 24 1 3 2 2 5 5 3 8 7 4 10 10 5 13 12 6 15 14 7 18 17 8 20 19 9 23 22 23 22 1 2 2 2 5 4 3 7 7 4 9 9 5 12 11 6 14 13 7 16 15 8 18 18 9 21 20 21 1 2 2 4 3 6 4 8 5 11 6 13 7 15 8 17 9 19 0.5446 0.5592 0.5736 0.5878 0.6018 0.6157 0.6293 0.6428 0.6561 0.6691 0.6820 0.6947 60' 50' 40' 30' 20' 10' 0' ~ NATURAL COSINES 3 TABLE II. NATURAL COSINES 0' 10' 20' 30' 40' 50' 60' 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 1.0000 0.9998 94 86 76 62 45 25 0.9903 0.9877 0.9848 0.9816 0.9781 0.9744 0.9703 0.9659 0.9613 0.9563 0.9511 0.9455 0.9397 0.9336 0.9272 0.9205 0.9135 0.9063 0.8988 0.8910 0.8829 0.8746 0.8660 0.8572 0.8480 0.8387 0.8290 0.8192 0.8090 0.7986 0.7880 0.7771 0.7660 0.7547 0.7431 0.7314 0.7193 1. 1. 9998 9997 93 92 85 83 74 71 59 57 42 39 9922 9918 9899 9894 9872 9868 9843 9838 9811 9805 9775 9769 9737 9730 9696 9689 9652 9644 9605 9596 9555 9546 9502 9492 9446 9436 9387 9377 9325 9315 9261 9250 9194 9182 9124 9112 9051 9038 8975 8962 8897 8884 8816 8802 8732 8718 8646 8631 8557 8542 8465 8450 8371 8355 8274 8258 8175 8158 8073 8056 7969 7951 7862 7844 7753 7735 7642 7623 7528 7509 7412 7392 7294 7274 7173 7153 1. 9997 90 81 69 54 36 9914 9890 9863 9833 9799 9763 9724 9681 9636 9588 9537 9483 9426 9367 9304 9239 9171 9100 9026 8949 8870 8788 8704 8616 8526 8434 8339 8241 8141 8039 7934 7826 7716 7604 7490 7373 7254 7133 9999 9999 96 95 89 88 80 78 67 64 51 48 32 29 9911 9907 9886 9881 9858 9853 9827 9822 9793 9787 9757 9750 9717 9710 9674 9667 9628 9621 9580 9572 9528 9520 9474 9465 9417 9407 9356 9346 9293 9283 9228 9216 9159 9147 9088 9075 9013 9001 8936 8923 8857 8843 8774 8760 8689 8675 8601 8587 8511 8496 8418 8403 8323 8307 8225 8208 8124 8107 8021 8004 7916 7898 7808 7790 7698 7679 7585 7566 7470 7451 7353 7333 7234 7214 7112 7092 9998 94 86 76 62 45 25 9903 9877 9848 9816 9781 9744 9703 9659 9613 9563 9511 9455 9397 9336 9272 9205 9135 9063 8988 8910 8829 8746 8660 8572 8480 8387 8290 8192 8090 7986 7880 7771 7660 7547 7431 7314 7193 7071 89 88 87 86 85 84 83 82 81 80 79' 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 P. P. m; 1 1 12 22 22 3 3 4 u:^ r' 4 4 5 0.'45 6 6 6 7 7 7 8 8 8 9 10 0 9 11 10 13 14 1 1 2 3 3 3 4 4 4 5 6 5 7 7 6 8 8 7 9 10 8 10 11 9 12 13 15 16 1 2 2 2 3 3 3 5 5 4 6 6 5 8 8 6 9 10 7 11 11 8 12 13 9 14 14 17 18 1 2 2 2 3 4 3 5 5 4 7 7 5 9 9 6 10 11 7 12 13 8 14 14 9 15 16 19 21 1 2 2 2 4 4 3 6 6 4 8 8 5 10 11 6 11 13 7 13 15 8 15 17 9 17 19 60' 50' 40' 30' 20' 10' 0' o NATURAL SINES 4 TABLE III. NATURAL COTANGENTS o ' 0' 1' 2' 3' 4' 5' 6' 7' 8' 9' v J A rs j s q j | _~~~~~~~~~~~~~~~~~~~~~~~~~ 0 1 10 20 30 40 50 1 0 10 20 30 40 50 2 0 10 20 30 40 50 3 0 10 20 30 540 50 6 o 10 20 30 40 50 7 0 10 20 30 40 50 8 o 10 20 30 40 50 9 0 10 20 30 40 50 10 0 10 20 30 40 50 343.8 171.9 114.6 85.94 68.75 57.29 49.10 42.96 38.19 34.37 31.24 28.64 26.43 24.54 22.90 21.47 20.21 19.08 18.07 17.17 16.35 3438 312.5 163.7 110.9 83.84 67.40 56.35 48.41 42.43 37.77 34.03 30.96 28.40 26.23 24.37 22.75 21.34 20.09 18.98 17.98 17.08 16.27 1719 286.5 156.3 107.4 81.85 66.11 55.44 47.74 41.92 37.36 33.69 30.68 28.17 26.03 24.20 22.60 21.20 19.97 18.87 17.89 17.00 16.20 1146 264.4 149.5 104.2 79.94 64.86 54.56 47.09 41.41 36.96 33.37 30.41 27.94 25.83 24.03 22.45 21.07 19.85 18.77 17.79 16.92 16.12 859.4 245.6 143.2 101.1 78.13 63.66 53.71 46.45 40.92 36.56 33.05 30.14 27.71 25.64 23.86 22.31 20.95 19.74 18.67 17.70 16.83 16.04 687.5 229.2 137.5 98.22 76.39 62.50 52.88 45.83 40.44 36.18 32.73 29.88 27.49 25.45 23.69 22.16 20.82 19.63 18.56 17.61 16.75 15.97 573.0 491.1 429.7 382. 214.9 202.2 191.0 180.9 132.2 127.3 122.8 118.5 95.49 92.91 90.46 88.14 74.73 73.14 71.62 70.15 61.38 60.31 59.27 58.26 52.08 51.30 50.55 49.82 45.23 44.64 44.07 43.51 39.97 39.51 39.06 38.62 35.80 35.43 35.07 34.72 32.42 32.12 31.82 31.53 29.62 29.37 29.12 28.88 27.27 27.06 26.84 26.64 25.26 25.08 24.90 24.72 23.53 23.37 23.21 23.06 22.02 21.88 21.74 21.61 20.69 20.57 20.45 20.33 19.52 19.41 19.30 19.19 18.46 18.37 18.27 18.17 17.52 17.43 17.34 17.26 16.67 16.59 16.51 16.43 15.89 15.82 15.75 15.68 50 40 30 20 ' 10 89 o 50 40 30 20 10 88 0 50 40 30 20 10 87 o 50 40 30 86 20 10.078 9.788 9.514 9.255 9.010 8.777 8.556 8.345 8.144 7.953 7.770 7.596 7.429 7.269 7.115 6.968 6.827 6.691 6.561 6.435 6.314 6.197 6.084 5.976 5.871 5.769 5.671 5.576 5.485 5.396 5.309 5.226 10.048 10.019 9.760 9.732 9.488 9.461 9.230 9.205 8.986 8.962 8.754 8.732 8.534 8.513 8.324 8.304 8.125 8.105 7.934 7.916 7.753 7.735 7.579 7.562 7.412 7.396 7.253 7.238 7.100 7.085 6.954 6.940 6.813 6.799 6.678 6.665 6.548 6.535 6.423 6.410 6.302 6.290 6.186 6.174 6.073 6.062 5.965 5.954 5.861 5.850 5.759 5.749 5.662 5.652 5.567 5.558 5.475 5.466 5.387 5.378 5.301 5.292 5.217 5.209 9.989 9.704 9.435 9.180 8.939 8.709 8.491 8.284 8.086 7.897 7.717 7.545 7.380 7.222 7.071 6.925 6.786 6.651 6.522 6.398 6.278 6.163 6.051 5.944 5.840 5.740 5.642 5.549 5.458 5.369 5.284 5.201 9.960 9.931 9.677 9.649 9.409 9.383 9.156 9.131 8.915 8.892 8.687 8.665 8.470 8.449 8.264 8.243 8.067 8.048 7.879 7.861 7.700 7.682 7.528 7.511 7.364 7.348 7.207 7.191 7.056 7.041 6.911 6.897 6.772 6.758 6.638 6.625 6.510 6.497 6.386 6.374 6.267 6.255 6.152 6.140 6.041 6.030 5.933 5.923 5.830 5.820 5.730 5.720 5.633 5.623 5.539 5.530 5.449 5.440 5.361 5.352 5.276 5.267 5.193 5.185 9.902 9.622 9.357 9.106 8.869 8.643 8.428 8.223 8.028 7.842 7.665 7.495 7.332 7.176 7.026 6.883 6.745 6.612 6.485 6.362 6.243 6.129 6.019 5.912 5.810 5.710 5.614 5.521 5.431 5.343 5.259 5.177 9.873 9.595 9.332 9.082 8.846 8.621 8.407 8.204 8.009 7.824 7.647 7.478 7.316 7.161 7.012 6.869 6.731 6.599 6.472 6.350 6.232 6.118 6.008 5.902 5.799 5.70.0 5.605 5.512 5.422 5.335 5.250 5.169 9.845 9.816 9.568 9.541 9.306 9.281 9.058 9.034 8.823 8.800 8.599 8.577 8.386 8.366 8.184 8.164 7.991 7.972 7.806 7.788 7.630 7.613 7.462 7.445 7.300 7.284 7.146 7.130 6.997 6.983 6.855 6.841 6.718 6.704 6.586 6.573 6.460 6.447 6.338 6.326 6.220 6.209 6.107 6.096 5.997 5.986 5.892 5.881 5.789 5.779 5.691 5.681 5.595 5.586 5.503 5.494 5.413 5.404 5.326 5.318 5.242 5.234 5.161 5.153 10 84 0 50 40 30 20 10 83 o 50 40 30 20 10 82 O 50 40 30 20 81 0 50 40 30 20 10 80 0 50 40 30 20 710 79 0 10' 9' 8' 7' 6' 5' 4' 3' 2/ 1' ~ NATURAL TANGENTS 5 TABLE IV. NATURAL TANGENTS 0' 10' 20' 30' o 40' 50' 60' 10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 0.0000 0029 0058 0087 0.0175 0204 0233 0262 0.0349 0378 0407 0437 0.0524 0553 (0582 0612 0.0699 0729 '>75 0787 0.0875 0904 0934 0963 0.1051 1080 1110 1139 0.1228 1257 1287 1317 0.1405 1435 1465 1495 0.1584 1614 1644 1673 0.1763 1793 1823 1853 0.1944 1974 2004 2035 0.2126 2156 2186 2217 0.2309 2339 2370 2401 0.2493 2524 2555 2586 0.2679 2711 2742 2773 0.2867 2899 2931 2962 0.3057 3089 3121 3153 0.3249 3281 3314 3346 0.3443 3476 3508 3541 0.3640 3673 3706 3739 0.3839 3872 3906 3939 0.4040 4074 4108 4142 0.4245 4279 4314 4348 0.4452 4487 4522 4557 0.4663 4699 4734 4770 0.4877 4913 4950 4986 0.5095 5132 5169 5206 0.5317 5354 5392 5430 0.5543 5581 5619 5658 0.5774 5812 5851 5890 0.6009 6048 6088 6128 0.6249 6289 6330 6371 0.6494 6536 6577 6619 0.6745 6787 6830 6873 0.7002 7046 7089 7133 0.7265 7310 7355 7400 0.7536 7581 7627 7673 0.7813 7860 7907 7954 0.8098 8146 8195 8243 0.8391 8441 8491 8541 0.8693 8744 8796 8847 0.9004 9057 9110 9163 0.9325 9380 9435 9490 0.9657 9713 9770 9827 0116 0291 0466 0641 0816 0992 1169 1346 1524 1703 1883 2065 2247 2432 2617 2805 2994 3185 3378 3574 3772 3973 4176 4383 4592 4806 5022 5243 5467 5696 5930 6168 6412 6661 6916 7177 7445 7720 8002 8292 8591 8899 9217 9545 9884 0145 0175 0320 0349 0495 0524 0670 0699 0846 0875 1022 1051 1198 1228 1376 1405 1554 1584 1733 1763 1914 1944 2095 2126 2278 2309 2462 2493 2648 2679 2836 2867 3026 3057 3217 3249 3411 3443 3607 3640 3805 3839 4006 4040 4210 4245 4417 4452 4628 4663 4841 4877 5059 5095 5280 5317 5505 5543 5735 5774 5969 6009 6208 6249 6453 6494 6703 6745 6959 7002 7221 7265 7490 7536 7766 7813 8050 8098 8342 8391 8642 8693 8952 9004 9271 9325 9601 9657 9942 1. 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 P. P. 29 31 32 33 34 1 3 3 3 3 3 2 6 6 6 7 7 3 9 9 10 10 10 4 12 12 13 13 14 5 15 16 16 17 17 6 17 19 19 20 20 7 20 22 22 23 24 8 23 25 26 26 27 9 26 28 29 30 31 35 36 37 38 39 1 4 4 4 4 4 2 7 7 7 8 8 3 11 11 11 1 12 4 14 14 15 15 16 5 18 18 19 19 20 6 21 22 22 23 23 7 25 25 26 27 27 8 28 29 30 30 31 932 32 33 34 35 41 42 43 44 45 1 4 4 4 4 5 2 8 8 9 9 9 3 12 13 13 13 14 4 16 17 17 18 18 5 21 21 22 22 23 6 25 25 26 26 27 7 29 29 30 31 32 8 33 34 34 35 36 9 37 38 39 40 41 46 47 48 49 51 52 1 5 5 5 5 5 5 2 9 9 10 10 10 10 3 14 14 14 1 15 16 4 18 19 19 2 0 21 5 23 24 24 25 26 26 6 28 28 29 29 31 31 7 32 33 34 34 36 36 8 37 38 3 3941 42 9 41 42 43 44 46 47 53 54 55 56 57 58 1 5 5 6 66 66 6 2 1 1 11 11 11 12 3 16 16 17 17 17 17 4 21 22 22 22 23 23 527 27 28 28 29 29 6 32 32 33 34 34 35 7 37 38 39 39 40 41 8 42 43 44 45 46 46 9 48 49 50 50 51 52 60' 50' 40' 30' 20' 10' 0' NATURAL COTANGENTS 6 TABLE V. NATURAL COTANGENTS P. P. 79 77 74 72 68 1 8 8 7 7 7 2 16 16 15 14 14 3 24 23 22 22 20 4 32 31 30 29 27 5 40 39 37 36 34 6 48 46 44 43 41 7 56 54 52 51 48 8 64 61 59 58 54 9 71 69 67 65 61 67 64 63 62 59 1 7 '6 7 6 6 2 14 13 13 12 12 3 20 19 19 19 18 4 27 26 26 25 24 5 34 32 32 31 30 6 40 39 38 37 35 7 47 45 45 43 41 8 54 52 51 50 47 9 60 58 57 56 53 28 27 26 25 24 1 3 3 3 3 2 2 6 5 5 5 5 3 8 8 8 8 7 4 11 11 10 10 10 5 14 14 13 13 12 6 17 16 16 15 14 7 20 19 18 18 17 8 22 22 21 20 19 9 25 24 23 23 22 23 22 21 19 18 17 1 2 2 2 2 2 2 25 4 4 4 4 3 3' 7 7 6 6 5 5 4 9 9 8 8 7 7 5 12 11 11 10 9 9 6 14 13 13 11 11 10 7 16 15 15 13 13 12 8 18 18 17 15 14 14 9 21 20 19 17 16 15 16 15 14 13 12 11 o 0' 10' 20' 30' 40' 50' 60'... Do not interpolate between two numbers in italics; use Table III 19.08 18.07 17.17 16.35 15.60 14.92 14.30 14.30 13.73 13.20 12.71 12.25 11.83 11.43 11.43 11.06 10.71 10.39 10.08 9.788 9.514 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 9.514 8.144 7.115 6.314 5.671 5.145 4.705 4.331 4.011 3.732 3.487 3.271 3.078 2.904 2.747 2.605 2.475 2.356 2.246 2.145 2.050 1.963 1.881 1.804 1.732 1.664 1.600 1.540 1.483 1.428 1.376 1.327 1.280 1.235 1.192 1.150 1.111 1.072 1.036 9.255 7.953 6.968 6.197 5.576 5.066 4.638 4.275 3.962 3.689 3.450 3.237 3.047 2.877 2.723 2.583 2.455 2.337 2.229 2.128 2.035 1.949 1.868 1.792 1.720 1.653 1.590 1.530 1.473 1.419 1.368 1.319 1.272 1.228 1.185 1.144 1.104 1.066 1.030 9.010 7.770 6.827 6.084 5.485 4.989 4.574 4.219 3.914 3.647 3.412 3.204 3.018 2.850 2.699 2.560 2.434 2.318 2.211 2.112 2.020 1.935 1.855 1.780 1.709 1.643 1.580 1.520 1.464 1.411 1.360 1.311 1.265 1.220 1.178 1.137 1.098 1.060 1.024 8.777 7.596 6.691 5.976 5.396 4.915 4.511 4.165 3.867 3.606 3.376 3.172 2.989 2.824 2.675 2.539 2.414 2.300 2.194 2.097 2.006 1.921 1.842 1.767 1.698 1.632 1.570 1.511 1.455 1.402 1.351 1.303 1.257 1.213 1.171 1.130 1.091 1.054 1.018 8.556 7.429 6.561 5.871 5.309 4.843 4.449 4.113 3.821 3.566 3.340 3.140 2.960 2.798 2.651 2.517 2.394 2.282 2.177 2.081 1.991 1.907 1.829 1.756 1.686 1.621 1.560 1.501 1.446 1.393 1.343 1.295 1.250 1.206 1.164 1.124 1.085 1.048 1.012 8.345 7.269 6.435 5.769 5.226 4.773 4.390 4.061 3.776 3.526 3.305 3.108 2.932 2.773 2.628 2.496 2.375 2.264 2.161 2.066 1.977 1.894 1.816 1.744 1.675 1.611 1.550 1.492 1.437 1.385 1.335 1.288 1.242 1.199 1.157 1.117 1.079 1.042 1.006 8.144 7.115 6.314 5.671 5.145 4.705 4.331 4.011 3.732 3.487 3.271 3.078 2.904 2.747 2.605 2.475 2.356 2.246 2.145 2.050 1.963 1.881 1.804 1.732 1.664 1.600 1.540 1.483 1.428 1.376 1.327 1.280 1.235 1.192 1.150 1.111 1.072 1.036 1.000 i 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 -. 1 2 2 1 1 1 1 2 3 3 3. 3 2 2 35 5 4 4 4 3 4 6 6 6 5 5 4 5 8 8 7 7 6 6 6 10 9 8 8 7 7 7 11 11 10 9 8 8 8 13 12 11 10 10 9 9 14 14 13 12 11 10 60' 50' 40' 30' 20' 10' 0' ~ NATURAL TANGENTS 7 I TABLE VI. TRAVERSE TABLE 2~ 30 40 1~ 50 D D cos D sin D cos D sin D cos D sin DcosD sinD cos D sin D hyp. adj oppadj. opp. adj. opp. adj. opp ad. adj. opp. hyp. dis. lat. dep. lat. e t. dep. t. dep. lat. dep. adis. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00 30.00 31.00 32.00 32.99 33.99 34.99 35.99 36.99 37.99 38.99 39.99 40.99 41.99 42.99 43.99 44.99 45.99 46.99 47.99 48.99 49.99 50.99 51.99 52.99 53.99 54.99 55.99 56.99 57.99 58.99 0.19 0.21 0.23 0.24 0.26 0.28 0.30 0.31 0.33 0.35 0.37 0.38 0.40 0.42 0.44 0.45 0.47 0.49 0.51 0.52 0.54 0.56 0.58 0.59 0.61 0.63 0.65 0.66 0.68 0.70 0.72 0.73 0.75 0.77 0.79 0.80 0.82 0.84 0.86 0.87 0.89 0.91 0.92 0.94 0.96 0.98 0.99 1.01 1.03 10.99 11.99 12.99 13.99 14.99 15.99 16.99 17.99 18.99 19.99 20.99 21.99 22.99 23.99 24.98 25.98 26.98 27.98 28.98 29.98 30.98 31.98 32.98 33.98 34.98 35.98 36.98 37.98 38.98 39.98 40.98 41.97 42.97 43.97 44.97 45.97 46.97 47.97 48.97 49.97 50.97 51.97 52.97 53.97 54.97 55.97 56.97 57.96 58.96 0.38 0.42 0.45 0.49 0.52 0.56 0.59 0.63 0.66 0.70 0.73 0.77 0.80 0.84 0.87 0.91 0.94 0.98 1.01 1.05 1.08 1.12 1.15 1.19 1.22 1.26 1.29 1.33 1.36 1.40 1.43 1.47 1.50 1.54 1.57 1.61 1.64 1.68 1.71 1.74 1.78 1.81 1.85 1.88 1.92 1.95 1.99 2.02 2.06 10.98 11.98 12.98 13.98 14.98 15.98 16.98 17.98 18.97 19.97 20.97 21.97 22.97 23.97 24.97 25.96 26.96 27.96 28.96 29.96 30.96 31.96 32.95 33.95 34.95 35.95 36.95 37.95 38.95 39.95 40.94 41.94 42.94 43.94 44.94 45.94 46.94 47.93 48.93 49.93 50.93 51.93 52.93 53.93 54.92 55.92 56.92 57.92 58.92 0.58 0.63 0.68 0.73 0.79 0.84 0.89 0.94 0.99 1.05 1.10 1.15 1.20 1.26 1.31 1.36 1.41 1.47 1.52 1.57 1.62 1.67 1.73 1.78 1.83 1.88 1.94 1.99 2.04 2.09 2.15 2.20 2.25 2.30 2.36 2.41 2.46 2.51 2.56 2.62 2.67 2.72 2.77 2.83 2.88 2.93 2.98 3.04 3.09 10.97 11.97 12.97 13.97 14.96 15.96 16.96 17.96 18.95 19.95 20.95 21.95 22.94 23.94 24.94 25.94 26.93 27.93 28.93 29.93 30.92 31.92 32.92 33.92 34.91 35.91 36.91 37.91 38.90 39.90 40.90 41.90 42.90 43.89 44.89 45.89 46.89 47.88 48.88 49.88 50.88 51.87 52.87 53.87 54.87 55.86 56.86 57.86 58.86 0.77 0.84 0.91 0.98 1.05 1.12 1.19 1.26 1.33 1.40 1.46 1.53 1.60 1.67 1.74 1.81 1.88 1.95 2.02 2.09 2.16 2.23 2.30 2.37 2.44 2.51 2.58 2.65 2.72 2.79 2.86 2.93 3.00 3.07 3.14 3.21 3.28 3.35 3.42 3.49 3.56 3.63 3.70 3.77 3.84 3.91 3.98 4.05 4.12 10.96 11.95 12.95 13.95 14.94 15.94 16.94 17.93 18.93 19.92 20.92 21.92 22.91 23.91 24.90 25.90 26.90 27.89 28.89 29.89 30.88 31.88 32.87 33.87 34.87 35.86 36.86 37.86 38.85 39.85 40.84 41.84 42.84 43.83 44.83 45.82 46.82 47.82 48.81 49.81 50.81 51.80 52.80 53.79 54.79 55.79 56.78 57.78 58.78 0.96 1.05 1.13 1.22 1.31 1.39 1.48 1.57 1.66 1.74 1.83 1.92 2.00 2.09 2.18 2.27 2.35 2.44 2.53 2.61 2.70 2.79 2.88 2.96 3.05 3.14 3.22 3.31 3.40 3.49 3.57 3.66 3.75 3.83 3.92 4.01 4.10 4.18 4.27 4.36 4.44 4.53 4.62 4.71 4.79 4.88 4.97 5.06 5.14 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. ep lat.de p. lat. de p. lat. de p. lat. de p. de lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp adj. hyp. D Dsin D cos D sin D cos D sin D cos D sin D cos D sin D cos D. _ 89~ 88~ 87~ 8 86~ 85~ TABLE VI. TRAVERSE TABLE 9 1~ 2~ 3~ 4~ 50 D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 59.99 60.99 61.99 62.99 63.99 64.99 65.99 66.99 67.99 68.99 69.99 70.99 71.99 72.99 73.99 74.99 75.99 76.99 77.99 78.99 79.99 80.99 81.99 82.99 83.99 84.99 85.99 86.99 87.99 88.99 89.99 90.99 91.99 92.99 93.99 94.99 95.99 96.99 97.99 98.98 99.98 199.97 299.95 399.94 499.92 599.91 699.89 799.88 899.86 1.05 1.06 1.08 1.10 1.12 1.13 1.15 1.17 1.19 1.20 1.22 1.24 1.26' 1.27 1.29 1.31 1.33 1.34 1.36 1.38 1.40 1.41 1.43 1.45 1.47 1.48 1.50 1.52 1.54 1.55 1.57 1.59 1.61 1.62 1.64 1.66 1.68 1.69 1.71 1.73 1.75 3.49 5.24 6.98 8.73 10.47 12.22 13.96 15.71 59.96 60.96 61.96 62.96 63.96 64.96 65.96 66.96 67.96 68.96 69.96 70.96 71.96 72.96 73.95 74.95 75,95 76.95 77.95 78.95 79.95 80.95 81.95 82.95 83.95 84.95 85.95 86.95 87.95 88.95 89.95 90.94 91.94 92.94 93.94 94.94 95.94 96.94 97.94 98.94 99.94 199.88 299.82 399.76 499.70 599.63 699.57 799.51 899.45 2.09 2.13 2.16 2.20 2.23 2.27 2.30 2.34 2.37 2.41 2.44 2.48 2.51 2.55 2.58 2.62 2.65 2.69 2.72 2.76 2.79 2.83 2.86 2.90 2.93 2.97 3.00 3.04 3.07 3.11 3.14 3.18 3.21 3.25 3.28 3.32 3.35 3.39 3.42 3.46 3.49 6.98 10.47 13.96 17.45 20.94 24.43 27.92 31.41 59.92 60.92 61.92 62.91 63.91 64.91 65.91 66.91 67.91 68.91 69.90 70.90 71.90 72.90 73.90 74.90 75.90 76.89 77.89 78.89 79.89 80.89 81.89 82.89 83.88 84.88 85.88 86.88 87.88 88.88 89.88 90.88 91.87 92.87 93.87 94.87 95.87 96.87 97.87 98.86 99.86 199.73 299.59 399.45 499.31 599.18 699.04 798.90 898.77 3.14 3.19 3.24 3.30 3.35 3.40 3.45 3.51 3.56 3.61 3.66 3.72 3.77 3.82 3.87 3.93 3.98 4.03 4.08 4.13 4.19 4.24 4.29 4.34 4.40 4.45 4.50 4.55 4.61 4.66 4.71 4.76 4.81 4.87 4.92 4.97 5.02 5.08 5.13 5.18 5.23 10.47 15.70 20.93 26.17 31.40 36.64 41.87 47.10 59.85 60.85 61.85 62.85 63.84 64.84 65.84 66.84 67.83 68.83 69.83 70.83 71.82 72.82 73.82 74.82 75.81 76.81 77.81 78.81 79.81 80.80 81.80 82.80 83.80 84.79 85.79 86.79 87.79 88.78 89.78 90.78 91.78 92.77 93.77 94.77 95.77 96.76 97.76 98.76 99.76 199.51 299.27 399.03 498.78 598.54 698.29 798.05 897.81 4.19 4.26 4.32 4.39 4.46 4.53 4.60 4.67 4.74 4.81 4.88 4.95 5.02 5.09 5.16 5.23 5.3( 5.37 5.44 5.51 5.58 5.65 5.72 5.79 5.86 5.93 6.00 6.07 6.14 6.21 6.28 6.35 6.42 6.49 6.56 6.63 6.70 6.77 6.84 6.91 6.98 13.95 20.93 27.90 34.88 41.85 48.83 55.80 62.78 59.77 60.77 61.76 62.76 63.76 64.75 65.75 66.75 67.74 68.74 69.73 70.73 71.73 72.72 73.72 74.71 75.71 76.71 77.70 78.70 79.70 80.69 81.69 82.68 83.68 84.68 85.67 86.67 87.67 88.66 89.66 90.65 91.65 92.65 93.64 94.64 95.63 96.63 97.63 98.62 99.62 199.24 298.86 398.48 498.10 597.72 697.34 796.96 896.58 5.23 5.32 5.40 5.49 5.58 5.67 5.75 5.84 5.93 6.01 6.10 6.19 6.28 6.36 6.45 6.54 6.62 6.71 6.80 6.89 6.97 7.06 7.15 7.23 7.32 7.41 7.50 7.58 7.67 7.76 7.84 7.93 8.02 8.11 8.19 8.28 8.37 8.45 8.54 8.63 8.72 17.43 26.15 34.86 43.58 52.29 61.01 69.72 78.44 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. dep. at. dep. at dep. lat dep. lat. dep. lat. dis. hyp. opp. adj. pp. adj opp. adj opp. adj. opp. adj. hyp. D Dsin D cos D sin D cos Dsin D cos sin D cos sin cos D 8Dsin 880 870860i85 89~ 88~ 87~ 86~ 85~ 10 TABLE VI. TRAVERSE TABLE 6~ 7~ 8~ 90 10~ D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. - 11~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~ [~~~~~~~~~~~~~~~~~~ 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 I 10.94 11.93 12.93 13.92 14.92 15.91 16.91 17.90 18.90 19.89 20.88 21.88 22.87 23.87 24.86 25.86 26.85 27.85 28.84 29.84 30.83 31.82 32.82 33.81 34.81 35.80 36.80 37.79 38.79 39.78 40.78 41.77 42.76 43.76 44.75 45.75 46.74 47.74 48.73 49.73 50.72 51.72 52.71 53.70 54.70 55.69 56.69 57.68 58.68 1.15 1.25 1.36 1.46 1.57 1.67 1.78 1.88 1.99 2.09 2.20 2.30 2.40 2.51 2.61 2.72 2.82 2.93 3.03 3.14 3.24 3.34 3.45 3.55 3.66 3.76 3.87 3.97 4.08 4.18 4.29 4.39 4.49 4.60 4.70 4.81 4.91 5.02 5.12 5.23 5.33 5.44 5.54 5.64 5.75 5.85 5.96 6.06 6.17 I 10.92 11.91 12.90 13.90 14.89 15.88 16.87 17.87 18.86 19.85 20.84 21.84 22.83 23.82 24.81 25.81 26.80 27.79 28.78 29.78 30.77 31.76 32.75 33.75 34.74 35.73 36.72 37.72 38.71 39.70 40.70 41.69 42.68 43.67 44.67 45.66 46.65 47.64 48.63 49.63 50.62 51.61 52.60 53.60 54.59 55.58 56.58 57.57 58.56 1.34 1.46 1.58 1.71 1.83 1.95 2.07 2.19 2.32 2.44 2.56 2.68 2.80 2.92 3.05 3.17 3.29 3.41 3.53 3.66 3.78 3.90 4.02 4.14 4.27 4.39 4.51. 4.63 4.75 4.87 5.00 5.12 5.24 5.36 5.48 5.61 5.73 5.85 5.97 6.09 6.22 6.34 6.46 6.58 6.70 6.82 6.95 7.07 7.19 10.89 11.88 12.87 13.86 14.85 15.84 16.83 17.82 18.82 1.53 1.67 1.81 1.95 2.09 2.23 2.37 2.51 2.64 19.81 20.80 21.79 22.78 23.77 24.76 25.75 26.74 27.73 28.72 29.71 30.70 31.69 32.68 33.67 34.66 35.65 36.64 37.63 38.62 39.61 40.60 41.59 42.58 43.57 44.56 45.55 46.54 47.53 48.52 49.51 50.50 51.49 52.48 53.47 54.46 55.46 56.45 57.44 58.43 2.78 2.92 3.06 3.20 3.34 3.48 3.62 3.76 3.90 4.04 4.18 4.31 4.45 4.59 4.73 4.87 5.01 5.15 5.29 5.43 5.57 5.71 5.85 5.98 6.12 6.26 6.40 6.54 6.68 6.82 6.96 7.10 7.24 7.38 7.52 7.65 7.79 7.93 8.07 8.21 I 10.86 11.85 12.84 13.83 14.82 15.80 16.79 17.78 18.77 19.75 20.74 21.73 22.72 23.70 24.69 25.68 26.67 27.66 28.64 29.63 30.62 31.61 32.59 33.58 34.57 35.56 36.54 37.53 38.52 39.51 40.50 41.48 42.47 43.46 44.45 45.43 46.42 47.41 48.40 49.38 50.37 51.36 52.35 53.34 54.32 55.31 56.30 57.29 58.27 1.72 1.88 2.03 2.19 2.35 2.50 2.66 2.82 2.97 3.13 3.29 3.44 3.60 3.75 3.91 4.07 4.22 4.38 4.54 4.69 4.85 5.01 5.16 5.32 5.48 5.63 5.79 5.94 6.10 6.26 6.41 6.57 6.73 6.88 7.04 7.20 7.35 7.51 7.67 7.82 7.98 8.13 8.29 8.45 8.60 8.76 8.92 9.07 9.23 10.83 11.82 12.80 13.79 14.77 15.76 16.74 17.73 18.71 19.70 20.68 21.67 22.65 23.64 24.62 25.61 26.59 27.57 28.56 29.54 30.53 31.51 32.50 33.48 34.47 35.45 36.44 37.42 38.41 39.39 40.38 41.36 42.35 43.33 44.32 45.30 46.29 47.27 48.26 49.24 50.23 51.21 52.19 53.18 54.16 55.15 56.13 57.12 58.10 1.91 2.08 2.26 2.43 2.60 2.78 2.95 3.13 3.30 3.47 3.65 3.82 3.99 4.17 4.34 4.51 4.69 4.86 5.04 5.21 5.38 5.56 5.73 5.90 6.08 6.25 6.42 6.60 6.77 6.95 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 7.12 7.29 7.47 7.64 7.81 7.99 8.16 8.34 8.51 8.68 8.86 9.03 9.20 9.38 9.55 9.72 9.90 10.07 10.25 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. dep. lat. dep. deat. ddep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp adj. hyp. D D sin D cos D sin D cos D sin D cosD sin D cos D sin D cos D 84~ 83~ 82~ 81~ 80~ TABLE VI. TRAVERSE TABLE 11 70 80 9~0 100 v I I D D cos Dsin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. i 11 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 I 59.67 60.67 61.6C 62.6P 63.65 64.64 65.64 66.63 67.63 68.62 69.62 70.61 71.61 72.6C 73.5c 74.5M 75.52 76.52 77.57 78.57 79.56 80.56 81.55 82.55 83.54 84.53 85.53 86.52 87.52 88.51 89.51 90.50 91.50 92.49 93.49 94.48 95.47 96.47 97.46 98.46 99.45 198.90 298.36 397.81 497.26 596.71 696.17 795.62 895.07 6.27 6.38 6.48 6.59 6.69 6.79 6.90 7.00 7.11 7.21 7.32 7.42 7.53 7.63 7.74 7.84 7.94 8.05 8.15 8.26 8.36 8.47 8.57 8.68 8.78 8.88 8.99 9.09 9.20 9.30 9.41 9.51 9.62 9.72 9.83 9.93 10.03 10.14 10.24 10.35 10.45 20.91 31.36 41.81 52.26 62.72 73.17 83.62 94.08 59.55 60.55 61.54 62.53 63.52 64.52 65.51 66.50 67.49 68.49 69.48 70.47 71.46 72.46 73.45 74.44 75.43 76.43 77.42 78.41 79.40 80.40 81.39 82.38 83.37 84.37 85.36 86.35 87.34 88.34 89.33 90.32 91.31 92.31 93.30 94.29 95.28 96.28 97.27 98.26 99.25 198.51 297.76 397.02 496.27 595.53 694.78 794.04 893.29 7.31 7.43 7.56 7.68 7.80 7.92 8.04 8.17 8.29 8.41 8.53 8.65 8.77 8.90 9.02 9.14 9.26 9.38 9.51. 9.63 9.75 9.87 9.99 10.12 10.24 10.36 10.48 10.60 10.72 10.85 10.97 11.09 11.21 11.33 11.46 11.58 11.70 11.82 11.94 12.07 12.19 24.37 36.56 48.75 60.93 73.12 85.31 97.50 109.68 59.42 60.41 61.4C 62.39 63.38 64.37 65.36 66.35 67.34 68.33 69.32 70.31 71.30 72.29 73.28 74.27 75.26 76.25 77.24 78.23 79.22 80.21 81.20 82.19 83.18 84.17 85.16 86.15 87.14 88.13 89.12 90.11 91.10 92.09 93.09 94.08 95.07 96.06 97.05 98.04 99.03 198.05 297.08 396.11 495.13 594.16 693.19 792.21 891.24 dep. 8.35 8.49 8.63 8.77 8.91 9.05 9.19 9.32 9.46 9.60 9.74 9.88 10.02 10.16 10.30 10.44 10.58 10.72 10.86 10.99 11.13 11.27 11.41 11.55 11.69 11.83 11.97 12.11 12.25 12.39 12.53 12.66 12.80 12.94 13.08 13.22 13.36 13.50 13.64 13.78 13.92 27.83 41.75 55.67 69.59 83.50 97.42 111.34 125.26 lat. I I 59.26 60.25 61.24 62.22 63.211 64.20 65.19 66.18 67.16 68.15 69.14 70.13 71.11 72.10 73.09 74.08 75.06 76.05 77.04 78.03 79.02 80.00 80.99 81.98 82.97 83.95 84.94 85.93 86.92 87.90 88.89 89.88 90.87 91.86 92.84 93.83 94.82 95.81 96.79 97.78 98.77 197.54 296.31 395.08 493.84 592.61 691.381: 790.15 888.921 dep. 9.39 9.54 9.70 9.86 10.01 10.17 10.32 10.48 10.64 10.79 10.95 11.11 11.26 11.42 11.58 11.73 11.89 12.05 12.20 12.36 12.51 12.67 12.83 12.98 13.14 13.30 13.45 13.61 13.77 13.92 14.08 14.24 14.39 14.55 14.70 14.86 15.02 15.17 15.33 15.49 15.64 31.29 46.93 62.57 78.22 93.86 109.50 125.15 140.79 lat. I 59.09 60.07 61.06 62.04 63.03 64.01 65.00 65.98 66.97 67.95 68.94 69.92 70.91 71.89 72.88 73.86 74.85 75.83 76.82 77.80 78.78 79.77 80.75 81.74 82.72 83.71 84.69 85.68 86.66 87.65 88.63 89.62 90.60 91.59 92.57 93.56 94.54 95.53 96.51 97.50 98.48 196.96 295.44 393.92 492.40 590.88 689.37 787.85 886.33 dep. 10.42 10.59 10.77 10.94 11.11 11.29 11.46 11.63 11.81 11.98 12.16 12.33 12.50 12.68 12.85 13.02 13.20 13.37 13.54 13.72 13.89 14.07 14.24 14.41 14.59 14.76 14.93 15.11 15.28 15.45 15.63 15.80 15.98 16.15 16.32 16.50 16.67 16.84 17.02 17.19 17.36 34.73 52.09 69.46 86.82 104.19 121.55 138.92 156.28 lat. 11II - 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. dis. dep. FI lat. II dep. lat. 'II II I I --- —-- I 11 hyp. op adj. pp aj. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj hyp. D | Dsin Dcos Di D sin D c os D sin s in Din Dcos D -- -6 in I Cs -b sn 6co 84~ 830 820 810 80~ 12 TABLE VI, TRAVERSE TABLE 11~ 12~ 13~ 14~ 15~ D os sin Dcos Dsin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. ad j. opp. adj. opp. adj. opp. 1| hyp. = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - - - -:.. dis. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 lat. dep. 1I 10.80 11.78 12.76 13.74 14.72 15.71 16.69 17.67 18.65 19.63 20.61 21.60 22.58 23.56 24.54 25.52 26.50 27.49 28.47 29.45 30.43 31.41 32.39 33.38 34.36 35.34 36.32 37.30 38.28 39.27 40.25 41.23 42.21 43.19 44.17 45.15 46.14 47.12 48.10 49.08 50.06 51.04 52.03 53.01 53.99 54.97 55.95 56.93 57.92 2.10 2.29 2.48 2.67 2.86 3.05 3.24 3.43 3.63 3.82 4.01 4.20 4.39 4.58 4.77 4.96 5.15 5.34 5.53 5.72 5.92 6.11 6.30 6.49 6.68 6.87 7.06 7.25 7.44 7.63 7.82 8.01 8.20 8.40 8.59 8.78 8.97 9.16 9.35 9.54 9.73 9.92 10.11 10.30 10.49 10.69 10.88 11.07 11.26 I I lat. dep..1 10.76 11.74 12.72 13.69 14.67 15.65 16.63 17.61 18.58 19.56 20.54 21.52 22.50 23.48 24.45 25.43 26.41 27.39 28.37 29.34 30.32 31.30 32.28 33.26 34.24 35.21 36.19 37.17 38.15 39.13 40.10 41.08 42.06 43.04 44.02 44.99 45.97 46.95 47.93 48.91 49.89 50.86 51.84 52.82 53.80 54.78 55.75 56.73 57.71 2.29 2.49 2.70 2.91 3.12 3.33 3.53 3.74 3.95 4.16 4.37 4.57 4.78 4.99 5.20 5.41 5.61 5.82 6.03 6.24 6.45 6.65 6.86 7.07 7.28 7.48 7.69 7.90 8.11 8.32 8.52 8.73 8.94 9.15 9.36 9.56 9.77 9.98 10.19 10.40 10.60 10.81 11.02 11.23 11.44 11.64 11.85 12.06 12.27 I lat. dep. I. 10.72 11.69 12.67 13.64 14.62 15.59 16.57 17.54 18.51 19.49 20.46 21.44 22.41 23.38 24.36 25.33 26.31 27.28 28.26 29.23 30.21 31.18 32.15 33.13 34.10 35.08 36.05 37.03 38.00 38.97 39.95 40.92 41.90 42.87 43.85 44.82 45.80 46.77 47.74 48.72 49.69 50.67 51.64 52.62 53.59 54.56 55.54 56.51 57.49 2.47 2.70 2.92 3.15 3.37 3.60 3.82 4.05 4.27 4.50 4.72 4.95 5.17 5.40 5.62 5.85 6.07 6.30 6.52 6.75 6.97 7.20 7.42 7.65 7.87 8.10 8.32 8.55 8.77 9.00 9.22 9.45 9.67 9.90 10.12 10.35 10.57 10.80 11.02 11.25 11.47 11.70 11.92 12.15 12.37 12.60 12.82 13.05 13.27. lat. _I dep. Li lat. dep. i 10.67 11.64 12.61 13.58 14.55 15.52 16.50 17.47 18.44 19.41 20.38 21.35 22.32 23.29 24.26 25.23 26.20 27.17 28.14 29.11 30.08 31.05 32.02 32.99 33.96 34.93 35.90 36.87 37.84 38.81 39.78 40.75 41.72 42.69 43.66 44.63 45.60 46.57 47.54 48.51 49.49 50.46 51.43 52.4C 53.37 54.34 55.31 56.28 57.2r 2.66 2.90 3.15 3.39 3.63 3.87 4.11 4.35 4.60 4.84 5.08 5.32 5.56 5.81 6.05 6.29 6.53 6.77 7.02 7.26 7.50 7.74 7.98 8.23 8.47 8.71 8.95 9.19 9.44 9.68 9.92 10.16 10.40 10.64 10.89 11.13 11.37 11.61 11.85 12.10 12.34 12.58 12.82 13.06 13.31 13.55 13.79 14.03 14.27 10.63 11.59 12.56 13.52 14.49 15.45 16.42 17.39 18.35 19.32 20.28 21,25 22.22 23.18 24.15 25.11 26.08 27.05 28.01 28.98 29.94 30.91 31.88 32.84 33.81 34.77 35.74 36.71 37.67 38.64 39.60 40.57 41.53 42.50 43.47 44.43 45.40 46.36 47.33 48.30 49.26 50.23 51.19 52.16 53.13 54.09 55.06 56.02 56.99 2.85 3.11 3.36 3.62 3.88 4.14 4.40 4.66 4.92 5.18 5.44 5.69 5.95 6.21 6.47 6.73 6.99 7.25 7.51 7.76! 8.02 8.28 8.54 8.80 9.06 9.32 9.58 9.84 10.09 10.35 10.61 10.87 11.13 11.39 11.65 11.91 12.16 12.42 12.68 12.94 13.20 13.46 13.72 13.98 14.24 14.49 14.75 15.01 15.27 I dis. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. de lat. deplep. lt. de. lat. dep. t. dep. la dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. D D sin D cos D sin D cos D sin D cos D sin D cs DD sin D cos D I0 79~ 78~ 77~ 76~ 75~ TABLE VI. TRAVERSE TABLE 13 11~ 12~ 13~ 14~ 15~ D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. 1lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. ~~~~l ~ I.......... I..... J __ — 11 = - — 1~ - - 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 I 58.90 59.88 60.86 61.84 62.82 63.81 64.79 65.77 66.75 67.73 68.71 69.70 70.68 71.66 72.64 73.62 74.60 75.59 76.57 77.55 78.53 79.51 80.49 81.48 82.46 83.44 84.42 85.40 86.38 87.36 88.35 89.33 90.31 91.29 92.27 93.25 94.24 95.22 96.20 97.18 98.16 196.33 294.49 392.65 490.81 588.98 687.14 785.30 883.46 11.45 11.64 11.83 12.02 12.21 12.40 12.59 12.78 12.98 13.17 13.36 13.55 13.74 13.93 14.12 14.31 14.50 14.69 14.88 15.07 15.26 15.46 15.65 15.84 16.03 16.22 16.41 16.60 16.79 16.98 17.17 17.36 17.55 17.75 17.94 18.13 18.32 18.51 18.70 18.89 19.08 38.16 57.24 76.32 95.40 114.49 133.57 152.65 171.73 58.69 59.67 60.65 61.62 62.60 63.58 64.56 65.54 66.51 67.49 68.47 69.45 70.43 71.40 72.38 73.36 74.34 75.32 76.30 77.27 78.25 79.23 80.21 81.19 82.16 83.14 84.12 85.10 86.08 87.06 88.03 89.01 89.99 90.97 91.95 92.92 93.90 94.88 95.86 96.84 97.81 195.63 293.44 391.26 489.07 586.89 684.70 '782.52 880.33 12.47 12.68 12.89 13.10 13.31 13.51 13.72 13.93 14.14 14.35 14.55 14.76 14.97 15.18 15.39 15.59 15.80 16.01 16.22 16.43 16.63 16.84 17.05 17.26 17.46 17.67 17.88 18.09 18.30 18.50 18.71 18.92 19.13 19.34 19.54 19.75 19.96 20.17 20.38 20.58 20.79 41.58 62.37 83.16 103.96 124.75 145.54 166.33 187.12 58.46 59.44 60.41 61.39 62.36 63.33 64.31 65.28 66.26 67.23 68.21 69.18 70.15 71.13 72.10 73.08 74.05 75.03 76.00 76.98 77.95 78.92 79.90 80.87 81.85 82.82 83.80 84.77 85.74 86.72 87.69 88.67 89.64 90.62 91.59 92.57 93.54 94.51 95.49 96.46 97.44 194.87 292.31 389.75 487.19 584.62 682.06 779.50 876.93 13.50 13.72 13.95 14.17 14.40 14.62 14.85 15.07 15.30 15.52 15.75 15.97 16.20 16.42 16.65 16.87 17.10 17.32 17.55 17.77 18.00 18.22 18.45 18.67 18.90 19.12 19.35 19.57 19.80 20.02 20.25 20.47 20.70 20.92 21.15 21.37 21.60 21.82 22.05 22.27 22.50 44.99 67.49 89.98 112.48 134.97 157.47 179.96 202.46 58.22 59.19 60.16 61.13 62.10 63.07 64.04 65.01 65.98 66.95 67.92 68.89 69.86 70.83 71.80 72.77 73.74 74.71 75.68 76.65 77.62 78.59 79.56 80.53 81.50 82.48 83.45 84.42 85.39 86.36 87.33 88.30 89.27 90.24 91.21 92.18 93.15 94.12 95.09 96.06 97.03 194.06 291.09 388.12 485.15 582.18 679.21 776.24 873.27 14.52 14.76 15.00 15.24 15.48 15.72 15.97 16.21 16.45 16.69 16.93 17.18 17.42 17.66 17.90 18.14 18.39 18.63 18.87 19.11 19.35 19.60 19.84 20.08 20.32 20.56 20.81 21.05 21.29 21.53 21.77 22.01 22.26 22.50 22.74 22.98 23.22 23.47 23.71 23.95 24.19 48.38 72.58 96.77 120.96 145.15 169.35 193.54 217.73 57.96 58.92 59.89 60.85 61.82 62.79 63.75 64.72 65.68 66.65 67.61 68.58 69.55 70.51 71.48 72.44 73.41 74.38 '75.34 76.31 77.27 78.24 79.21 80.17 81.14 82.10 83.07 84.04 85.00 85.97 86.93 87.90 88.87 89.83 90.80 91.76 92.73 93.69 94.66 95.63 96.59 193.19 289.78 386.37 482.96 579.56 676.15 772.74 869.33 15.53 15.79 16.05 16.31 16.56 16.82 17.08 17.34 17.60 17.86 18.12 18.38 18.63 18.89 19.15 19.41 19.67 19.93 20.19 20.45 20.71 20.96 21.22 21.48 21.74 22.00 22.26 22.52 22.78 23.03 23.29 23.55 23.81 24.07 24.33 24.59 24.85 25.11 25.36 25.62 25.88 51.76 77.65 103.53 129.41 155.29 181.17 207.06 232.94 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. dep. lat. dep. _ at. dep. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. D D sin D cos sin Dcos D sin D cos D sin D c os D sin D cos D 790Dsi cs1 78 770 766 co750 79~ 78~ 77~ 76~ 75~ 14 TABLE VI. TRAVERSE TABLE 16~ 17~ 18~ 19~ 20~ D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. l ~ ~ ~ 11 1,.. I -~ ~ 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 10.57 11.54 12.50 13.46 14.42 15.38 16.34 17.30 18.26 19.23 20.19 21.15 22.11 23.07 24.03 24.99 25.95 26.92 27.88 28.84 29.80 30.76 31.72 32.68 33.64 34.61 35.57 36.53 37.49 38.45 39.41 40.37 41.33 42.30 43.26 44.22 45.18 46.14 47.10 48.06 49.02 49.99 50.95 51.91 52.87 53.83 54.79 55.75 56.71 3.03 3.31 3.58 3.86 4.13 4.41 4.69 4.96 5.24 5.51 5.79 6.06 6.34 6.62 6.89 7.17 7.44 7.72 7.99 8.27 8.54 8.82 9.10 9.37 9.65 9.92 10.20 10.47 10.75 11.03 11.30 11.58 11.85 12.13 12.40 12.68 12.95 13.23 13.51 13.78 14.06 14.33 14.61 14.88 15.16 15.44 15.71 15.99 16.26 10.52 11.48 12.43 13.39 14.34 15.30 16.26 17.21 18.17 19.13 20.08 21.04 21.99 22.95 23.91 24.86 25.82 26.78 27.73 28.69 29.65 30.60 31.56 32.51 33.47 34.43 35.38 36.34 37.30 38.25 39.21 40.16 41.12 42.08 43.03 43.99 44.95 45.90 46.86 47.82 48.77 49.73 50.68 51.64 52.60 53.55 54.51 55.47 56.42 3.22 3.51 3.80 4.09 4.39 4.68 4.97 5.26 5.56 5.85 6.14 6.43 6.72 7.02 7.31 7.60 7.89 8.19 8.48 8.77 9.06 9.36 9.65 9.94 10.23 10.53 10.82 11.11 11.40 11.69 11.99 12.28 12.57 12.86 13.16 13.45 13.74 14.03 14.33 14.62 14.91 15.20 15.50 15.79 16.08 16.37 16.67 16.96 17.25 10.46 11.41 12.36 13.31 14.27 15.22 16.17 17.12 18.07 19.02 19.97 20.92 21.87 22.83 23.78 24.73 25.68 26.63 27.58 28.53 29.48 30.43 31.38 32.34 33.29 34.24 35.19 36.14 37.09 38.04 38.99 39.94 40.90 41.85 42.80 43.75 44.70 45.65 46.60 47.55 48.50 49.45 50.41 51.36 52.31 53.26 54.21 55.16 56.11 3.40 3.71 4.02 4.33 4.64 4.94 5.25 5.56 5.87 6.18 6.49 6.80 7.11 7.42 7.73 8.03 8.34 8.65 8.96 9.27 9.58 9.89 10.20 10.51 10.82 11.12 11.43 11.74 12.05 12.36 12.67 12.98 13.29 13.60 13.91 14.21 14.52 14.83 15.14 15.45 15.76 16.07 16.38 16.69 17.00 17.30 17.61 17.92 18.23 10.40 11.35 12.29 13.24 14.18 15.13 16.07 17.02 17.96 18.91 19.86 20.80 21.75 22.69 23.64 24.58 25.53 26.47 27.42 28.37 29.31 30.26 31.20 32.15 33.09 34.04 34.98 35.93 36.88 37.82 38.77 39.71 40.66 41.60 42.55 43.49 44.44 45.38 46.33 47.28 48.22 49.17 50.11 51.06 52.00 52.95 53.89 54.84 55.79 3.58 3.91 4.23 4.56 4.88 5.21 5.53 5.86 6.19 6.51 6.84 7.16 7.49 7.81 8.14 8.46 8.79 9.12 9.44 9.77 10.09 10.42 10.74 11.07 11.39 11.72 12.05 12.37 12.70 13.02 13.35 13.67 14.00 14.32 14.65 14.98 15.30 15.63 15.95 16.28 16.60 16.93 17.26 17.58 17.91 18.23 18.56 18.88 19.21 10.34. 11.28 12.22 13.16 14.10 15.04 15.97 16.91 17.85 18.79 19.73 20.67 21.61 22.55 23.49 24.43 25.37 26.31 27.25 28.19 29.13 30.07 31.01 31.95 32.89 33.83 34.77 35.71 36.65 37.59 38.53 39.47 40.41 41.35 42.29 43.23 44.17 45.11 46.04 46.98 47.92 48.86 49.80 50.74 51.68 52.62 53.56 54.50 55.44 3.76 4.10 4.45 4.79 5.13 5.47 5.81 6.16 6.50 6.84 7.18 7.52 7.87 8.21 8.55 8.89 9.23 9.58 9.92 10.26 10.60 10.94 11.29 11.63 11.97 12.31 12.65 13.00 13.34 13.68 14.02 14.36 14.71 15.05 15.39 15.73 16.07 16.42 16.76 17.10 17.44 17.79 18.13 18.47 18.81 19.15 19.50 19.84 20.18 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. depep. l.t. p. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. D ID sin D cos D sin D cos D sin D cos D sin D cos D sin D cos D 74~ 73~ 72~ 71~ 70~ TABLE VI. TRAVERSE TABLE 15 16~ 17~ 18~ 19~ 20~ D D cos D sin D cos D sin D cos D sin D cos D sin hyp. ajadj. opp.adj. opp. adj. opp. dis. lat. dep. lat. dep. lat. dep. lat. dep. ~l I........ -11~.s _.,. _s.,, 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 57'.68 58.64 59.60 60.56 61.52 62.48 63.44 64.40 65.37 66.33 67.29 68.25 69.21 70.17 71.13 72.09 73.06 74.02 74.98 75.94 76.90 77.86 78.82 79.78 80.75 81.71 82.67 83.63 84.59 85.55 86.51 87.47 88.44 89.40 90.36 91.32 92.28 93.24 94.20 95.16 96.13 192.25 288.38 384.50 480.63 576.76 672.88 769.01 865.14 16.54 16.81 17.09 17.37 17.64 17.92 18.19 18.47 18.74 19.02 19.29 19.57 19.85 20.12 20.40 20.67 20.95 21.22 21.50 21.78 22.05 22.33 22.60 22.88 23.15 23.43 23.70 23.98 24.26 24.53 24.81 25.08 25.36 25.63 25.91 26.19 26.46 26.74 27.01 27.29 27.56 55.13 82.69 110.25 137.82 165.38 192.95 220.51 248.07 57.38 58.33 59.29 60.25 61.2( 62.1( 63.12 64.07 65.03 65.99 66.94 67.9C 68.8C 69.81 70.77 71.72 72.68 73.64 74.59 75.55 76.50 77.46 78.42 79.37 80.33 81 29 82.24 83.20 84.15 85.11 86.07 87.02 87.98 88.94 89.89 90.85 91.81 92.76 93.72 94.67 95.63 191.26 286.89 382.52 478.15 573.78 669.41 765.04 860.67 17.54 17.83 18.13 18.42 18.71 19.00 19.30 19.59 19.88 20.17 20.47 20.76 21.05 21.34 21.64 21.93 22.22 22.51 22.80 23.10 23.39 23.68 23.97 24.27 24.56 24.85 25.14 25.44 25.73 26.02 26.31 26.61 26.90 27.19 27.48 27.78 28.07 28.36 28.65 28.94 29.24 58.47 87.71 116.95 146.19 175.42 204.66 233.90 263.13 57.06 58.01 58.97 59.92 60.87 61.82 62.77 63.72 64.67 65.62 66.57 67.53 68.48 69.43 70.38 71.33 72.28 73.23 74.18 75.13 76.08 77.04 77.99 78.94 79.89 80.84 81.79 82.74 83.69 84.64 85.60 86.55 87.50 88.45 89.40 90.35 91.30 92.25 93.20 94.15 95.11 190.21 285.32 380.42 475.53 570.63 665.74 760.85 855.95 18.54 18.85 19.16 19.47 19.78 20.09 20.40 20.70 21.01 21.32 21.63 21.94 22.25 22.56 22.87 23.18 23.49 23.79 24.10 24.41 24.72 25.03 25.34 25.65 26.96 26.27 26.58 26.88 27.19 27.50 27.81 28.12 28.43 28.74 29.05 29.36 29.67 29.97 30.28 30.59 30.90 61.80 92.71 123.61 154.51 185.41 216.31 247.21 278.12 56.73 57.68 58.62 59.57 60.51 61.46 62.40 63.35 64.30 65.24 66.19 67.13 68.08 69.02 69.97 70.91 71.86 72.80 73.75 74.70 75.64 76.59 77.53 78.48 79.42 80.37 81.31 82.26 83.21 84.15 85.10 86.04 86.99 87.93 88.88 89.82 90.77 91.72 92.66 93.61 94.55 189.10 283.66 378.21 472.76 567.31 661.86 756.42 850.97 19.53 19.86 20.19 20.51 20.84 21.16 21.49 21.81 22.14 22.46 22.79 23.12 23.44 23.77 24.09 24.42 24.74 25.07 25.39 25.72 26.05 26.37 26.70 27.02 27.35 27.67 28.00 28.32 28.65 28.98 29.30 29.63 29.95 30.28 30.60 30.93 31.25 31.58 31.91 32.23 32.56 65.11 97.67 130.23 162.78 195.34 227.90 260.45 293.01 D cos D sin adj. opp. lat. dep. 56.38 20.52 57.32 20.86 58.26 21.21 59.20 21.55 60.14 21.89 61.08 22.23 62.02 22.57 62.96 22.92 63.90 23.26 64.84 23.60 65.78 23.94 66.72 24.28 67.66 24..63 68.60 24.97 69.54 25.31 70.48 25.65 71.42 25.99 72.36 26.34 73.30 26.68 74.24 27.02 75.18 27.36 76.12 27.70 77.05 28.05 77.99 28.39 78.93 28.73 79.87 29.07 80.81 29.41 81.75 29.76 82.69 30.10 83.63 30.44 84.57 30.78 85.51 31.12 86.45 31.47 87.39 31.81 88.33 32.15 89.27 32.49 90.21 32.83 91.15 33.18 92.09 33.52 93.03 33.86 93.97 34.20 187.94 68.40 281.91 102.61 375.88 136.81 469.85171.01 563.82 205.21 657.79 239.41 751.75273.62 845.72 307.82 D hyp. dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 -- I,, dis. dep, lat. dep. lat dep. lat. dep, lat, depp. lat. dis. hyp. opp. adj. opp aadj. opp. adj. opp, adj. opp. adj. hyp. D Dsin Dcos Dsin Dcos Dsin Dcos Dsin Dcos Dsin Dcos D 74~ 73~ 72~ 71~ 700 16 TABLE VI. TRAVERSE TABLE 210 22~ 230 24~ 250 D Dcos D sin Dcos Dsin D cos D sin D cos Dsin Dcos Dsin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. ____ I I 1 ~ ~ 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 10.27 11.20 12.14 13.07 14.00 14.94 15.87 16.80 17.74 18.67 19.61 20.54 21.47 22.41 23.34 24.27 25.21 26.14 27.07 28.01 28.94 29.87 30.81 31.74 32.68 33.61 34.54 35.48 36.41 37.34 38.28 39.21 40.14 41.08 42.01 42.94 43.88 44.81 45.75 46.68 47.61 48.55 49.48 50.41 51.35 52.28 53.21 54.15 55.08 3.94 4.30 4.66 5.02 5.38 5.73 6.09 6.45 6.81 7.17 7.53 7.88 8.24 8.60 8.96 9.32 9.68 10.03 10.39 10.75 11.11 11.47 11.83 12.18 12.54 12.90 13.26 13.62 13.98 14.33 14.69 15.05 15.41 15.77 16.13 16.48 16.84 17.20 17.56 17.92 18.28 18.64 18.99 19.35 19.71 20.07 20.43 20.79 21.14 10.20 11.13 12.05 12.98 13.91 14.83 15.76 16.69 17.62 18.54 19.47 20.40 21.33 22.25 23.18 24.11 25.03 25.96 26.89 27.82 28.74 29.67 30.60 31.52 32.45 33.38 34.31 35.23 36.16 37.09 38.01 38.94 39.87 40.80 41.72 42.65 43.58 44.50 45.43 46.36 47.29 48.21 49.14 50.07 51.00 51.92 52.85 53.78 54.70 4.12 4.50 4.87 5.24 5.62 5.99 6.37 6.74 7.12 7.49 7.87 8.24 8.62 8.99 9.37 9.74 10.11 10.49 10.86 11.24 11.61 11.99 12.36 12.74 13.11 13.49 13.86 14.24 14.61 14.98 15.36 15.73 16.11 16.48 16.86 17.23 17.61 17.98 18.36 18.73 19.10 19.48 19.85 20.23 20.60 20.98 21.35 21.73 22.10 10.13 11.05 11.97 12.89 13.81 14.73 15.65 16.57 17.49 18.41 19.33 20.25 21.17 22.09 23.01 23.93 24.85 25.77 26.69 27.62 28.54 29.46 30.38 31.30 32.22 33.14 34.06 34.98 35.90 36.82 37.74 38.66 39.58 40.50 41.42 42.34 43.26 44.18 45.10 46.03 46.95 47.87 48.79 49.71 50.63 51.55 52.47 53.39 54.31 4.30 4.69 5.08 5.47 5.86 6.25 6.64 7.03 7.42 7.81 8.21 8.60 8.99 9.38 9.77 10.16 10.55 10.94 11.33 11.72 12.11 12.50 12.89 13.28 13.68 14.07 14.46 14.85 15.24 15.63 16.02 16.41 16.80 17.19 17.58 17.97 18.36 18.76 19.15 19.54 19.93 20.32 20.71 21.10 21.49 21.88 22.27 22.66 23.05 10.05 10.96 11.88 12.79 13.70 14.62 15.53 16.44 17.36 18.27 19.18 20.10 21.01 21.93 22.84 23.75 24.67 25.58 26.49 27.41 28.32 29.23 30.15 31.06 31.97 32.89 33.80 34.71 35.63 36.54 37.46 38.37 39.28 40.20 41.11 42.02 42.94 43.85 44.76 45.68 46.59 47.50 48.42 49.33 50.24 51.16 52.07 52.99 53.90 4.47 4.88 5.29 5.69 6.10 6.51 6.91 7.32 7.73 8.13 8.54 8.95 9.35 9.76 10.17 10.58 10.98 11.39 11.80 12.20 12.61 13.02 13.42 13.83 14.24 14.64 15.05 15.46 15.86 16.27 16.68 17.08 17.49 17.90 18.30 18.71 19.12 19.52 19.93 20.34 20.74 21.15 21.56 21.96 22.37 22.78 23.18 23.59 24.00 9.97 10.88 11.78 12.69 13.59 14.50 15.41 16.31 17.22 18.13 19.03 19.94 20.85 21.75 22.66 23.56 24.47 25.38 26.28 27.19 28.10 29.00 29.91 30.81 31.72 32.63 33.53 34.44 35.35 36.25 37.16 38.06 38.97 39.88 40.78 41.69 42.60 43.50 44.41 45.32 46.22 47.13 48.03 48.94 49.85 50.75 51.66 52.57 53.47 4.65 5.07 5.49 5.92 6.34 6.76 7.18 7.61 8.03 8.45 8.87 9.30 9.72 10.14 10.57 10.99 11.41 11.83 12.26 12.68 13.10 13.52 13.95 14.37 14.79 15.21 15.64 16.06 16.48 16.90 17.33 17.75 18.17 18.60 19.02 19.44 19.86 20.29 20.71 21.13 21.55 21.98 22.40 22.82 23.24 23.67 24.09 24.51 24.93 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. Ist. atdep. lat. dep. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. op. adj. opp. adj. opp. adj. hyp. D D sin D cos D sin D cos D sin D cos D sin D cos D sin Dcos D 69~ 68~ 67~ 66~ 65~ TABLE VI. TRAVERSE TABLE 17 21~ 22~ 23~ 24~ 250 D D cos D sin II D cos ) sin D cos D sin D cos D sin D cos D sin D hyp. Idj. p. adj. opp. adj. op adj. opp. adj. opp. hyp. dis. lat. I dep. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 56.01 56.95 57.88 58.82 59.75 60.68 61.62 62.55 63.48 64.42 65.35 66.28 67.22 68.15 69.08 70.02 70.9[ 71.89 72.82 73.75 74.69 75.62 76.55 77.49 78.42 79.35 80.29 81.22 82.16 83.09 84.02 84.96 85.89 86.82 87.76 88.69 89.62 90.56 91.49 92.42 93.36 186.72 280.07 373.43 466.79 560.15 653.51 746.86 840.22 21.50 21.86 22.22 22.58 22.94 23.29 23.65 24.01 24.37 24.73 25.09 25.44 25.80 26.16 26.52 26.88 27.24 27.59 27.95 28.31 28.67 29.03 29.39 29.74 30.10 30.46 30.82 31.18 31.54 31.89 32.25 32.61 32.97 33.33 33.69 34.04 34.40 34.76 35.12 35.48 35.84 71.67 107.51 143.35 179.18 215.02 250.86 286.69 322.53 I I I I lat. I dep. lat. dep. lat. dep. lat. dep. 55.63 56.56 57.49 58.41 59.34 60.27 61.19 62.12 63.05 63.98 64.90 65.83 66.76 67.68 68.61 69.54 70.47 71.39 72.32 73.25 74.17 75.10 76.03 76.96 77.88 78.81 79.74 80.66 81.59 82.52 83.45 84.37 85.30 86.23 87.16 88.08 89.01 89.94 90.86 91. 79 92.72 185.44 278.16 370.87 463.59 556.31 649.03 741.75 834.47 22.48 22.85 23.23 23.6C 23.97 24.35 24.72 25.1C 25.47 25.85 26.22 26.6C 26.97 27.35 27.72 28.1C 28.47 28.84 29.22 29.59 29.97 30.34 30.72 31. 09 31.47 31.84 32.22 32.59 32.97 33.34 33.71 34.09 34.46 34.84 35.21 35.59 35.96 36.34 36.71 37.09 37.46 74.92 112.38 149.84 187.30 224.76 262.22 299.69 337.15 55.23 56.15 57.07 57.99 58.91 59.83 60.75 61.67 62.59 63.51 64.44 65.36 66.28 67.20 68.12 69.04 69.96 70.88 71.80 72.72 73.64 74.56 75.48 76.40 77.32 78.24 79.16 80.08 81.00 81.92 82.85 83.77 84.69 85.61 86.53 87.45 88.37 89.29 90.21 91.13 92.05 184.10 276.15 368.20 460.25 552.30 644.35 736.40 828.45 23.44 23.83 24.23 24.62 25.01 25.40 25.79 26.18 26.57 26.96 27.35 27.74 28.13 28.52 28.91 29.30 29.70 30.09 30.48 30.87 31.26 31.65 32.04 32.43 32.82 33.21 33.60 33.99 34.38 34.78 35.17 35.56 35.95 36.34 36.73 37.12 37.51 37.90 38.29 38.68 39.07 78.15 117.22 156.29 195.37 234.44 273.51 312.58 351.66 54.81 55.73 56.64 57.55 58.47 59.38 60.2c 61.21 62.12 63.03 63.95 64.86 65.78 66.69 67.6C 68.52 69.43 70.34 71.26 72.17 73.08 74.00 74.91 75.82 76.74 77.65 78.56 79.48 80.39 81.31 82.22 83.13 84.05 84.96 85.87 86.79 87.70 88.61 89.53 90.44 91.35 182.71 274.06 365.42 456.77 548.13 639.48 730.84 822.19 24.40 24.81 25.22 25.62 26.03 26.44 26.84 27.25 27.66 28.06 28.47 28.88 29.28 29.69 30.10 30.51 30.91 31.32 31.73 32.13 32.54 32.95 33.35 33.76 34.17 34.57 34.98 35.39 35.79 36.20 36.61 37.01 37.42 37.83 38.23 38.64 39.05 39.45 39.86 40.27 40.67 81.35 122.02 162.69 203.37 244.04 284.72 325.39 366.06 54.38 55.28 56.19 57.10 58.00 58.91 59.82 60.72 61.63 62.54 63.44 64.35 65.25 66.16 67.07 67.97 68.88 69.79 70.69 71.60 72.50 73.41 74.32 75.22 76.13 77.04 77.94 78.85 79.76 80.66 81.57 82.47 83.38 84.29 85.19 86.10 87.01 87.91 88.82 89.72 90.63 181.26 271.89 362.52 453.15 543.78 634.42 725.05 815.68 25.36 25.78 26.20 26.62 27.05 27.47 27.89 28.32 28.74 29.16 29.58 30.01 30.43 30.85 31.27 31.70 32.12 32.54 32.96 33.39 33.81 34.23 34.65 35.08 35.50 35.92 36.35 36.77 37.19 37.61 38.04 38.46 38.88 39.30 39.73 40.15 40.57 40.99 41.42 41.84 42.26 84.52 126.79 169.05 211.31 253.57 295.83 338.09 380.36 dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 -- 11 11 I I,, 11 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp. ad opp.adj. o opp. adj. opp. adj. opp. adj. hyp. D- Dsin D cos s D si n Dcos sin osD sin D cos D 69~ 680 67~ 66~ 65~ 18 TABLE VI. TRAVERSE TABLE 26~ 270 280 290 300 D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.89 10.79 11.68 12.58 13.48 14.38 15.28 16.18 17.08 17.98 18.87 19.77 20.67 21.57 22.47 23.37 24.27 25.17 26.07 26.96 27.86 28.76 29.66 30.56 31.46 32.36 33.26 34.15 35.05 35.95 36.85 37.75 38.65 39.55 40.45 41.34 42.24 43.14 44.04 44.94 45.84 46.74 47.64 48.53 49.43 50.33 51.23 52.13 53.03 4.82 5.26 5.70 6.14 6.58 7.01 7.45 7.89 8.33 8.77 9.21 9.64 10.08 10.52 10.96 11.40 11.84 12.27 12.71 13.15 13.59 14.03 14.47 14.90 15.34 15.78 16.22 16.66 17.10 17.53 17.97 18.41 18.85 19.29 19.73 20.17 20.60 21.04 21.48 21.92 22.36 22.80 23.23 23.67 24.11 24.55 24.99 25.43 25.86 lat. dep. 9.80 10.69 11.58 12.47 13.37 14.26 15.15 16.04 16.93 17.82 18.71 19.60 20.49 21.38 22.28 23.17 24.06 24.95 25.84 26.73 27.62 28.51 29.40 30.29 31.19 32.08 32.97 33.86 34.75 35.64 36.53 37.42 38.31 39.20 40.10 40.99 41.88 42.77 43.66 44.55 45.44 46.33 47.22 48.11 49.01 49.90 50.79 51.68 52.57 4.99 5.45 5.90 6.36 6.81 7.26 7.72 8.1.7 8.63 9.08 9.53 9.99 10.44 10.90 11.35 11.80 12.26 12.71 13.17 13.62 14.07 14.53 14.98 15.44 15.89 16.34 16.80 17.25 17.71 18.16 18.61 19.07 19.52 19.98 20.43 20.88 21.34 21.79 22.25 22.70 23.15 23.61 24.06 24.52 24.97 25.42 25.88 26.33 26.79 lat. dep. lat. dep. lat. dep. dis. 9.71 10.60 11.48 12.36 13.24 14.13 15.01 15.89 16.78 17.66 18.54 19.42 20.31 21.19 22.07 22.96 23.84 24.72 25.61 26.49 27.37 28.25 29.14 30.02 30.90 31.79 32.67 33.55 34.43 35.32 36.20 37.08 37.97 38.85 39.73 40.62 41.50 42.38 43.26 44.15 45.03 45.91 46.80 47.68 48.56 49.45 50.33 51.21 52.09 5.16 5.63 6.10 6.57 7.04 7.51 7.98 8.45 8.92 9.39 9.86 10.33 10.80 11.27 11.74 12.21 12.68 13.15 13.61 14.08 14.55 15.02 15.49 15.96 16.43 16.90 17.37 17.84 18.31 18.78 19.25 19.72 20.19 20.66 21.13 21.60 22.07 22.53 23.00 23.47 23.94 24.41 24.88 25.35 25.82 26.29 26.76 27.23 27.70 9.62 10.50 11.37 12.24 13.12 13.99 14.87 15.74 16.62 17.49 18.37 19.24 20.12 20.99 21.87 22.74 23.61 24.49 25.36 26.24 27.11 27.99 28.86 29.74 30.61 31.49 32.36 33.24 34.11 34.98 35.86 36.73 37.61 38.48 39.36 40.23 41.11 41.98 42.86 43.73 44.61 45.48 46.35 47.23 48.10 48.98 49.85 50.73 51.60 5.33 5.82 6.30 6.79 7.27 7.76 8.24 8.73 9.21 9.70 10.18 10.67 11.15 11.64 12.12 12.61 13.09 13.57 14.06 14.54 15.03 15.51 16.00 16.48 16.97 17.45 17.94 18.42 18.91 19.39 19.88 20.36 20.85 21.33 21.82 22.30 22.79 23.27 23.76 24.24 24.73 25.21 25.69 26.1.8 26.66 27.15 27.63 28.12 28.60 9.53 10.39 11.26 12.12 12.99 13.86 14.72 15.59 16.45 17.32 18.19 19.05 19.92 20.78 21.65 22.52 23.38 24.25 25.11 25.98 26.85 27.71 28.58 29.44 30.31 31.18 32.04 32.91 33.77 34.64 35.51 36.37 37.24 38.11 38.97 39.84 40.70 41.57 42.44 43.30 44.17 45.03 45.90 46.77 47.63 48.50 49.36 50.23 51.10 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.50 13.00 13.50 14.00 14.50 15.00 15.50 16.00 16.50 17.00 17.50 18.00 18.50 19.00 19.50 20.00 20.50 21.00 21.50 22.00 22.50 23.00 23.50 24.00 24.50 25.00 25.50 26.00 26.50 27.00 27.50 28.00 28.50 29.00 29.50 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. D D sin D cosI D sin D cos D sin D cos D sin D cos D sin D cos I D 640P 630s 620n 610 600 64~ 63~ 62~ 61~ 60~ TABLE VI. TRAVERSE TABLE 19 26~ 27~ 28~ 29~ 30~ D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. o pp. a d. opp. p adj. opp. adj. opp. hyp. dis. lat dep at. dep. lat. dep. lat. dep. l dep. lat. dep. dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 53.93 54.83 55.73 56.62 57.52 58.42 59.32 60.22 61.12 62.02 62.92 63.81 64.71 65.61 66.51 67.41 68.31 69.21 70.11 71.00 71.90 72.80 73.70 74.60 75.50 76.40 77.30 78.20 79.09 79.99 80.89 81.79 82.69 83.59 84.49 85.39 86.28 87.18 88.08 88.98 89.88 179.76 269.64 359.52 449.40 539.28 629.16 719.04 808.91 26.30 26.74 27.18 27.62 28.06 28.49 28.93 29.37 29.81 30.25 30.69 31.12 31.56 32.00 32.44 32.88 33.32 33.75 34.19 34.63 35.07 35.51 35.95 36.38 36.82 37.26 37.70 38.14 38.58 39.01 39.45 39.89 40.33 40.77 41.21 41.65 42.08 42.52 42.96 43.40 43.84 87.67 131.51 175.35 219.19 263.02 306.86 350.7C 394.53 I I I I I I I I I I I I j 53.46 54.35 55.24 56.13 57.02 57.92 58.81 59.70 60.59 61.48 62.37 63.26 64.15 65.04 65.93 66.83 67.72 68.61 69.50 70.39 71.28 72.17 73.06 73.95 74.84 75.74 76.63 77.52 78.41 79.30 80.19 81.08 81.97 82.86 83.75 84.65 85.54 86.43 87.32 88.21 89.10 178.20 267.30 356.40 445.50 534.60 623.70 712.81 801.91 27.24 27.69 28.15 28.6C 29.06 29.51 29.96 30.42 30.87 31.32 31.78 32.22 32.6c 33.14 33.6( 34.03 34.5C 34.96 35.41 35.81 36.32 36.77 37.23 37.68 38.14 38.59 39.04 39.5C 39.93 40.41 40.86 41.31 41.71 42.22 42.68 43.1: 43.58 44.04 44.49 44.93 45.4( 90.8( 136.2( 181.6( 227.0( 272.39 317.79 363.19 408.5c 52.98 53.86 54.74 55.63 56.51 57.39 58.27 59.16 60.04 60.92 61.81 62.69 63.57 64.46 65.34 66.22 67.10 67.99 68.87 69.75 70.64 71.52 72.40 73.28 74.17 75.05 75.93 76.82 77.70 78.58 79.47 80.35 81.23 82.11 83.00 83.88 84.76 85.65 86.53 87.41 88.29 176.59 264.88 353.18 441.47 529.77 618.06 706.36 794.65 28.17 28.64 29.11 29.58 30.05 30.52 30.99 31.45 31.92 32.39 32.86 33.33 33.80 34.27 34.74 35.21 35.68 36.15 38.62 37.09 37.56 38.03 38.50 38.97 39.44 39.91 40.37 40.84 41.31 41.78 42.25 42.72 43.19 43.66 44.13 44.60 45.07 45.54 46.01 46.48 46.95 93.89 140.84 187.79 234.74 281.68 328.63 375.58 422.52 11 52.48 53.35 54.23 55.10 55.98 56.85 57.72 58.60 59.47 60.35 61.22 62.10 62.97 63.85 64.72 65.60 66.47 67.35 68.22 69.09 69.97 70.84 71.72 72.59 73.47 74.34 75.22 76.09 76.97 77.84 78.72 79.59 80.46 81.34 82.21 83.09 83.96 84.84 85.71 86.59 87.46 174.92 262.39 349.85 437.31 524.77 612.23 699.70 787.16 29.09 29.57 30.06 30.54 31.03 31.51 32.00 32.48 32.97 33.45 33.94 34.42 34.91 35.39 35.88 36.36 36.85 37.33 37.82 38.30 38.78 39.27 39.75 40.24 40.72 41.21 41.69 42.18 42.66 43.15 43.63 44.12 44.60 45.09 45.57 46.06 46.54 47.03 47.51 48.00 48.48 96.96 145.44 193.92 242.4C 290.89 339.37 387.83 436.33 51.96 52.83 53.69 54.56 55.43 56.29 57.16 58.02 58.89 59.76 60.62 61.49 62.35 63.22 64.09 64.95 65.82 66.68 67.55 68.42 69.28 70.15 71.01 71.88 72.75 73.61 74.48 75.34 76.21 77.08 77.94 78.81 79.67 80.54 81.41 82.27 83.14 84.00 84.87 85.74 86.60 173.21 259.81 346.41 433.01 519.62 606.22 692.82 779.42 30.00 30.50 31.00 31.50 32.00 32.50 33.00 33.50 34.00 34.50 35.00 35.50 36.00 36.50 37.00 37.50 38.00 38.50 39.00 39.50 40.00 40.50 41.00 41.50 42.00 42.50 43.00 43.50 44.00 44.50 45.00 45.50 46.00 46.50 47.00 47.50 48.00 48.50 49.00 49.50 50.00 100.00 150.00 200.00 250.00 300.00 350.00 400.00 450.00 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. dep. dep. l a t. de p. la t. d ep. la t. d ep. la t. dep. at. dis. hyp. opp. adj. opp. adj. op j opp. adj. opp. adj. hyp. D D sin D cos D sin D cos D sin Dcos D sin D cos D sin D cos D 640n 630 62061060 64~ 63~ 62~ 61~ 69~ 20 TABLE VI. TRAVERSE TABLE 31~ 32~ 33~ 340 35~ D D cos D sin 1( D cos D sin j D cos _ n D sin D cos | D sin 11 D cos j D sin D hyp. adj. opp. adj. opp. adj. opp. adjo opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 9.43 10.29 11.14 12.00 12.86 13.71 14.57 15.43 16.29 17.14 18.00 18.86 19.71 20.57 21.43 22.29 23.14 24.00 24.86 25.71 26.57 27.43 28.29 29.14 30.00 30.86 31.72 32.57 33.43 34.29 35.14 36.00 36.86 37.72 38.57 39.43 40.29 41.14 42.00 42.86 43.72 44.57 45.43 46.29 47.14 48.00 48.86 49.72 50.57 5.67 6.18 6.70 7.21 7.73 8.24 8.76 9.27 9.79 10.30 10.82 11.33 11.85 12.36 12.88 13.39 13.91 14.42 14.94 15.45 15.97 16.48 17.00 17.51 18.03 18.54 19.06 19.57 20.09 20.60 21.12 21.63 22.15 22.66 23.18 23.69 24.21 24.72 25.24 25.75 26.27 26.78 27.30 27.81 28.33 28.84 29.36 29.87 30.39 9.33 10.18 11.02 11.87 12.72 13.57 14.42 15.26 16.11 16.96 17.81 18.66 19.51 20.35 21.20 22.05 22.90 23.75 24.59 25.44 26.29 27.14 27.99 28.83 29.68 30.53 31.38 32.23 33.07 33.92 34.77 35.62 36.47 37.31 38.16 39.01 39.86 40.71 41.55 42.40 43.25 44.10 44.95 45.79 46.64 47.49 48.34 49.19 50.03 5.83 6.36 6.89 7.42 7.95 8.48 9.01 9.54 10.07 10.60 11.13 11.66 12.19 12.72 13.25 13.78 14.31 14.84 15.37 15.90 16.43 16.96 17.49 18.02 18.55 19.08 19.61 20.14 20.67 21.20 21.73 22.26 22.79 23.32 23.85 24.38 24.91 25.44 25.97 26.50 27.03 27.56 28.09 28.62 29.15 29.68 30.21 30.74 31.27 9.23 10.06 10.90 11.74 12.58 13.42 14.26 15.10 15.93 16.77 17.61 18.45 19.29 20.13 20.97 21.81 22.64 23.48 24.32 25.16 26.00 26.84 27.68 28.51 29.35 30.19 31.03 31.87 32.71 33.55 34.39 35.22 36.06 36.90 37.74 38.58 39.42 40.26 41.09 41.93 42.77 43.61 44.45 45.29 46.13 46.97 47.80 48.64 49.48 5.99 6.54 7.08 7.62 8.17 8.71 9.26 9.80 10.35| 10.89 11.44 11.98 12.53 13.07 13.62 14.16 14.71 15.25 15.79 16.34 16.88 17.43 17.97 18.52 19.06 19.61 20.15 20.70 21.24 21.79 22.33 22.87 23.42 23.96 24.51 25.05 25.60 26.14 26.69 27.23 27.78 28.32 28.87 29.41 29.96 30.50 31.04 31.59 32.13 9.12 9.95 10.78 11.61 12.44 13.26 14.09 14.92 15.75 16.58 17.41 18.24 19.07 19.90,20.73 21.55 22.38 23.21 24.04 24.87 25.70 26.53 27.36 28.19 29.02 29.85 30.67 31.50 32.33 33.16 33.99 34.82 35.65 36.48 37.31 38.14 38.96 39.79 40.62 41.45 42.28 43.11 43.94 44.77 45.60 46.43 47.26 48.08 48.91 6.15 6.71 7.27 7.83 8.39 8.95 9.51 10.07 10.62 11.18 11.74 12.30 12.86 13.42 13.98 14.54 15.10 15.66 16.22 16.78 17.33 17.89 18.45 19.01 19.57 20.13 20.69 21.25 21.81 22.37 22.93 23.49 24.05 24.60 25.16 25.72 26.28 26.84 27.40 27.96 28.52 29.08 29.64 30.20 30.76 31.31 31.87 32.43 32.99 9.01 9.83 10.65 11.47 12.29 13.11 13.93 14.74 15.56 16.38 17.20 18.02 18.84 19.66 20.48 21.30 22.12 22.94 23.76 24.57 25.39 26.21 27.03 27.85 28.67 29.49 30.31 31.13 31.95 32.77 33.59 34.40 35.22 36.04 36.86 37.68 38.50 39.32 40.14 40.96 41.78 42.60 43.42 44.23 45.05 45.87 46.69 47.51 48.33 6.31 6.88 7.46 8.03 8.60 9.18 9.75 10.32 10.90 11.47 12.05 12.62 13.19 13.77 14.34 14.91 15.49 16.06 16.63 17.21 17.78 18.35 18.93 19.50 20.08 20.65 21.22 21.80 22.37 22.94 23.52 24.09 24.66 25.24 25.81 26.38 26.96 27.53 28.11 28.68 29.25 29.83 30.40 30.97 31.55 32.12 32.69 33.27 33.84 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp.. adj. opp. adj. opp. adj. opp. adj. hyp. D D sin D cos D sin D cos D sin D cos D sin D cos D sin D cos D 590 580 570 56- i 59~ 58~ 57~ 56~ 55~ TABLE VI. TRAVERSE TABLE 21 R10 32~ 330 '340 350 D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. la. dep. lat. dep. lat. dep. lat. dep. dis.........I 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 51.43 52.29 53.14 54.00 54.86 55.72 56.57 57.43 58.29 59.14 60.00 60.86 61.72 62.57 63.43 64.29 65.14 66.00 66.86 67.72 68.57 69.43 70.29 71.14 72.00 72.86 73.72 74.57 75.43 76.29 77.15 78.00 78.86 79.72 80.57 81.43 82.29 83.15 84.00 84.86 85.72 171.43 257.15 342.87 428.58 514.30 600.02 685.73 771.45 30.90 31.42 31.93 32.45 32.96 33.48 33.99 34.51 35.02 35,54 36.05 36.57 37.08 37.60 38.11 38.63 39.14 39.66 40.1.7 40.69 41.20 41.72 42.23 42.75 43.26 43.78 44.29 44.81 45.32 45.84 46.35 46.87 47.38 47.90 48.41 48.93 49.44 49.96 50.47 50.99 51.50 103.01 154.51 206.02 257.52 309.02 360.53 412.03 463.53 50.88 51.73 52.58 53.43 54.28 55.12 55.97 56.82 57.67 58.52 59.36 60.21 61.06 61.91 62.76 63.60 64.45 65.30 66.15 67.00 67.84 68.69 69.54 70.39 71.24 72.08 72.93 73.78 74.63 75.48 76.32 77.17 78.02 78.87 79.72 80.56 81.41 82.26 83.11 83.96 84.80 169.61 254.41 339.22 424.02 508.83 593.63 678.44 763.24 31.80 32.33 32.85 33.38 33.91 34.44 34.97 35.50 36.03 36.56 37.09 37.62 38.15 38.68 39.21 39.74 40.27 40.80 41.33 41.86 42.39 42.92 43.45 43.98 44.51 45.04 45.57 46.10 46.63 47.16 47.69 48.22 48.75 49.28 49.81 50.34 50.87 51.40 51.93 52.46 52.99 105.98 158.98 211.97 264.96 317.95 370.94 423.94 476.93 I I 50.32 51.16 52.00 52.84 53.67 54.51 55.35 56.19 57.03 57.87 58.71 59.55 60.38 61.22 62.06 62.90 63.74 64.58 65.42 66.25 67.09 67.93 68.77 69.61 70.45 71.29 72.13 72.96 73.80 74.64 75.48 76.32 77.16 78.00 78.83 79.67 80.51 81.35 82.19 83.03 83.87 167.73 251.60 335.47 419.34 503.20 587.07 670.94 754.80 32.68 33.22 33.77 34.31. 34.86 35.40 35.95 36.49 37.04 37.58 38.12 38.67 39.21 39.76 40.30 40.85 41.39 41.94 42.48 43.03 43.57 44.12 44.66 45.20 45.75 46.29 46.84 47.38 47.93 48.47 49.02 49.56 50.11 50.65 51.20 51.74 52.29 52.83 53.37 53.92 54.46 108.93 163.39 217.86 272.32 326.78 381.25 435.71 490.18 49.74 50.57 51.40 52.23 53.06 53.89 54.72 55.55 56.37 57.20 58.03 58.86 59.69 60.52 61.35 62.18 63.01 63.84 64.66 65.49 66.32 67.15 67.98 68.81 69.64 70.47 71.30 72.13 72.96 73.78 74.61 75.44 76.27 77.10 77.93 78.76 79.59 80.42 81.25 82.07 82.90 165.81 248.71 331.62 414.52 497.42 580.33 663.23 746.13 33.55 34.11 34.67 35.23 35.79 36.35 36.91 37.46 38.03 38.58 39.14 39.70 40.26 40.82 41.38 41.94 42.50 43.06 43.62 44.18 44.74 45.29 45.85 46.41 46.97 47.53 48.09 48.65 49.21 49.77 50.33 50.89 51.45 52.00 52.56 53.12 53.68 54.24 54.80 55.36 55.92 111.84 167.76 223.68 279.60 335.52 391.44 447.35 503.27 49.15 49.97 50.79 51.61 52.43 53.24 54.06 54.88 55.70 56.52 57.34 58.16 58.98 59.80 60.62 61.44 62.26 63.07 63.89 64.71 65.53 66.35 67.17 67.99 68.81 69.63 70.45 71.27 72.09 72.90 73.72 74.54 75.36 76.18 77.00 77.82 78.64 79.46 80.28 81.10 81.92 163.83 245.75 327.66 409.58 491.49 573.41 655.32 737.24 34.41 34.99 35.56 36.14 36.71 37.28 37.86 38.43 39.00 39.58 40.15 40.72 41.30 41.87 42.44 43.02 43.59 44.17 44.74 45.31 45.89 46.46 47.03 47.61 48.18 48.75 49.33 49.90 50.47 51.05 51.62 52.20 52.77 53.34 53.92 54.49 55.06 55.64 56.21 56.78 57.36 114.72 172.07 229.43 286.79 344.15 401.50 458.86 516.22 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. D D sin Dcos D sico scos Dsin Dcos Dsin Dcos Dsin D cos D 59~ 580 570 560 55~ 22 TABLE VI. TRAVERSE TABLE 360 37~ 38~ 390 400 D D cos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adjadj.adj. opp. adj. opp adj.. a dj. opp a. opp hyp. dis. lat. dep.at. ep at. dep. lat. dep. lat. dep. dis. ~~11 ~,-, 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30. 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.90 9.71 10.52 11.33 12.14 12.94 13.75 14.56 15.37 16.18 16.99 17.80 18.61 19.42 20.23 21.03 21.84 22.65 23.46 24.27 25.08 25.89 26.70 27.51 28.32 29.12 29.93 30.74 31.55 32.36 33.17 33.98 34.79 35.60 36.41 37.21 38.02 38.83 39.64 40.45 41.26 42.07 42.88 43.69 44.50 45.30 46.11 46.92 47.73 6.47 7.05 7.64 8.23 8.82 9.40 9.99 10.58 11.17 11.76 12.34 12.93 13.52 14.11 14.69 15.28 15.87 16.46 17.05 17.63 18.22 18.81 19.40 19.98 20.57 21.16 21.75 22.34 22.92 23.51 24.10 24.69 25.27 25.86 26.45 27.04 27.63 28.21 28.80 29.39 29.98 30.56 31.15 31.74 32.33 32.92 33.50 34.09 34.68 8.78 9.58 10.38 11.18 11.98 12.78 13.58 14.38 15.17 15.97 16.77 17.57 18.37 19.17 19.97 20.76 21.56 22.36 23.16 23.96 24.76 25.56 26.35 27.15 27.95 28.75 29.55 30.35 31.15 31.95 32.74 33.54 34.34 35.14 35.94 36.74 37.54 38.33 39.13 39.93 40.73 41.53 42.33 43.13 43.92 44.72 45.52 46.32 47.12 6.62 7.22 7.82 8.43 9.03 9.63 10.23 10.83 11.43 12.04 12.64 13.24 13.84 14.44 15.05 15.65 16.25 16.85 17.45 18.05 18.66 19.26 19.86 20.46 21.06 21.67 22.27 22.87 23.47 24.07 24.67 25.28 25.88 26.48 27.08 27.68 28.29 28.89 29.49 30.09 30.69 31.29 31.90 32.50 33.10 33.70 34.30 34.91 35.51 8.67 9.46 10.24 11.03 11.82 12.61 13.40 14.18 14.97 15.76 16.55 17.34 18.12 18.91 19.70 20.49 21.28 22.06 22.85 23.64 24.43 25.22 26.00 26.79 27.58 28.37 29.16 29.94 30.73 31.52 32.31 33.10 33.88 34.67 35.46 36.25 37.04 37.82 38.61 39.40 40.19 40.98 41.76 42.55 43.34 44.13 44.92 45.70 46.49 6.77 7.39 8.00 8.62 9.23 9.85 10.47 11.08 11.70 12.31 12.93 13.54 14.16 14.78 15.39 16.01 16.62 17.24 17.85 18.47 19.09 19.70 20.32 20.93 21.55 22.16 22.78 23.40 24.01 24.63 25.24 25.86 26.47 27.09 27.70 28.32 28.94 29.55 30.17 30.78 31.40 32.01 32.63 33.25 33.86 34.48 35.09 35.71 36.32 8.55 9.33 10.10 10.88 11.66 12.43 13.21 13.99 14.77 15.54 16.32 17.10 17.87 18.65 19.43 20.21 20.98 21.76 22.54 23.31 24.09 24.87 25.65 26.42 27.20 27.98 28.75 29.53 30.31 31.09 31.86 32.64 33.42 34.19 34.97 35.75 36.53 37.30 38.08 38.86 39.63 40.41 41.19 41.97 42.74 43.52 44.30 45.07 45.85 6.92 7.55 8.18 8.81 9.44 10.07 10.70 11.33 11.96 12.59 13.22 13.85 14.47 15.10 15.73 16.36 16.99 17.62 18.25 18.88 19.51 20.14 20.77 21.40 22.03 22.66 23.28 23.91 24.54 25.17 25.80 26.43 27.06 27.69 28.32 28.95 29.58 30.21 30.84 31.47 32.10 32.72 33.35 33.98 34.61 35.24 35.87 36.50 37.13 8.43 9.19 9.96 10.72 11.49 12.26 13.02 13.79 14.55 15.32 16.09 16.85 17.62 18.39 19.15 19.92 20.68 21.45 22.22 22.98 23.75 24.51 25.28 26.05 26.81 27.58 28.34 29.11 29.88 30.64 31.41 32.17 32.94 33.71 34.47 35.24 36.00 36.77 37.54 38.30 39.07 39.83 4Q.60 41.37 42.13 42.90 43.66 44.43 45.20 7.07 7.71 8.36 9.00 9.64 10.28 10.93 11.57 12.21 12.86 13.50 14.14 14.78 15.43 16.07 16.71 17.36 18.00 18.64 19.28 19.93 20.57 21.21 21.85 22.50 23.14 23.78 24.43 25.07 25.71 26.35 27.00 27.64 28.28 28.93 29.57 30.21 30.85 31.50 32.14 32.78 33.42 34.07 34.71 35.35 36.00 36.64 37.28 37.92 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp ad. opp. app dj. opp. adj. opp. adj. hyp. D Dysin D cosDsinD cosoDsin DcosDsin D.. psin cosD D D sin D cos D sin D cos D sin D cos D sin D cos D sin D cos D 540.30 520 510 500 54~ 53~ 52~ 51~ 50O TABLE VI. TRAVERSE TABLE 23 36~ 370 38~ 390 400 I I D Dcos D sin D cos D sin D cos D sin D cos D sin D cos D sin D hyp. adj. opp. adj. opp. adj. opp. adj. opp. adj. opp. hyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis..... 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 48.54 49.35 50.16 50.97 51.78 52.59 53.40 54.20 55.01 55.82 56.63 57.44 58.25 59.06 59.87 60.68 61.49 62.29 63.10 63.91 64.72 65.53 66.34 67.15 67.96 68.77 69.58 70.38 71.19 72.00 72.81 73.62 74.43 75.24 76.05 76.86 77.67 78.47 79.28 80.09 80.90 161.80 242.71 323.61 404.51 485.41 566.31 647.21 728.12 35.27 35.8P 36.44 37.03 37.62 38.21 38.79 39.38 39.97 40.5( 41.14 41.72 42.32 42.91 43.5C 44.0 44.67 45.20 45.85 46.43 47.02 47.61 48.20 48.79 49.37 49.96 50.55 51.14 51.73 52.31 52.90 53.49 54.08 54.66 55.25 55.84 56.43 57.02 57.60 58.19 58.78 117.56 176.34 235.11 293.89 352.67 411.45 470.23 529.01 7 II I I I I I ] ). ) 47.92 48.72 49.52 50.31 51.11 51.91 52.71 53.51 54.31 55.11 55.90 56.70 57.50 58.30 59.10 59.90 60.70 61.49 62.29 63.09 63.89 64.69 65.49 66.29 67.09 67.88 68.68 69.48 70.28 71.08 71.88 72.68 73.47 74.27 75.07 75.87 76.67 77.47 78.27 79.06 79.86 159.73 239.59 319.45 399.32 479.18 559.04 638.91 718.77 i I I I I I I 36.11 36.71 37.31 37.91 38.52 39.12 39.72 40.32 40.92 41.53 42.13 42.73 43.33 43.93 44.53 45.14 45.74 46.34 46.94 47.54 48.15 48.75 49.35 49.95 50.55 51.15 51.76 52.36 52.96 53.56 54.16 54.77 55.37 55.97 56.57 57.17 57.77 58.38 58.98 59.58 60.18 120.36 180.54 240.73 300.91 361.09 421.27 481.45 541.63 47.28 48.07 48.86 49.64 50.43 51.22 52.01 52.80 53.58 54.37 55.16 55.95 56.74 57.52 58.31 59.10 59.89 60.68 61.46 62.25 63.04 63.83 64.62 65.40 66.19 66.98 67.77 68.56 69.34 70.13 70.92 71.71 72.50 73.28 74.07 74.86 75.65 76.44 77.22 78.01 78.80 157.60 236.40 315.20 394.01 472.81 551.61 630.41 709.21 36.94 37.56 38.17 38.79 39.4C 40.02 40.63 41.25 41.86 42.48 43.1C 43.71 44.33 44.94 45.56 46.17 46.79 47.41 48.02 48.64 49.25 49.87 50.48 51.10 51.72 52.33 52.95 53.56 54.18 54.79 55.41 56.03 56.64 57.26 57.87 58.49 59.10 59.72 60.33 60.95 61.57 123.13 184.70 246.26 307.83 369.40 430.96 492.53 554.09 II I I I I I I i I 46.63 47.41 48.18 48.96 49.74 50.51 51.29 52.07 52.85 53.52 54.40 55.18 55.95 56.73 57.51 58.29 59.06 59.84 60.62 61.39 62.17 62.95 63.73 64.50 65.28 66.06 66.83 67.61 68.39 69.17 69.94 70.72 71.50 72.27 73.05 73.83 74.61 75.38 76.16 76.94 77.71 155.43 233.14 310.86 388.57 466.29 544.00 621.72 699.43 37.76 38.39 39.02 39.65 40.28 40.91 41.54 42.16 42.79 43.42 44.05 44.68 45.31 45.94 46.57 47.20 47.83 48.46 49.09 49.72 50.35 50.97 51.60 52.23 52.86 53.49 54.12 54.75 55.38 56.01 56.64 57.27 57.90 58.53 59.16 59.79 60.41 61.04 61.67 62.30 62.93 125.86 188.80 251.73 314.66 377.59 440.52 503.46 566.39 45.9( 46.73 47.49 48.26 49.03 49.79 50.56 51.32 52.09 52.86 53.62 54.39 55.16 55.92 56.69 57.45 58.22 58.99 59.75 60.52 61.28 62.05 62.82 63.58 64.35 65.11 65.88 66.65 67.41 68.18 68.94 69.71 70.48 71.24 72.01 72.77 73.54 74.31 75.07 75.84 76.60 153.21 229.81 306.42 383.02 459.63 536.23 612.84 689.44 I II I II I II I I I 0 i I 38.57 39.21 39.85 40.50 41.14 41.78 42.42 43.07 43.71 44.35 45.00 45.64 46.28 46.92 47.57 48.21 48.85 49.49 50.14 50.78 51.42 52.07 52.71 53.35 53.99 54.64 55.28 55.92 56.57 57.21 57.85 58.49 59.14 59.78 60.42 61.06 61.71 62.35 62.99 63.64 64.28 128.56 192.84 257.12 321.39 385.67 449.95 514.23 578.51 i I 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 _ l.,,,, _ dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. ihyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. PD Dsin os Dsin D sin D cos D sin D cos D sin Dcos D sin D cos D 540~~~~~~ ~ Do 530n 520 s 5D10 540 53~ 52~ 51~ 500 24 TABLE VI. TRAVERSE TABLE 41~ 42~ 430 440 450 D D cos D sin D Cos Dsin D cos D sin D cos D sin D cos D sin D hyp..adj. opp. adj. op p. adj. opp. adhyp. dis. lat. dep. lat. dep. lat. dep. lat. dep. lat. dep. dis. - 11 ~~~~ ~ ~~~~~~~~~~~~~~~~~~~~1~~ 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 8.30 9.06 9.81 10.57 11.32 12.08 12.83 13.58 14.34 15.09 15.85 16.60 17.36 18.11 18.87 19.62 20.38 21.13 21.89 22.64 23.40 24.15 24.91 25.66 26.41 27.17 27.92 28.68 29.43 30.19 30.94 31.70 32.45 33.21 33.96 34.72 35.47 36.23 36.98 37.74 38.49 39.24 40.00 40.75 41.51 42.26 43.02 43.77 44.53 7.22 7.87 8.53 9.18 9.84 10.50 11.15 11.81 12.47 13.12 13.78 14.43 15.09 15.75 16.40 17.06 17.71 18.37 19.03 19.68 20.34 20.99 21.65 22.31 22.96 23.62 24.27 24.93 25.59 26.24 26.90 27.55 28.21 28.87 29.52 30.18 30.83 31.49 32.15 32.80 33.46 34.12 34.77 35.43 36.08 36.74 37.40 38.05 38.71 8.17 8.92 9.66 10.40 11.15 11.89 12.63 13.38 14.12 14.86 15.61 16.35 17.09 17.84 18.58 19.32 20.06 20.81 21.55 22.29 23.04 23.78 24.52 25.27 26.01 26.75 27.50 28.24 28.98 29.73 30.47 31.21 31.96 32.70 33.44 34.18 34.93 35.67 36.41 37.16 37.90 38.64 39.39 40.13 40.87 41.62 42.36 43.10 43.85 7.36 8.03 8.70 9.37 10.04 10.71 11.38 12.04 12.71 13.38 14.05 14.72 15.39 16.06 16.73 17.40 18.07 18.74 19.40 20.07 20.74 21.41 22.08 22.75 23.42 24.09 24.76 25.43 26.10 26.77 27.43 28.10 28.77 29.44 30.11 30.78 31.45 32.12 32.79 33.46 34.13 34.79 35.46 36.13 36.80 37.47 38.14 38.81 39.48 8.04 8.78 9.51 10.24 10.97 11.70 12.43 13.16 13.90 14.63 15.36 16.09 16.82 17.55 18.28 19.02 19.75 20.48 21.21 21.94 22.67 23.40 24.13 24.87 25.60 26.33 27.06 27.79 28.52 29.25 29.99 30.72 31.45 32.18 32.91 33.64 34.37 35.10 35.84 36.57 37.30 38.03 38.76 39.49 40.22 40.96 41.69 42.42 43.15 7.50 8.18 8.87 9.55 10.23 10.91 11.59 12.28 12.96 13.64 14.32 15.00 15.69 16.37 17.05 17.73 18.41 19.10 19.78 20.46 21.14 21.82 22.51 23.19 23.87 24.55 25.23 25.92 26.60 27.28 27.96 28.64 29.33 30.01 30.69 31.37 32.05 32.74 33.42 34.10 34.78 35.46 36.15 36.83 37.51 38.19 38.87 39.56 40.24 7.91 8.63 9.35 10.07 10.79 11.51 12.23 12.95 13.67 14.39 15.11 15.83 16.54 17.26 17.98 18.70 19.42 20.14 20.86 21.58 22.30 23.02 23.74 24.46 25.18 25.90 26.62 27.33 28.05 28.77 29.49 30.21 30.93 31.65 32.37 33.09 33.81 34.53 35.25 35.97 36.69 37.41 38.12 38.84 39.56 40.28 41.00 41.72 42.44 7.64 8.34 9.03 9.73 10.42 11.11 11.81 12.50 13.20 13.89 14.59 15.28 15.98 16.67 17.37 18.06 18.76 19.45 20.15 20.84 21.53 22.23 22.92 23.62 24.31 25.01 25.70 26.40 27.09 27.79.28.48 29.18 29.87 30.56 31.26 31.95 32.65 33.34 34.04 34.73 35.43 36.12 36.82 37.51 38.21 38.90 39.60 40.29 40.98 7.78 8.49 9.19 9.90 10.61 11.31 12.02 12.73 13.44 14.14 14.85 15.56 16.26 16.97 17.68 18.38 19.09 19.80 20.51 21.21 21.92 22.63 23.33 24.04 24.75 25.46 26.16 26.87 27.58 28.28 28.99 29.70 30.41 31.11 31.82 32.53 33.23 33.94 34.65 35.36 36.06 36.77 37.48 38.18 38.89 39.60 40.31 41.01 41.72 7.78 8.49 9.19 9.90 10.61 11.31 12.02 12.73 13.44 14.14 14.85 15.56 16.26 16.97 17.68 18.38 19.09 19.80 20.51 21.21 21.92 22.63 23.33 24.04 24.75 25.46 26.16 26.87 27.58 28.28 28.99 29.70 30.41 31.11 31.82 32.53 33.23 33.94 34.65 35.36 36.06 36.77 37.48 38.18 38.89 39.60 40.31 41.01 41.72 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 dis. dep. lat. dep. lat. dep. lat. dep. lat. dep. lat. dis. hyp. opp. adj. opp. adj. opp. adj. opp. adj. opp. adj. hyp. D sin D co si cos D sin Dcos Dsin Dcos D sin Dcos D 49~ 48~ 47~ 46~ 45~ TABLE VI. TRAVERSE TABLE 25 41~ 420 430 440 450 D hyp. dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dis. hyp. D D cos adj. lat. 45.28 46.04 46.79 47.55 48.30 49.06 49.81 50.57 51.32 52.07 52.83 53.58 54.34 55.09 55.85 56.60 57.36 58.11 58.87 59.62 60.38 61.13 61.89 62.64 63.40 64.15 64.90 65.66 66.41 67.17 67.92 68.68 69.43 70.19 70.94 71.70 72.45 73.21 73.96 74.72 75.47 150.94 226.41 301.88 377.35 452.83 528.30 603.77 679.24 dep. oppo D sin D sin opp. dep. 39.36 40.02 40.68 41.33 41.99 42.64 43.30 43.96 44.61 45.27 45.92 46.58 47.24 47.89 48.55 49.20 49.86 50.52 51.17 51.83 52.48 53.14 53.80 54.45 55.11 55.76 56.42 57.08 57.73 58.39 59.05 59.70 60.36 61.01 61.67 62.33 62.98 63.64 64.29 64.95 65.61 131.21 196.82 262.42 328.03 393.64 459.24 524.85 590.45 lat. adj. i D cos D sin adj. opp. lat. dep. 44.59 40.15 45.33 40.82 46.07 41.49 46.82 42.16 47.56 42.82 48.30 43.49 49.05 44.16 49.79 44.83 50.53 45.50 51.28 46.17 52.02 46.84 52.76 47.51 53.51 48.18 54.25 48.85 54.99 49.52 55.74 50.18 56.48 50.85 57.22 51.52 57.97 52.19 58.71 52.86 59.45 53.53 60.19 54.20 60.94 54.87 61.68 55.54 62.42 56.21 63.17 56.88 63.91 57.55 64.65 58.21 65.40 58.88 66.14 59.55 66.88 60.22 67.63 60.89 68.37 61.56 69.11 62.23 69.86 62.90 70.60 63.57 71.34 64.24 72.08 64.91 72.83 65.57 73.57 66.24 74.31 66.91 148.63 133.83 222.94200.74 297.26 267.65 371.57 334.57 445.89 401.48 520.20 468.39 594.52 535.30 668.83 602.22 dep. lat. opp. adj. I D cos adj. lat. D sin opp. dep. D cos adj. lat. D sin opp. dep. - 43.88 44.61 45.34 46.08 46.81 47.54 48.27 49.00 49.73 50.46 51.19 51.93 52.66 53.39 54.12 54.85 55.58 56.31 57.05 57.78 58.51 59.24 59.97 60.70 61.43 62.17 62.90 63.63 64.36 65.09 65.82 66.55 67.28 68. 02 68.75 69.48 70.21 70.94 71.67 72.40 73.14 146.27 219.41 292.54 365.68 438.81 511.95 585.08 658.22 40.92 41.60 42.28 42.97 43.65 44.33 45.01 45.69 46.38 47.06 47.74 48.42 49.10 49.79 50.47 51.15 51.83 52.51 53.20 53.88 54.56 55.24 55.92 56.61 57.29 57.97 58.65 59.33 60.02 60.70 61.38 62.06 62.74 63.43 64.11 64.79 65.47 66.15 66.84 67.52 68.20 136.40 204.60 272.80 341.00 409.20 477.40 545.60 613.80 43.16 43.88 44.60 45.32 46.04 46.76 47.48 48.20 48.92 49.63 50.35 51.07 51.79 52.51 53.23 53.95 54.67 55.39 56.11 56.83 57.55 58.27 58.99 59.71 60.42 61.14 61.86 62.58 63.30 64.02 64.74 65.46 66.18 66.90 67.62 68.34 69.06 69.78 70.50 71.21 71.93 143.87 215.80 287.74 359.67 431.60 503.54 575.47 647.41 41.68 42.37 43.07 43.76 44.46 45.15 45.85 46.54 47.24 47.93 48.63 49.32 50.02 50.71 51.40 52.10 52.79 53.49 54.18 54.88 55.57 56.27 56.96 57.66 58.35 59.05 59.74 60.44 61.13 61.82 62.52 63.21 63.91 64.60 65.30 65.99 66.69 67.38 68.08 68.77 69.47 138.93 208.40 277.86 347.33 416.80 486.26 555.73 625.19.1 D cos adj. lat. D sin opp. dep. I 42.43 43.13 43.84 44.55 45.25 45.96 46.67 47.38 48.08 48.79 49.50 50.20 50.91 51.62 52.33 53.03 53.74 54.45 55.15 55.86 56.57 57.28 57.98 58.69 59.40 60.10 60.81 61.52 62.23 62.93 63.64 64.35 65.05 65.76 66.47 67.18 67.88 68.5c 69.30 70.00 70.71 141.42 212.13 282.84 353.55 424.26 494.97 565.69 636.40 42.43 43.13 43.84 44.55 45.25 45.96 46.67 47.38 48.08 48.79 49.50 50.20 50.91 51.62 52.33 53.03 53.74 54.45 55.15 55.86 56.57 57.28 57.98 58.69 59.40 60.10 60.81 61.52 62.23 62.93 63.64 64.35 65.05 65.76 66.47 67.18 67.88 68.59 69.30 70.00 70.71 141.42 212.13 282.84 353.55 424.26 494.97 565.69 636.40, D hyp. dis. 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 200 300 400 500 600 700 800 900 dep. lat. dep. at. dep. lat. dis. opp. adj. opp. adj. opp. adj. hyp., 1 D cos II D sin D cos D cos Dsin D cos I) Dsin D cos F q m.r 49~ 48~ 47~ 46~ 45~ TABLE VII. LOGARITHMS OF NUMBERS N 0 1 2 3 4 5 6 7 8 9 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 0004 0048 0090 0133 0175 0216 0257 0298 0338 0378 0418 0457 0496 0535 0573 0611 0648 0686 0722 0759 0795 0831 0867 0903 0938 0973 1007 1041 1075 1109 1143 1176 1209 1242 1274 1307 1339 1370 1402 1433 1464 1495 1526 1556 1587 1617 1647 1676 1706 1735 0009 0052 0095 0137 0179 0220 0261 0302 0342 0382 0422 0461 0500 0538 0577 0615 0652 0689 0726 0763 0799 0835 0871 0906 0941 0976 1011 1045 1079 1113 1146 1179 1212 1245 1278 1310 1342 1374 1405 1436 1467 1498 1529 1559 1590 1620 1649 1679 1708 1738 0013 0056 0099 0141 0183 0224 0265 0306 0346 0386 0426 0465 0504 0542 0580 0618 0656 0693 0730 0766 0803 0839 0874 0910 0945 0980 1014 1048 1082 1116 1149 1183 1216 1248 1281 1313 1345 1377 1408 1440 1471 1501 1532 1562 1593 1623 1652 1682 1711 1741 0017 0060 0103 0145 0187 0228 0269 0310 0350 0390 0430 0469 0508 0546 0584 0622 0660 0697 0734 0770 0806 0842 0878 0913 0948 0983 1017 1052 1086 1119 1153 1186 1219 1252 1284 1316 1348 1380 1411 1443 1474 1504 1535 1565 1596 1626 1655 1685 1714 1744 0022 0026 0065 0069 0107 0111 0149 0154 0191 0195 0233 0237 0273 0278 0314 0318 0354 0358 0394 0398 0434 0438 0473 0477 0512 0515 0550 0554 0588 0592 0626 0630 0663 0667 0700 0704 0737 0741 0774 0777 0810 0813 0846 0849 0881 0885 0917 0920 0952 0955 0986 0990 1021 1024 1055 1059 1089 1092 1123 1126 1156 1159 1189 1193 1222 1225 1255 1258 1287 1290 1319 1323 1351 1355 1383 1386 1414 1418 1446 1449 1477 1480 1508 1511 1538 1541 1569 1572 1599 1602 1629 1632 1658 1661 1688 1691 1717 1720 1746 1749 0030 0073 0116 0158 0199 0241 0282 0322 0362 0402 0441 0481 0519 0558 0596 0633 0671 0708 0745 0781 0817 0853 0888 0924 0959 0993 1028 1062 1096 1129 1163 1196 1229 1261 1294 1326 1358 1389 1421 1452 1483 1514 1544 1575 1605 1635 1664 1694 1723 1752 0035 0039 0077 0082 0120 0124 0162 0166 0204 0208 0245 0249 0286 0290 0326 0330 0366 0370 0406 0410 0445 0449 0484 0488 0523 0527 0561 0565 0599 0603 0637 0641 0674 0678 0711 0715 0748 0752 0785 0788 0821 0824 0856 0860 0892 0896 0927 0931 0962 0966 0997 1000 1031 1035 1065 1069 1099 1103 1133 1136 1166 1169 1199 1202 1232 1235 1265 1268 1297 1300 1329 1332 1361 1364 1392 1396 1424 1427 1455 1458 1486 1489 1517 1520 1547 1550 1578 1581 1608 1611 1638 1641 1667 1670 1697 1700 1726 1729 1755 1758 0 1 2 3 4 5 6 7 8 9 26 TABLE VII. LOGARITHMS OF NUMBERS 27 N 0 1 2 3 4 5 6 7 8 9 l 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 1761 1764 1790 1793 1818 1821 1847 1850 1875 1878 1903 1906 1931 1934 1959 1962 1987 1989 2014 2017 2041 2044 2068 2071 2095 2098 2122 2125 2148 2151 2175 2177 2201 2204 2227 2230 2253 2256 2279 2281 2304 2307 2330 2333 2355 2358 2380 2383 2405 2408 2430 2433 2455 2458 2480 2482 2504 2507 2529 2531 2553 2555 2577 2579 2601 2603 2625 2627 2648 2651 2672 2674 2695 2697 2718 2721 2742 2744 2765 2767 2788 2790 2810 2813 2833 2835 2856 2858 2878 2880 2900 2903 2923 2925 2945 2947 2967 2969 2989 2991 1767 1770 1796 1798 1824 1827 1853 1855 1881 1884 1909 1912 1937 1940 1965 1967 1992 1995 2019 2022 2047 2049 2074 2076 2101 2103 2127 2130 2154 2156 2180 2183 2206 2209 2232 2235 2258 2261 2284 2287 2310 2312 2335 2338 2360 2363 2385 2388 2410 2413 2435 2438 2460 2463 2485 2487 2509 2512 2533 2536 2558 2560 2582 2584 2605 2608 2629 2632 2653 2655 2676 2679 2700 2702 2723 2725 2746 2749 2769 2772 2792 2794 2815 2817 2838 2840 2860 2862 2882 2885 2905 2907 2927 2929 2949 2951 2971 2973 2993 2995 1772 1801 1830 1858 1886 1915 1942 1970 1998 2025 2052 2079 2106 2133 2159 2185 2212 2238 2263 2289 2315 2340 2365 2390 2415 2440 2465 2490 2514 2538 2562 2586 2610 2634 2658 2681 2704 2728 2751 2774 2797 2819 2842 2865 2887 2909 2931 2953 2975 2997 1775 1804 1833 1861 1889 1917 1945 1973 2000 2028 2055 2082 2109 2135 2162 2188 2214 2240 2266 2292 2317 2343 2368 2393 2418 2443 2467 2492 2516 2541 2565 2589 2613 2636 2660 2683 2707 2730 2753 2776 2799 2822 2844 2867 2889 2911 2934 2956 2978 2999 1778 1807 1836 1864 1892 1920 1948 1976 2003 2030 2057 2084 2111 2138 2164 2191 2217 2243 2269 2294 2320 2345 2370 2395 2420 2445 2470 2494 2519 2543 2567 2591 2615 2639 2662 2686 2709 2732 2755 2778 2801 2824 2847 2869 2891 2914 2936 2958 2980 3002 1781 1810 1838 1867 1895 1923 1951 1978 2006 2033 2060 2087 2114 2140 2167 2193 2219 2245 2271 2297 2322 2348 2373 2398 2423 2448 2472 2497 2521 2545 2570 2594 2617 2641 2665 2688 2711 2735 2758 2781 2804 2826 2849 2871 2894 2916 2938 2960 2982 3004 1784 1787 1813 1816 1841 1844 1870 1872 1898 1901 1926 1928 1953 1956 1981 1984 2009 2011 2036 2038 2063 2066 2090 2092 2117 2119 2143 2146 2170 2172 2196 2198 2222 2225 2248 2251 2274 2276 2299 2302 2325 2327 2350 2353 2875 2378 2400 2403 2425 2428 2450 2453 2475 2477 2499 2502 2524 2526 2548 2550 2572 2574 2596 2598 2620 2622 2643 2646 2667 2669 2690 2693 2714 2716 2737 2739 2760 2762 2783 2785 2806 2808 2828 2831 2851 2853 2874 2876 2896 2898 2918 2920 2940 2942 2962 2964 2984 2986 3006 3008 0 1 2 3 4 5 6 7 8 9 28 TABLE VII. LOGARITHMS OF NUMBERS N 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 0000 0043 0414 0453 0792 0828 1139 1173 1461 1492 1761 1790 2041 2068 2304 2330 2553 2577 2788 2810 3010 3032 3222 3243 3424 3444 3617 3636 3802 3820 3979 3997 4150 4166 4314 4330 4472 4487 4624 4639 4771 4786 4914 4928 5051 5065 5185 5198 5315 5328 5441 5453 5563 5575 5682 5694 5798 5809 5911 5922 6021 6031 6128 6138 6232 6243 6335 6345 6435 6444 6532 6542 6628 6637 6721 6730 6812 6821 0086 0492 0864 1206 1523 1818 2095 2355 2601 2833 3054 3263 3464 3655 3838 4014 4183 4346 4502 4654 4800 4942 5079 5211 5340 5465 5587 5705 5821 5933 6042 6149 6253 6355 6454 6551 6646 6739 6830 0128 0531. 0899 1239 1553 1847 2122 2380 2625 2856 3075 3284 3483 3674 3856 4031 4200 4362 4518 4669 4814 4955 5092 5224 5353 5478 5599 5717 5832 5944 6053 6160 6263 6365 6464 6561 6656 6749 6839 0170 0569 0934 1271 1584 1875 2148 2405 2648 2878 3096 3304 3502 3692 3874 4048 4216 4378 4533 4683 4829 4969 5105 5237 5366 5490 5611 5729 5843 5955 6064 6170 6274 6375 6474 6571 6665 6758 6848 0212 0253 0607 0645 0969 1004 1303 1335 1614 1644 1903 1931 2175 2201 2430 2455 2672 2695 2900 2923 3118 3139 3324 3345 3522 3541 3711 3729 3892 3909 4065 4082 4232 4249 4393 4409 4548 4564 4698 4713 4843 4857 4983 4997 5119 5132 5250 5263 5378 5391 5502 5514 5623 5635 5740 5752 5855 5866 5966 5977 6075 6085 6180 6191 6284 6294 6385 6395 6484 6493 6580 6590 6675 6684 767 6776 6857 6866 0294 0682 1038 1367 1673 1959 2227 2480 2718 2945 3160 3365 3560 3747 3927 4099 4265 4425 4579 4728 4871 5011 5145 5276 5403 5527 5647 5763 5877 5988 6096 6201 6304 6405 6503 6599 6693 6785 6875 0334 0374 0719 0755 1072 1106 1399 1430 1703 1732 ~ 1987 2014 2253 2279 2504 2529 o \2742 2765 2967 2989 3181 3201 3385 3404 3579 3598 3766 3784 3945 3962 4116 4133 4281 4298 4440 4456 4594 4609 4742 4757 4886 4900 5024 5038 5159 5172 5289 5302 5416 5428 5539 5551 5658 5670 5775 5786 5888 5899 5999 6010 6107 6117 6212 6222 6314 6325 6415 6425 6513 6522 6609 6618 6702 6712 6794 6803 6884 6893 PROPORTIONAL PARTS 22 21 19 18 17 16 15 14 13 12 11 9 8 7 6 1 2 3 4 5 6 7 8 9 2 2 2 2 2 2 2 1 1 1 1 4 4 4 -4 3 3 3 3 3 2 2 7 6 6 5 5 5 5 4 4 4 3 9 8 8 7 7 6 6 6 5 5 4 11 11 10 9 9 8 8 7 7 6 6 13 13 11 11 10 10 9 8 8 7 7 15 15 13 13 12 11 11 10 9 8 8 18 17 15 14 14 13 12 11 10 10 9 20 19 17 16 15 14 14 13 12 11 10 1 1 1 1 2 2 1 1 3 2 2 2 4 3 3 2 5 4 4 3 5 5 4 4 6 6 5 4 7 6 6 5 8 7 6 5 1 2 3 4 5 6 7 8 9 TABLE VII. LOGARITHMS OF NUMBERS 29 N 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 0 1 2 3 4 5 6 7 8 9 6902 6990 7076 7160 7243 7324 7404 7482 7559 7634 7709 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 6911 6920 6928 6998 7007 7016 7084 7093 7101 7168 7177 7185 7251 7259 7267 7332 7340 7348 7412 7419 7427 7490 7497 7505 7566 7574 7582 7642 7649 7657 7716 7723 7731 7789 7796 7803 7860 7868 7875 7931 7938 7945 8000 8007 8014 8069 8075 8082 8136 8142 8149 8202 8209 8215 8267 8274 8280 8331 8338 8344 8395 8401 8407 8457 8463 8470 8519 8525 8531 8579 8585 8591 8639 8645 8651 8698 8704 8710 8756 8762 8768 8814 8820 8825 8871 8876 8882 8927 8932 8938 8982 8987 8993 9036 9042 9047 9090 9096 9101 9143 9149 9154 9196 9201 9206 9248 9253 9258 9299 9304 9309 9350 9355 9360 9400 9405 9410 9450 9455 9460 9499 9504 9509 9547 9552 9557 9595 9600 9605 9643 9647 9652 9689 9694 9699 9736 9741 9745 9782 9786 9791 9827 9832 9836 9872 9877 9881 9917 9921 9926 9961 9965 9969 6937 6946 7024 7033 7110 7118 7193 7202 7275 7284 7356 7364 7435 7443 7513 7520 7589 7597 7664 7672 7738 7745 7810 7818 7882 7889 7952 7959 8021 8028 8089 8096 8156 8162 8222 8228 8287 8293 8351 8357 8414 8420 8476 8482 8537 8543 8597 8603 8657 8663 8716 8722 8774 8779 8831 8837 8887 8893 8943 8949 8998 9004 9053 9058 9106 9112 9159 9165 9212 9217 9263 9269 9315 9320 9365 9370 9415 9420 9465 9469 9513 9518 9562 9566 9609 9614 9657 9661 9703 9708 9750 9754 9795 9800 9841 9845 9886 9890 9930 9934 9974 9978 6955 7042 7126 7210 7292 7372 7451 7528 7604 7679 7752 7825 7896 7966 8035 8102 8169 8235 8299 8363 8426 8488 8549 8609 8669 8727 8785 8842 8899 8954 9009 9063 9117 9170 9222 9274 9325 9375 9425 9474 9523 9571 9619 9666 9713 9759 9805 9850 9894 9939 9983 6964 7050 7135 7218 7300 7380 7459 7536 7612 7686 7760 7832 7903 7973 8041 8109 8176 8241 8306 8370 8432 8494 8555 8615 8675 8733 8791 8848 8904 8960 9015 9069 9122 9175 9227 9279 9330 9380 9430 9479 9528 9576 9624 9671 9717 9763 9809 9854 9899 9943 9987 6972 6981 7059 7067 7143 7152 7226 7235 7308 7316 7388 7396 7466 7474 7543 7551 7619 7627 7694 7701 7767 7774 7839 7846 7910 7917 7980 7987 8048 8055 8116 8122 8182 8189 8248 8254 8312 8319 8376 8382 8439 8445 8500 8506 8561 8567 8621 8627 8681 8686 8739 8745 8797 8802 8854 8859 8910 8915 8965 8971 9020 9025 9074 9079 9128 9133 9180 9186 9232 9238 9284 9289 9335 9340 9385 9390 9435 9440 9484 9489 9533 9538 9581 9586 9628 9633 9675 9680 9722 9727 9768 9773 9814 9818 9859 9863 9903 9908 9948 9952 9991 9996 N 0 1 2 3 4 5 6 7 8 9 TABLE VIII. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS log sin log o 0' 1' 2' 3' 4' 5' 6' 7' 8' 9' cos 0 0 6.4637 7648 9408 7.0658 1627 2419 3088 3668 4180 10.0000 50 10 7.4637 5051 5429 5777 6099 6398 6678 6942 7190 7425 10.0000 40 20 7.7648 7859 8061 8255 8439 8617 8787 8951 9109 9261 10.0000 30 30 7.9408 9551 9689 9822 9952 8.0078 0200 0319 0435 0548 10.0000 20 40 8.0658 0765 0870 0972 1072 1169 1265 1358 1450 1539 10.0000 10 50 8.1627 1713 1797 1880 1961 2041 2119 2196 2271 2346 10.0000 89 0 1 0 8.2419 2490 2561 2630 2699 2766 2832 2898 2962 3025 9.9999 50 10 8.3088 3150 3210 3270 3329 3388 3445 3502 3558 3613 9.9999 40 20 8.3668 3722 3775 3828 3880 3931 3982 4032 4082 4131 9.9999 30 30 8.4179 4227 4275 4322 4368 4414 4459 4504 4549 4593 9.9999 20 40 8.4637 4680 4723 4765 4807 4848 4890 4930 4971 5011 9.9998 10 50 8.5050 5090 5129 5167 5206 5243 5281 5318 5355 5392 9.99988 0 2 0 8.5428 5464 5500 5535 5571 5605 5640 5674 5708 5742 9.9997 50 10 8.5776 5809 5842 5875 5907 5939 5972 6003 6035 6066 9.9997 40 20 8.6097 6128 6159 6189 6220 6250 6279 6309 6339 6368 9.9996 30 30 8.6397 6426 6454 6483 6511 6539 6567 6595 6622 6650 9.9996 20 40 8.6677 6704 6731 6758 6784 6810 6837 6863 6889 6914 9.9995 10 50 8.6940 6965 6991 7016 7041 7066 7090 7115 7140 7164 9.999587 o 3 0 8.7188 7212 7236 7260 7283 7307 7330 7354 7377 7400 9.9994 50 10 8.7423 7445 7468 7491 7513 7535 7557 7580 7602 7623 9.9993 40 20 8.7645 7667 7688 7710 7731 7752 7773 7794 7815 7836 9.9993 30 30 8.7857 7877 7898 7918 7939 7959 7979 7999 8019 8039 9.9992 20 40 8.8059 8078 8098 8117 8137 8156 8175 8194 8213 8232 9.9991 10 50 8.8251 8270 8289 8307 8326 8345 8363 8381 8400 8418 9.999086 0 10' 9' 8' 7' 6' 5' 4' 3' 2' 1' log log cos sin log tan o 0' 1' 2' 3' 4' 5' 6' 7' 8' 9' 0 0 6.4637 7648 9408 7.0658 1627 2419 3088 3668 4180 50 10 7.4637 5051 5429 5777 6099 6398 6678 6942 7190 7425 40 20 7.7648 7860 8062 8255 8439 8617 8787 8951 9109 9261 30 30 7.9409 9551 9689 9823 9952 8.0078 0200 0319 0435 0548 20 40 8.0658 0765 0870 0972 1072 1170 1265 1359 1450 1540 10 50 8.1627 1713 1798 1880 1962 2041 2120 2196 2272 2346 89 0 1 0 8.2419 2491 2562 2631 2700 2767 2833 2899 2963 3026 50 10 8.3089 3150 3211 3271 3330 3389 3446 3503 3559 3614 40 20 8.3669 3723 3776 3829 3881 3932 3983 4033 4083 4132 30 30 8.4181 4229 4276 4323 4370 4416 4461 4506 4551 4595 20 40 8.4638 4682 4725 4767 4809 4851 4892 4933 4973 5013 10 50 8.5053 5092 5131 5170 5208 5246 5283 5321 5358 5394 88 0 2 0 8.5431 5467 5503 5538 5573 5608 5643 5677 5711 5745 50 10 8.5779 5812 5845 5878 5911 5943 5975 6007 6038 6070 40 20 8.6101 6132 6163 6193 6223 6254 6283 6313 6343 6372 30 30 8.6401 6430 6459 6487 6515 6544 6571 6599 6627 6654 20 40 8.6682 6709 6736 6762 6789 6815 6842 6868 6894 6920 10 50 8.6945 6971 6996 7021 7046 7071 7096 7121 7145 7170 87 0 3 0 8.7194 7218 7242 7266 7290 7313 7337 7360 7383 7406 50 10 8.7429 7452 7475 7497 7520 7542 7565 7587 7609 7631 40 20 8.7652 7674 7696 7717 7739 7760 7781 7802 7823 7844 30 30 8.7865 7886 7906 7927 7947 7967 7988 8008 8028 8048 20 40 8.8067 8087 8107 8126 8146 8165 8185 8204 8223 8242 10 50 8.8261 8280 8299 8317 8336 8355 8373 8392 8410 8428 86 0 10' 9' 8'.7' 6' 5' 4' 3' 2' 1' log cot 30 TABLE VIII. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 31 log sin log tan log cot log cos o / log sin log tan log cot log cos o f 9. 9. 10. 9. 4 0 5 10 15 20 25 30 35 40 45 50 55 5 0 5 10 15 20 25 30 35 40 45 50 55 6 O 5 10 15 20 25 30 35 40 45 50 55 7 0 8.8436 525 613 699 783 865 8.8946 8.9026 104 181 256 330 403 475 545 614 682 750 816 881 8.9945 9.0008 070 132 192 252 311 369 426 483 539 594 648 702 755 807 9.0859 8.8446 11.1554 536 464 624 376 711 289 795 205 878 122 8.8960 11.1040 8.9040 11.0960 118 882 196 272 346 420 492 563 633 701 769 836 901 8.9966 9.0030 093 155 216 277 336 395 453 510 567 622 678 732 786 839 9.0891 804 728 654 580 508 437 367 299 231 164 099 11.0034 10.9970 907 845 784 723 664 605 547 490 433 378 322 268 214 161 10.9109 9.9989 89 89 88 88 87 87 86 86 85 85 84 83 83 82 82 81 81 80 79 79 78 77 77 76 75 75 74 73 73 72 71 71 70 69 68 9.9968 86 0 55 50 45 40 35 30 25 20 15 10 5 85 0 55 50 45 40 35 30 25 20 15 10 5 84 0 55 50 45 40 35 30 25 20 15 10 83 83 o 7 0 0859 5 0910 10 0961 15 1011 20 1060 25 1109 30 1157 35 1205 40 1252 45 1299 50 1345 55 1390 8 01 436 5 1480 10 1525 15 1568 20 1612 25 1655 30 1697 35 1739 40 1781 45 1822 50 1863 55 1903 9 0 1943 5 1983 10 2022 15 2061 20 2100 25 2138 30 2176 35 2214 40 2251 45 2288 50 2324 55 2361 10 02397 0891 0943 0995 1045 1096 1145 1194 1243 1291 1338 1385 1432 1478 1524 1569 1613 1658 1702 1745 1788 1831 1873 1915 1956 1997 2038 2078 2118 2158 2197 2236 2275 2313 2351 2389 2426 2463 9109 9057 9005 8955 8904 8855 8806 8757 8709 8662 8615 8568 8522 8476 8431 8387 8342 8298 8255 8212 8169 8127 8085 8044 8003 7962 7922 7882 7842 7803 7764 7725 7687 7649 7611 7574 7537 9968 67 66 65 64 64 63 62 61 60 59 58 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 9934 83 O 55 50 45 40 35 30 25 20 15 10 5 82 O 55 50 45 40 35 30 25 20 15 10 5 81 0 55 50 45 40 35 30 25 20 15 10 80 o 0o log cos log cot log tan log sin 9. 9. 10. 9. 0~ log cos log cot log tan log sin P. P. FOR DIFFERENCE OF 5' 81 80 78 77 76 75 74 73 72 71 7 90 89 88 87 86 84 83 82 '0 69 68 67 66 65 64 63 1 18 18 18 17 17 17 17 16 16 16 16 15 15 1 15 15 15 14 14 14 14 14 13 13 13 13 13 1 2 36 36 35 35 34 34 33 33 32 32 31 31 30 2 30 30 29 29 28 28 28 27 27 26 26 26 25 2 3 54 53 53 52 52 50 50 49 49 48 47 46 46 3 45 4444 43 4342 41 41 40 40 39 38 38 3 4 72 71 70 70 69 67 66 66 65 64 62 62 61 4 60 59 58 58 57 56 55 54 54 53 52 51 50 4 62 61 60 59 1 12 12 12 12 2 25 24 24 24 3 37 37 36 35 4 50 49 48 47 58 57 56 55 54 53 52 51 49 48 12 11 11 11 11 11 10 10 101 10 23 23 22 22 22 21 21 20 20 2 19 35 34 34 33 32 32 31 31 29 3 29 46 46 45 44 43 42 42 41 39 4 38 9 9 9 9 19 18 18 18 28 28 27 26 38 37 36 35 47 46 45 44 43 42 41 40 39 38 37 36 9 8 8 8 8 17 17 16 16 16 26 25 25 24 23 34 34 33 32 31 8771 15 15 14 2 23 22 22 3 30 30 29 4 32 TABLE VIII. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS log sin log tan log cot log cos o t 9. 9. 10. 9. log sin log tan log cot log cos o / 9. 9. 10. 9. 10 0 10 20 30 40 50 11 o 10 20 30 40 50 12 0 10 20 30 40 50 13 0 10 20 30 40 50 14 O 10 20 30 40 50 15 0 2397 2468 2538 2606 2674 2740 2806 2870 2934 2997 3058 3119 3179 3238 3296 3353 3410 3466 3521 3575 3629 3682 3734 3786 3837 3887 3937 3986 4035 4083 4130 2463 2536 2609 2680 2750 2819 2887 2953 3020 3085 3149 3212 3275 3336 3397 3458 3517 3576 3634 3691 3748 3804 3859 3914 3968 4021 4074 4127 4178 4230 4281 7537 7464 7391 7320 7250 7181 7113 7047 6980 6915 6851 6788 6725 6664 6603 6542 6483 6424 6366 6309 6252 6196 6141 6086 6032 5979 5926 5873 5822 5770 5719 9934 9931 9929 9927 9924 9922 9919 9917 9914 9912 9909 9907 9904 9901 9899 9896 9893 9890 9887 9884 9881 9878 9875 9872 9869 9866 9863 9859 9856 9853 9849 80 o 50 40 30 20 10 50 79 o 40 30 20 10 78 0 50 40 30 20 10 77 0 50 40 30 20 10 76 0 50 40 30 20 710 75 o 10 15 o 20 30 40 50 16 0 10 20 30 40 50 17 o 10 20 30 40 50 18 0 o10 20 30 40 50 19 o s10 20 30 40 50 20 0 10 30 40 1 50 21 0 4130 4177 4223 4269 4314 4359 4403 4447 4491 4533 4576 4618 4659 4700 4741 4781 4821 4861 4900 4939 4977 5015 5052 5090 5126 5163 5199 5235 5270 5306 5341 5375 5409 5443 5477 5510 5543 4281 4331 4381 4430 4479 4527 4575 4622 4669 4716 4762 4808 4853 4898 4943 4987 5031 5075 5118 5161 5203 5245 5287 5329 5370 5411 5451 5491 5531 5571 5611 5650 5689 5727 5766 5804 5842 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 719 9849 75 0 669 9846 50 619 9843 40 570 9839 30 521 9836 20 473 9832 10 425 9828 74 O 378 9825 50 331 9821 40 284 9817 30 238 9814 20 192 9810 10 147 9806 73 0 102 9802 50 057 9798 40 013 9794 30 969 9790 20 925 9786 10 882 9782 72 0 839 9778 50 797 9774 40 755 9770 30 713 9765 20 671 9761 10 630 9757 71 0 589 9752 50 549 9748 40 509 9743 30 L469 9739 20 L429 9734 10 389 9730 70 0 350 9725 50 L311 9721 40 L273 9716 30 234 9711 20 196 9706 10 158 9702 69 10. 9. o / 9. 9. 10. 9. o log cos log cot log tan log sin P. P. 73 71 69 68 67 66 65 64 63 61 59 7 7 7 7 7 7 7 6 6 6 6 15 14 14 14 13 13 13 13 13 12 12 22 21 21 20 20 20 20 19 19 18 18 29 28 28 27 27 26 26 26 25 24 24 37 36 35 34 34 33 33 32 32 31 30 44 43 41 41 40 40 39 38 38 37 35 51 50 48 48 47 46 46 45 44 43 41 58 57 55 54 54 53 52 51 50 49 47 66 64 62 61 60 59 59 58 57 55 53 58 57 56 55 54 53 52 51 49 48 47 46 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 I ' 9. 9. og cos log cot log tan log sin 45 44 43 42 41 39 38 37 36 35 34 33 6 6 6 6 5 5 5 5 5 5 12 11 11 11 11 11 10 10 10 10 9 9 17 17 17 17 16 16 16 15 15 14 14 14 23 23 22 22 22 21 21 20 20 19 19 18 29 29 28 28 27 27 26 26 25 24 24 23 35 34 34 33 32 32 31 31 29 29 28 28 41 40 39 39 38 37 36 36 34 34 33 32 46 46 45 44 43 42 42 41 39 38 38 37 52 51 50 50 49 48 47 46 44 43 42 41 1 2 3 4 5 6 7 8 9 5 4 4 4 4 4 4 4 4 4 3 9 99 8 8 8 8 7 7 77 7 14 13 13 13 12 12 11 11 11 11 10 10 18 18 17 17 16 16 15 15 14 14 14 13 23 22 22 21 21 20 19 19 18 18 17 17 27 26 26 25 25 23 23 22 22 21 20 20 32 31 30 29 29 27 27 26 25 25 24 23 36 35 34 34 33 31 30 30 29 28 27 26 41 40 39 38 37 35 34 33 32 32 31 30 I 2 3 4 5 6 7 8 9 TABLE VIII. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 33 log sin log tan log cot log cos o / 9. 9. 10. 9. log sin log tan log cot log cos o / 9. 9. 10. 9. 21 0 5543 10 5576 20 5609 30 5641 40 5673 50 5704 22 0 5736 10 5767 20 5798 30 5828 40 5859 50 5889 23 0 5919 10 5948 20 5978 30 6007 40 6036 50 6065 24 0 6093 10 6121 20 6149 30 6177 40 6205 50 6232 25 0 6259 10 6286 20 6313 30 6340 40 6366 50 6392 26 0 6418 10 6444 20 6470 30 6495 40 6521 50 6546 27 06570 5842 5879 5917 5954 5991 6028 6064 6100 6136 6172 6208 6243 6279 6314 6348 6383 6417 6452 6486 6520 6553 6587 6620 6654 6687 6720 6752 6785 6817 6850 6882 6914 6946 6977 7009 7040 7072 4158 4121 4083 4046 4009 3972 3936 3900 3864 3828 3792 3757 3721 3686 3652 3617 3583 3548 3514 3480 3447 3413 3380 3346 3313 3280 3248 3215 3183 3150 3118 3086 3054 3023 2991 2960 2928 9702 69 0 9697 50 92 40 87 30 82 20 77 10 7268 0 67 50 61 40 56 30 51 20 46 10 4067 0 35 50 29 40 24 30 18 20 13 10 0766 o 9602 50 9596 40 90 30 84 20 79 10 7365 0 67 50 61 40 55 30 49 20 43 10 3764 0 30 50 24 40 18 30 12 20 9505 10 9499 63 27 0 6570 10 6595 20 6620 30 44 40 68 28 0 6716 10 40 20 63 30 6787 40 6810 50 33 29 0 56 10 6878 20 6901 30 23 40 46 50 68 30 0 6990 10 7012 20 33 30 55 40 76 50 7097 31 07118 10 39 20 60 30 7181 40 7201 50 22 32 0 42 10 62 20 7282 30 7302 40 22 50 42 33 0 7361 7072 7103 34 65 7196 7226 7257 7287 7317 7348 7378 7408 38 67 7497 7526 7556 7585 7614 7644 7673 7701 30 59 7788 7816 7845 7873 7902 30 58 7986 8014 42 70 8097 8125 2928 2897 66 35 2804 2774 2743 2713 2683 2652 2622 2592 62 33 2503 2474 2444 2415 2386 2356 2327 2299 70 41 2212 2184 2155 2127 2098 70 42 2014 1986 58 30 1903 1875 9499 63 0 92 50 86 40 79 30 73 20 66 10 59 62 0 53 50 46 40 39 30 32 20 25 10 18 61 0 11 50 9404 40 9397 30 90 20 83 10 7560 0 68 50 61 40 53 30 46 20 38 10 31 59 23 50 15 40 08 30 9300 20 9292 10 8458 0 76 50 68 40 60 30 52 20 44 10 9236 57 0 9. 9. 10. 9. ~ log cos log cot log tan log sin 9. 9. 10. 9. o / log cos log cot log tan log sin P. P. 38 37 36 35 34 33 32 31 29 28 27 26 25 24 23 22 21 19 I ' 1 2 3 4 5 6 7 8 9 4 4 8 7 11 11 15 15 19 19 23 22 27 26 30 30 34 33 4 4 3 7 7 7 11 11 10 14 14 14 18 18 17 22 21 20 25 25 24 29 28 27 32 32 31 3 3 3 3 7 6 6 6 10 10 9 9 13 13 12 12 17 16 16 15 20 19 19 17 23 22 22 20 26 26 25 23 30 29 28 26 1 3 3 2 6 5 3 8 8 4 11 11 5 14 14 6 17 16 7 20 19 8 22 22 9 25 24 6 6 2 2 5 5 5 5 8 8 7 7 10 10 10 9 13 13 12 12 16 15 14 14 18 18 17 16 21 20 19 18 23 23 22 21 2 2 2 4 4 4 7 6 6 9 8 8 11 11 11 13 13 13 15 15 15 18 17 17 20 19 19 1 2 3 4 5 6 7 8 9 34 TABLE VIII. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS log sin log tan log cot log cos / 9. 9. 10. 9. log sin log tan log cot log cos 0 / 9. 9. 10. 9. 33 O 10 20 30 40 50 34 O 10 20 30 40 50 35 O 10 20 30 40 50 36 0 10 20 30 40 50 37 O 10 20 30 40 50 38 O 10 20 30 40 50 39 0 7361 7380 7400 19 38 57 76 7494 7513 31 50 68 7586 7604 22 40 57 75 7692 7710 27 44 61 78 7795 7811 28 44 61 77 7893 7910 26 41 57 73 7989 8125 8153 8180 8208 35 63 8290 8317 44 71 8398 8425 8452 8479 8506 33 59 8586 8613 39 66 8692 8718 45 71 8797 8824 8850 8876 8902 28 54 8980 9006 32 58 9084 1875 1847 1820 1792 65 37 1710 1683 56 29 1602 1575 1548 1521 1494 67 41 1414 1387 61 34 1308 1282 55 29 1203 1176 1150 1124 1098 72 46 1020 0994 68 42 0916 - - - - I 28 19 11 9203 9194 86 77 69 60 51 42 34 25 16 9107 9098 89 80 70 61 52 42 33 23 14 9004 8995 85 75 65 55 45 35 25 15 8905 57 0 50 40 30 20 10 56 0 50 40 30 20 10 55 0 50 40 30 20 10 54 5 50 40 30 20 10 53 0 50 40 30 20 10 52 0 50 40 30 20 10 51 o 39 O 10 20 30 40 50 40 O 10 20 30 40 50 41 0 10 20 30 40 50 42 O 10 20 30 40 50 43 0~ 10 20 30 40 50 44 0 10 20 30 40 50 45 o 7989 9084 8004 9110 20 35 35 61 50 9187 66 9212 81 38 8096 64 8111 9289 25 9315 40 41 55 66 69 9392 84 9417 8198 43 8213 68 27 9494 41 9519 55 44 69 70 83 9595 8297 9621 8311 46 24 71 38 9697 51 9722 65 47 78 72 8391 9798 8405 9823 18 48 31 74 44 9899 57 9924 69 9949 82 9.9975 8495 10.0000 0916 0890 65 39 0813 0788 62 36 0711 0685 59 34 0608 0583 57 32 0506 0481 56 30 0405 0379 54 29 0303 0278 53 28 0202 0177 52 26 0101 0076 51. 25 0000 8905 8895 84 74 64 53 43 32 21 10 8800 8789 78 67 56 45 33 22 8711 8699 88 76 65 53 41 29 18 8606 8594 82 69 57 45 32 20 8507 8495 51 5 50 40 30 20 10 50 o 50 40 30 20 10 49. 0 50 40 30 20 10 48 0 50 40 30 20 10 47 O 50 40 30 20 10 46 O 50 40 30 20 10 45 0 9. 9. 10. 9. o / log cos log cot log tan log sin 9. 9. 10. 9. / log cos log cot log tan log sin p. p. 28 27 26 25 24 23 22 21 19 18 17 16 15 14 13 12 11 1 2 3 4 5 6 7 8 9 3 3 3 3 2 2 2 2 2 1 6 5 5 5 5 5 4 4 4 2 8 8 8 8 7 7 7 6 6 3 11 11 10 10 10 9 9 8 4 14 14 13 13 12 12 11 11 11 5 17 16 16 15 14 14 13 13 13 6 20 19 18 18 17 16 15 15 15 7 22 22 21 20 19 18 18 17 17 8 25 24 23 23 22 21 20 19 19 9 2 2 2 2 1 1 1 1 4 3 3 3 3 3 2 2 2 5 5 5 5 4 4 4 3 3 7 7 6 6 6 5 5 4 4 9 9 8 8 7 7 6 6 5 11 10 10 9 8 8 7 7 6 13 12 11 11 10 9 8 7 14 14 13 12 11 10 10 9 8 16 15 14 14 13 12 11 10 TABLE IX. MERIDIONAL PARTS 1~ 59.6 6~ 358.2 11~ 659.6 16~ 966.3 21~ 1280.8 26" 1606.2 2~ 119.2 7~ 418.2 12~ 720.5 17~ 1028.5 22~ 1344.9 27~ 1672.9 3~ 178.9 8~ 478.3 13~ 781.5 18~ 1091.0 23~ 1409.5 28~ 1740.2 4~ 238.6 9~ 538.6 14~ 842.8 19~ 1153.9 24~ 1474.5 29~ 1808.1 5~ 298.3 10~ 599.0 15~ 904.4 20~ 1217.1 25~ 1540.1 30~ 1876.7 0' 10' 20' 30' 40' 50' 60' 30 1876.7 1888.2 1899.7 1911.2 1922.8 1934.4 1946.0 31 1946.0 1957.6 1969.2 1980.9 1992.6 2004.3 2016.0 32 2016.0 2027.7 2039.5 2051.3 2063.1 2074.9 2086.8 33 2086.8 2098.7 2110.6 2122.5 2134.4 2146.4 2158.4 34 2158.4 2170.4 2182.5 2194.5 2206.6 2218.7 2230.9 35 2230.9 2243.0 2255.2 2267.4 2279.7 2291.9 2304.2 36 2304.2 2316.5 2328.9 2341.3 2353.7 2366.1 2378.5 37 2378.5 2391.0 2403.5 2416.1 2428.6 2441.2 2453.8 38 2453.8 2466.5 2479.2 2491.9 2504.6 2517.4 2530.2 39 2530.2 2543.0 2555.9 2568.8 2581.7 2594.7 2607.6 40 2607.6 2620.7 2633.7 2646.8 2659.9 2673.1 2686.2 41 2686.2 2699.5 2712.7 2726.0 2739.3 2752.7 2766.0 42 2766.0 2779.5 2792.9 2806.4 2820.0 2833.5 2847.1 43 2847.1 2860.8 2874.4 2888.2 2901.9 2915.7 2929.5 44 2929.5 2943.4 2957.3 2971.3 2985.3 2999.3 3013.4 45 3013.4 3027.5 3041.7 3055.9 3070.1 3084.4 3098.7 46 3098.7 3113.1 3127.5 3141.9 3156.4 3171.0 3185.6 47 3185.6 3200.2 3214.9 3229.6 3244.4 3259.3 3274.1 48 3274.1 3289.0 3304.0 3319.0 3334.1 3349.2 3364.4 49 3364.4 3379.6 3394.9 3410.2 3425.6 3441.0 3456.5 50 3456.5 3472.1 3487.7 3503.3 3519.0 3534.8 3550.6 51 3550.6 3566.5 3582.4 3598.4 3614.5 3630.6 3646.7 52 3646.7 3663.0 3679.3 3695.6 3712.0 3728.5 3745.1 53 3745.1 3761.7 3778.3 3795.1 3811.9 3828.7 3845.7 54 3845.7 3862.7 3879.8 3896.9 3914.1 3931.4 3948.8 55 3948.8 3966.2 3983.7 4001.3 4018.9 4036.7 4054.5 56 4054.5 4072.4 4090.3 4108.4 4126.5 4144.7 4163.0 57 4163.0 4181.3 4199.8 4218.3 4236.9 4255.6 4274.4 58 4274.4 4293.3 4312.3 4331.3 4350.5 4369.7 4389.1 59 4389.1 4408.5 4428.0 4447.6 4467.3 4487.2 4507.1 60 4507.1 4527.1 4547.2 4567.4 4587.8 4608.2 4628.7 61 4628.7 4649.4 4670.1 4691.0 4712.0 4733.1 4754.3 62 4754.3 4775.6 4797.1 4818.6 4840.3 4862.1 4884.1 63 4884.1 4906.1 4928.3 4950.6 4973.1 4995.6 5018.4 64 5018.4 5041.2 5064.2 5087.3 5110.6 5134.0 5157.6 65 5157.6 5181.3 5205.1 5229.1 5253.3 5277.6 5302.1 66 5302.1 5326.7 5351.5 5376.5 5401.6 5427.0 5452.4 67 5452.4 5477.1 5503.9 5529.9 5556.1 5582.5 5609.1 68 5609.1 5635.9 5662.8 5690.0 5717.3 5744.9 5772.7 69 5772.7 5800.7 5828.9 5857.3 5885.9 5914.8 5943.9 70 5943.9 5973.2 6002.8 6032.6 6062.7 6093.0 6123.5 71 6123.5 6154.4 6185.5 6216.8 6248.4 6280.4 6312.5 72 6312.5 6345.0 6377.8 6410.9 6444.3 6478.0 6512.0 73 6512.0 6546.4 6581.0 6616.1 6651.4 6687.1 6723.2 74 6723.2 6759.7 6796.5 6833.7 6871.3 6909.3 6947.7 35