PLANE AND SPHERICAL TRIGONOMETRY BOOKS BY C. I. PALMER (Published by McGraw-Hill Book Company, Inc.) PALMER'S Practical Mathematics' Part I-Arithmetic with Applications Part II-Algebra with Applications Part III-Geometry with Applications Part IV-Trigonometry and Logarithms PALMER'S Practical Mathematics for Home Study PALMER'S Practical Calculus for Home Study PALMER AND LEIGH'S Plane and Spherical Trigonometry with Tables PALMER AND KRATHWOHL'S Analytic Geometry PALMER AND MISER'S College Algebra (PuhliRherl hy Scott, Foresman and Company) PALMER, TAYLOR, AND FARNUM'S Plane Geometry Solid Geometry PALMER, TAYLOR, AND FARNUM'S Plane and Solid Geometry , In the earlier editions of Practical Mathematics, Geome- try with Applications was Part II and Algebra with Applica- tions was Part III. The Parts have bcen rearranged in response to many requests from users of the book. PLANE AND SPHERICAL TRIGONOMETRY BY CLAUDE IRWIN PALMER Late Professor of Mathematics and Dean of Students, Armour Institute of Technology; Author of a Series of Mathematics Texts AND CHARLES WILBER LEIGH Professor Emeritus of Analytic Mechanics, Armour Institute of Technology, Author of Practical 1\1echanics FOURTH EDITION NINTH IMPRESSION McGRAW-HILL BOOK COMPANY, INC. NEW YORK AND LONDON 1934 COPYRIGHT, 1914, 1916, 1925, 1934, BY THE MCGRAW-HILL BOOK COMPANY, INC. PRINTED IN THE UNITED STATES OF AMERICA All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. THE MAPLE PRESS COMPANY, YORK, PA. l PREFACE TO THE FOURTH EDITION This edition presents a new set of problems in Plane Trigo- nometry. The type of problem has been preserved, but the details have been changed. The undersigned acknowledges indebtedness to the members of the Department of Mathematics at the Armour Institute of Technology for valuable suggestions and criticisms. He is especially indebted to Profs. S. F. Bibb and W. A. Spencer for their contribution of many new identities and equations and also expresses thanks to Mr. Clark Palmer, son of the late Dean Palmer, for assisting in checking answers to problems and in proofreading and for offering many constructive criticisms. CHICAGO, June, 1934. CHARLES WILBER LEIGH. v r I PREFACE TO THE FIRST EDITION This text has been written because the authors felt the need of a treatment of trigonometry that duly emphasized those parts necessary to a proper understanding of the courses taken in schools of technology. Yet it is hoped that teachers of mathe- matics in classical colleges and universities as well will find it suited to their needs. It is useless to claim any great originality in treatment or in the selection of subject matter. No attempt has been made to be novel only; but the best ideas and treatment have been used, no matter how often they have appeared in other works on trigonometry. The following points are to be especially noted: (1) The measurement of angles is considered at the beginning. (2) The trigonometric functions are defined at once for any angle, then specialized for the acute angle; not first defined for acute angles, then for obtuse angles, and then for general angles. To do this, use is made of Cartesian coordinates, which are now almost universally taught in elementary algebra. (3) The treatment of triangles comes in its natural and logical unler and is not JOfced to the first pages 01 the book. (4) Considerable use is made of the line representation of the trigonometric functions. This makes the proof of certain theo- rems easier of comprehension and lends itself to many useful applications. (5) Trigonometric equations are introduced early and used often. (6) Anti-trigonometric functions are used throughout the work, not placed in a short chapter at the close. They are used in the solutions of equations and triangles. Much stress is laid upon the principal values of anti-trigonometric functions as used later in the more advanced subjects of mathematics. (7) A limited use is made of the so-called "laboratory method" to impress upon the student certain fundamental ideas. (8) Numerous carefully graded practical problems are given and an abundance of drill exercises. (9) There is a chapter on complex numbers, series, and hyper- bolic functions. vii , ' ,'/ ' I ' I ! viii PREFACE TO THE FIRST EDITION (10) A very complete treatment is given on the use of logarith- mic and trigonometric tables. This is printed in connection with the tables, and so does not break up the continuity of the trigo- nometry proper. (11) The tables are carefully compiled and are based upon those of Gauss. Particular attention has been given to the determination of angles near 0 and 90°, and to the functions of such angles. The tables are printed in an unshaded type, and the arrangement on the pages has received careful study. The authors take this opportunity to express their indebted- ness to Prof. D. F. Campbell of the Armour Institute of Tech- nology, Prof. N. C. Riggs of the Carnegie Institute of Technology, and Prof. W. B. Carver of Cornell University, who have read the work in manuscript and proof and have made many valuable suggestions and criticisms. THE AUTHORS. CHICAGO, September, 1914. l CONTENTS PAGE PREFACETO THE FOURTHEDITION. . . . . . . . . . . . . . V PREFACETOTHE FIRST EDITION. . . . . . . . . . . . . . . . . vii CHAPTER I INTRODUCTION ART. 1. Introductory remarks. . . . . . . . . . . . . . . . . . . . 2. Angles, definitions. . . . . . . . . . . . . . . . . . 3. Quadrants. . . . . . . . . . . . . . . . . . . . . . . . . 4. Graphical addition and subtraction of angles. . . . . . . . . . 5. Angle measurement. . . . . . . . . . . . . . . . . . . . . 6. The radian. . . . . . . . . . . . . . . . . . . . . . . . 7. Relations between radian and degree. . . . . . . . . . . . . 8. Relations between angle, arc, and radius. . . . . . . . . . . 9. Area of circular sector. . . . . . . . . . . . . . . . . . . 10. General angles. . . . . . . . . . . . . . . . . . . . . . . 11. Directed lines and segments. . . . . . . . . . . . . . . . . 12. Rectangular coordinates. . . . . . . . . . . . . . . . . . . 13. Polar coordinates. . . . . . . . . . . . . . . . . . . . . . 1 2 3 3 4 5 6 8 10 12 13 14 15 CHAPTER II TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 14. Functions of an angle. . . . . . . . . . . . . . . 15. Trigonometric ratios. . . . . . . . . . . . . . . . . . . 16. Correspondence between angles and trigonometric ratios. . . . 17. Signs of the trigonometric functions. . . . . . . . . . 18. Calculation from measurements. . . . . . . . . . . . . . . 19. Calculations from geometric relations. . . . . . . . . . . . . 20. Trigonometric functions of 30°. 21. Trigonometric functions of 45°. . . . . . . . . . . . . 22. Trigonometric functions of 120° . . . . . . . . . . . . 23. Trigonometric functions of 0° . . . . . . . . . . . . . 24. Trigonometric functions cf 90°. . . . . . . . . . . . . . . . 25. Exponents of trigonometric functions. . . . . . . . . . . . . 26. Given the function of an angle, to construct the angle. . . . . 27. Trigonometric functions applied to right triangles. . . . . 28. Relations between the functions of complementary angles. 29. Given the function of an angle in any quadrant, to construct the angle. . . . . . . . . . . . . . . . . . . . . . . . . . ix 17 17 18 19 20 21 21 22 22 23 23 25 26 28 30 31 r x CONTENTS CHAPTER III RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS ART. 30. Fundamental relations between the functions of an angle. . . 31. To express one function in terms of each of the other functions. 32. To express all the functions of an angle in terms of one functioI) of the angle, by means of a triangle. . . . . . . . . . 33. Transformation of trigonometric expressions. . . . . 34. Identities. . . . . . . . . . . . . . . . . . . . . . . . . 35. Inverse trigonometric functions. . . . . . . . . . . . 36. Trigonometric equations. . . . . . . . . . . . . CHAPTER IV RIGHT TRIANGLES 37. General statement. . . . . . . . . . . . 38. Solution of a triangle. . . . . . . . . . . . 39. The graphical solution. . . . . . . . . . . . . . . . . . 40. The solution of right triangles by computation. . . . . . . . . 41. Steps in the solution. . . . . . . . . . . . . . . . . 42. Remark on logarithms. . . . . . . . . . . . . . . . . . . 43. Solution of right triangles by logarithmic functions. . . . . . . 44. Definitions. . . . . . . . . . . . . . . . . . . . . . . . CHAPTER V FUNCTIONS OF LARGE ANGLES 46. Functions of !71' - e in terms of functions of e. . . . . . . . . 47. FUilptioni' of: + e in tnJJJ.j vi iUlldiuns of u. . . . . . 48. Functions of 71'- e in terms of functions of e . . . . . . . . . 49. Functions of 71'+ ein terms of functions of e . . . . . . . . . 50. Functions of ~71'- e in terms of functions of e. . . . . . . . . 51. Functions of !71' + ein terms of functions of e. . . . . . . . . 52. Functions of - e or 271'- e in terms of functions of e. . . . . . 53. Functions of an angle greater than 271'. . . . . . . . . . . . . 54. Summary of the reduction formulas. . . . . . . . . . . . . . 55. Solution of trigonometric equations. . . . . . . . . . . . . . CHAPTER VI GRAPHICAL REPRESENTATION OF TRIGONOMETRIC FUNCTIONS 56. Line representation of the trigonometric functions. " . 57. Changes in the value of the sine and cosine as the angle increases from 0 to 3600. . . . . . . . . . . . . . . . . . . . . . 58. Graph of y = sin e. . . . . . . . . . . . . . . . . . . . . 59. Periodic functions and periodic curves. . . . . . . . . . . . 60. Mechanical construction of graph of sin e. . . . . . . . . . . 61. Projection of point having uniform circular motion. . . . . . . PAGE 34 36 37 38 40 42 43 47 47 48 48 49 54 54 56 62 63 63 64 65 65 66 67 67 71 76 78 79 80 82 83 CONTENTS xi PAGE 85 86 87 87 ART. 62. Summary. . . . . . . . . . . . . . . . . . . . . . 63. Simple harmonic motion. . . . . . . . . . . . . . . . . . 64. Inverse functions. . . . . . . . . . . . . . . . . . . . . . 65. Graph of y = sin-l x, or y = arc sin x . . """ CHAPTER VII PRACTICAL APPLICATIONS AND RELATED PROBLEMS 66. Accuracy. . . . . . . . . . . . . . . . . . . . . . . . . 90 67. Tests of accuracy. . . . . . . . . . . . . . . . . . . . . . 91 68. Orthogonal projection. . . . . . . . . . . . . . . . . . . . 92 69. Vectors. . . . . . . . . . . . . . . . . " """ 93 70. Distance and dip of the horizon. . . . . . . . . . . 95 71. Areas of sector and segment. . . . . . 99 72. Widening of pavements on curves 97 73. Reflection of a ray of light. . . . . . . . 102 74. Refraction of a ray of light. . . " '" ""'" 102 75. Relation between sin e, e, and tan e, for small angles. . . . . . 103 76. Side opposite small angle given. . . . . . . . . . . . . . . 105 77. Lengths of long sides given 105 CHAPTER VIII FUNCTIONS INVOLVING MORE THAN ONE ANGLE 78. Addition and subtraction formulas. . . . . . . . . . . . . . 108 79. Derivation of formulas for sine and cosine of the sum of two angles 108 80. Derivation of the formulas for sine and cosine of the difference of two angles. . . . . . . . . . . . . . . . . . 100 01. .Pruof of the addition formulas for other values of the angles. . . 110 82. Proof of the subtraction formulas for other values of the angles. 110 83. Formulas for the tangents of the sum and the difference of two angles. . . . . . . . . . . . . . . . . . . . . . . . .113 84. Functions of an angle in terms of functions of half the angle. . . 114 85. Functions of an angle in terms of functions of twice the angle. . 117 86. Sum and difference of two like' trigonometric functions as a product. . . . . . . . . . . . . . . . . . . . . . . . . 119 87. To change the product of functions of angles to a sum. . . . . 122 88. Important trigonometric series. . . . . . . . . . . . . . . . 123 CHAPTER IX OBLIQUE TRIANGLES 89. General statement. . . . . . . . . . . . . . . . . . 130 90. Law of sines. . . . . . . . . . . . . . . . . . . . . . . . 130 91. Law of cosines. . . . . . . . . . . . . . . . . . . . . . . 132 92. Case 1. The solution of a triangle when one side and two angles are given. . . . . . . . . . . . . . . " . . '. 132 93. Case II. The solution of a triangle when two sides and an angle opposite one of them are given. . . . . . . . . . . . 136 xii CONTENTS ART. PAGE 94. Case III. The solution of a triangle when two sides and tne included angle are given. First niethod. "" 140 95. Case III. Second method. . . . . . . "" 140 96. Case IV. The solution of a triangle when the three sides are given 143 97. Case IV. Formulas adapted to the use of logarithms. . . . . . 144 CHAPTER X MISCELLANEOUS TRIGONOMETRIC EQUATIONS 98. Types of equations. . . . . . . . . . . . . . . . . . . . . 158 99. To solve r sin 0 + 8 cos 0 = t for 0 when r, 8, and t are known. 160 100. Equations in the form p sin a cos fJ = a, p sin a sin fJ = b, p cos a = c, where p, a, and fJ are variables. . . . . . . . . . . . . . 161 101. Equations in the form sin (a + fJ) = c sin a, where fJ and care known. . . . . . . . . . . . . . . . . . . . . . . . . 161 102. .Equations in the form tan (a + fJ) = c tan a, where fJ and care known. . . . . . . . . . . . . . . . . . . . . . . . . 162 103. Equations of the form t = 0 + '" sin t, where 0 and", are given angles. . . . . . . . . . . . . . . . . . . . . . . . . 162 CHAPTER XI COMPLEX NUMBERS, DEMOIVRE'S THEOREM, SERIES 104. Imaginary numbers. . . . . . . . . . . . . . . . . . . . . 165 105. Square root of a negative number. . . . . . . . . . . . . . 165 106. Operations with imaginary numbers. . . . . . . . . . . . . 166 107. Complex numhers . . . . . . . . . . . . . . . . . . . . . 166 108. Conjugate complex numbers. . . . . . . . . . . . . . . . . 167 109. Graphical representation of ('ompJex nurnncr:" Wi 110. Powers of i . . . . . . . . . . . . . . . . . . . . . . . . 169 111. Operations on complex numbers. . . . . . . . . . . . " 169 112. Properties of complex numbers. . . . . . . . . . . . . . . .171 113. Complex numbers and vectors. . . . . . . . . . . . . . . . 171 114. Polar form of complex numbers. . . . . . . . . . . . . . . 172 115. Graphical representation of addition. . . . . . . . . . . . . 174 116. Graphical representation of subtraction. . . . . . . . . . . . 175 117. Multiplication of complex numbers in polar form. . . . . . . . 176 118. Graphical representation of multiplication. . . . . . . . . . . 176 119. Division of complex numbers in polar form. . . . . . . " 176 120. Graphical representation of division. . . . . . . . . '. 177 121. In volution of complex numbers. . . . . . . . . . . . . . . 177 122. DeMoivre's theorem for negative and fractional exponents. . 178 123. Evolution of complex numbers. . . . . . . . . . . . . 179 124. Expansion of sin nO and cos nO. . . . . . . . . . . . " . 182 125. Computation of trigonometric functions. . . . . . . . . . . . 184 126. Exponential values of sin 0, cos 0, and tan O. . . . . . . . . . 184 127. Series for sinn 0 and cosn 0 in terms of sines or cosines of multiples of O. . . . . . . . . . . . . . . . . . . . . . . . . . . 185 ] 28. Hyperbolic functions. . . . . . . . . 187 CONTENTS xiii ART. PAGE 129. Relations between the hyperbolic functions. . . . . . . . . . 188 130. Relations between the trigonometric and the hyperbolic functions 188 131. Expression for sinh x and cosh x in a series. Computation 189 131'. Forces and velocities represented as complex numbers 189 CHAPTER XII SPHERICAL TRIGONOMETRY 132. Great circle, small circle, axis. . . . . . . . . . . . . . . . 193 133. Spherical triangle. . . . . . . . . . . . . . . . . . . . . 193 134. Polar triangles. . . . . . . . . . . . . . . . . . . . . . . 194 135. Right spherical triangle. . . . . . . . . . . . . . . . . . . 195 136. Derivation of formulas for right spherical triangles. . . . . . . 196 137. Napier's rules of circular parts. . . . . . . . . . . . . . . . 197 138. Species. . . . . . . . . . . . . . . . . . . . . . . . . . 198 139. Solution of right spherical triangles. . . . . . . . . . . . . . 198 140. Isosceles spherical triangles. . . . . . . . . . . . . . . . . 200 141. Quadrantal triangles. . . . . . . . . . . . . . . . . . . . 201 142. Sine theorem (law of sines) . . . . . . . . . . . . . . . . . 202 143. Cosine theorem (law of cosines) . . . . . . . . . . . . . . . 202 144. Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 204 145. Given the three sides to find the angles. . . . . . . . . . . . 204 146. Given the three angles to find the sides. . . . . . . . . . . . 205 147. Napier's analogies. . . . . . . . . . . . . . . . . . . . . 206 148. Gauss's equations. . . . . . . . . . . . . . . . . . . . . 208 149. Rules for species in oblique spherical triangles. . . . . . . . . 209 150. Cases. . . . . . . . . . . . . . . . . . . . . . . . . . . 210 151. Case I. Given the three sides to find the three an~le:" . 211 152. Case 11. Given the three angles to find the thrce sides. . . . . 212 153. Case III. Given two sides and the included angle. . . . . . . 212 154. Case IV. Given two angles and the included side. . . . . . . 213 155. Case V. Given two sides and the angle opposite one of them. . 213 156. Case VI. Given two angles and the side opposite one of them. . 215 157. Area of a spherical triangle. . . . . . . . . . . . . . . . . 215 158. L'Huilier's formula. . . . . . . . . . . . . . . . . . . . . 216 159. Definitions and notations. . . . . . . . . . . . . . . . . . 217 160. The terrestrial triangle. . . . . . . . . . . . . . . . . . . 217 161. Applications to astronomy. . . . . . . . . . . . . . . . . . 218 162. Fundamental points, circles of reference. . . . . . . . . . . . 219 Summary of formulas. . . . . . . . . . . . . . . . . 222 Useful constants. . . . . . . . . . . . . . . . . . . . . . 225 INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 The contents for the Logarithmic and Trigonometric Tables and Explanatory Chapter is printed with the tables. A, a. . . . . . . . Alpha N, II. . . . . . . . Nu B, (3. . . . . . . . Beta Z, ~. . . . . . . . Xi r, 'Y. . . . . . . . Gamma 0, o. . . . . . . . Omicron .1, O. . . . . . . . Delta IT, 7r. . . . . . . . Pi I PLANE AND SPHERICAL E, E. . . . . . . . Epsilon P, p. . . . . . . . Rho Z, t. . . . . . . . Zeta 1:, ()". . . . . . . . Sigma TRIGONOMErrRY H, 'T/. . . . . . . . Eta T, T. . . . . . . . Tau EI, (). . . . . . . . Theta T, u. . . . . . . . Upsilon I, L . . . . . . . . Iota <P,cf>. . . . . . . . Phi . CHAPTER I K, K. . . . . . . . Kappa X, x. . . . . . . . Chi A, A. . . . . . . . Lambda '1', if;. . . . . . . . Psi . INTRODUCTION M,}J . . . . . . . Mu 11, "'. . . . . . . . Omega . GEOMETRY xiv CONTENTS GREEK ALPHABET 1. Introductory remarks The word trigonometry is derived from two Greek words, TPL'Y"'IIOIl(trigonon), meaning triangle, and }J.ETpLa(metria), meaning measurement. While the derivation of the word would seem to confine the subject to triangles, the measurement of triangles is merely a part of the general subject which includes many other investigations involving angles. Trigonometry is both geometric and algebraic in nature. Historically, trigonometry developed in connection with astron- omy, where distances that could not be measured directly were computed by means oLaJlgltltLlicQ<il111e;:L1hat_~9JJld_bemeasill'gd. The beginning of these methods may be traced to Babylon and Ancient Egypt. The noted Greek astronomer Hipparchus is often called the founder of trigonometry. He did his chief work between 146 and 126 B. C. and developed trigonometry as an aid in measuring angles and lines in connection with astronomy. The subject of trigonometry was separated from astronomy and established as a distinct branch of mathematics by the great mathematician Leonhard Euler, who lived from 1707 to 1783. To pursue the subject of trigonometry successfully, the student should know the subjects usually treated in algebra up to and including quadratic equations, and be familiar with plane geom- etry, especially the theorems on triangles and circles. Frequent use is made of the protractor, compasses, and the straightedge in constructing figures. While parts of trigonometry can be applied at once to the solution of various interesting and practical problems, much of 1 I l - '2 PLANE AND SPHERICAL TRIGONOMETRY it is studied because it is very frequently used in more advanced subjects in mathematics. ANGLES 2. Definitions The definition of an angle as given in geometry admits of a clear conception of small angles only. In trigo- nometry, we wish to consider positive and negative angles and these Jf any size whatever; hence we need a more comprehensive definition of an angle. If a line, starting from the position OX (Fig. 1), is revolved about the point 0 and always kept in the same plane, we say the line generates an angle. If it revolves from the position OX to the position OA, in the direction indicated by the arrow, the YI angle XOA is generated. ,A The original position OX of the generating line is called the initial side, and the final position OA, the x terminal side of the angle. If the rotation of the generating line is counterclockwise, as already taken, the angle is said to be positive. If OX revolves in a clockwise direc- tion to a position, as OB, the angle generated is said to be negative. In reading an angle, the letter on the initial side is read first to give the proper sense of direction. If the angle is read in the opposite sense, the negative of the angle is meant. Thus, LAOX = -LXOA. It is easily seen that this conception of an angle makes it possible to think of an angle as being of any size whatever. Thus, the generating line, when it has reached the position OY, having made a quarter of a revolution in a counterclockwise direction, has generated a right angle; when it has reached the position OX' it has generated two right angles. A complete revolution gener- ates an angle containing four right angles; two revolutions, eight right angles; and so on for any amount of turning. The right angle is divided into 90 equal parts called degrees (°), each degree is divided into 60 equal parts called minutes ('), and each minute into 60 equal parts called seconds ("). Starting from any position as initial side, it is evident that for each position of the terminal side, there are two angles less x! C y' FIG. 1. INTRODUCTION 3 than 360°, one positive and one negative. Thus, in Fig. 1, oe is the terminal side for the positive angle xoe or for the negative angle xoe. 3. Quadrants It is convenient to divide the plane formed by a complete revolution of the generating line into four parts by the two perpendicular lines X' X and Y' Y. These parts are called first, second, third, and fourth quadrants, respectively. They are placed as shown by the x~ Roman numerals in Fig. 2. If OX is taken as the initial side of an angle, the angle is said to lie in the quadrant in which its terminal side lies. Thus, XOP1 (Fig. 2) lies in the third quadrant, and XOP2, formed by more than one revolution, lies in the first quadrant. An angle lies between two quadrants if its terminal side lies on the line between two quadrants. 4. Graphical addition and subtraction of angles Two angles are added by placing them in the same plane IL c with their vertices together and the initialside B of the "econd on the terminal "ide of the first. The sum is the angle from the initial side of . the first to the terminal side of the second. 0 Subtraction is performed by adding the 0 A negative of the subtrahend to the minuend. FIG. 3. Th ' F ' 3us, In Ig. , LAOB + LBOe = LAOe. LAOe - LBOe = LAOe + LeOB = LAOE. LBOe - LAOe = LBOe + LeOA = LBOA. Y II Pz III IV x fl IY' FIG. 2. EXERCISES Use the protractor in laying off the angles in the L>llowing exercises: 1. Choose an initial side and layoff the following angles, Indicate each angle by a circular arrow. 75°; 145°; 243°; 729°; 456°; 976°. State the quadrant in which each angle lies. 2. Layoff the following angles and state the quadrant that each is in: -40°; -147°; -295°; -456°; -1048°. 3. Layoff the following pairs of angles, using the same initial side for each pair: 170° and -190°; -40° and 320°; 150° and -210°. 4 PLANE AND SPHERICAL TRIGONOMETRY 4. Give a positive angle that has the same terminal side as each of the following: 30°; 165°; -90°; -210°; -45°; 395°; -390°. 5. Show by a figure the position of the revolving line when it has gener- ated each of the following: 3 right angles; 2i right angles; Ii right angles; 4i right angles. Unite graphically, using the protractor: 6. 40° + 70°; 25° + 36°; 95° + 125°; 243° + 725°. 7. 75° - 43°; 125° - 59°; 23° - 49°; 743° - 542°; 90° - 270°. 8. 45° + 30° + 25°; 125° + 46° + 95°; 327° + 25° + 400°. 9. 45° - 56° + 85°; 325° - 256° + 400°. 10. Draw two angles lying in the first quadrant put differing by 360°. Two negative angles in the fourth quadrant and differing by 360°. 11. Draw the following angles and their complements: 30°; 210°; 345°; -45°; -300°; -150°. 5. Angle measurement Several systems for measuring angles are in use. The system is chosen that is best adapted to the purpose for which it is used. (1) The right angle The most familiar unit of measure of an angle is the right angle. It is easy to construct, enters frequently into the practical uses of life, and is almost always used in geom- etry. It has no subdivisions and does not lend itself readily to computations. (2) The sexagesimal system The sexagesimal system has for its fundamental unit the degree, which is defined to be the angle formed by -do part of a revolution of the generating linp system used by eng;mecrs and others in making prac- tical numerical computations. The subdivisions of the degree are the minute and the second, as stated in Art. 2. The word "sexagesimal" is derived from the Latin word sexagesimus, meaning one-sixtieth. (3) The centesimal system Another system for measuring angles was proposed in France somewhat over a century ago. This is the centesimal system. In it the right angle is divided into 100 equal parts called grades, the grade into 100 equal parts called minutes, and the minute into 100 equal parts called seconds. While this system has many admirable features, its use could not become general without recomputing with a great expenditure of labor many of the existing tables. (4) The circular or natural system In the circular or natural system for measuring angles, sometimes called radian measure or .,,;-measure, the fundamental unit is the radian. The radian is defined to be the angle which, when placed with its vertex at the center of a circle, intercepts an arc equal in length l INTRODUCTION 5 to the radius of the circle. Or it itS defined as the positive angle generated when a point on the generating line has passed through an arc equal in length to the radius of the circle being formed by that point. In Fig. 4, the angles AOB, BOC, . . . FOG are each 1 radian, since the sides of each angle intercept an arc equal in length to the radius of the circle. The circular system lends itself nat- urally to the measurement of angles in many theoretical considerations. It is used almost exclusively in the calculus D and its applications. (5) Other systems Instead of divid- ing the degree into minutes and seconds, it is sometimes divided into tenths, hundredths, and thousandths. This decimal scale has been used more or less ever since decimal frac- tions were invented in the sixteenth century. The mil is a unit of angle used in artillery practice. The mil is lr roo revolution, or very nearly -rcrITOradian; hence its name. The scales by means of which the guns in the United States Field Artillery are aimed are graduated in this unit. . f"'T"I'I, ,'t - ,-;YDLemto UDein measuring an angle is apparent from a consider- ation of the geometrical basis for the definition of the radian. FIG. 5. FIG. 6. (1) Given several concentric circles and an angle AOB at the center as in Fig. 5, then arc PlQl arc P2Q2 arc PaQa t OPl = OP2 = OPa ' e c. [...]... meanin~." " l 24 PLANE AND SPHERICAL FREQUENTLY 0 in radians 0° 0° sin 0 0 30° 6 7r 46° 4 60° va T :3 7r 90° 1 2 0 2 0 2 2", 120° "3 3", 136° 4 160° fur 6" 180° 7r 7", 210° 6" 5", 226° 4 4", 240° "3 37r "2 5", 300° 316° 330° "3 7", 4 II", 6 360° 2", 1 2 0 1 -2 0 -2 0 -2 1 -2 0 -2 0 -2 -1 0 -2 0 -2 1 -2 -1 1 2 cot 0 0 0 1 00 2 0 1 V2 0 va a0 20 2 00 00 0 -a- -0 -a- -2 20 -r -1 -1 -0 -0 20 a-1 20 ""f FIG... tan - 1 = VI () = + 1 - t cot2 = 3 - 4 () = VI + 1\ = t ~ ~ =~ sin (J = csc () = t 5 113 () cas = - () = - = - sec t 5 Example 2.-Given sin () = !, and () in the second quadrant, determine the other functions by means of the fundamental formulas Solution.-By[1],cos()= -Vl-sin2()= -Yl-t= -h/3 () ! 1 By [7], t an () = sin () = = -3 v - r3 cas -! V3 By [6], 11cot (J = tan () = -tv3 = -v3 - 36 PLANE AND SPHERICAL. .. = (i = O r 0 -a0 0 a- 00 0 1 1 0 0 "3" 0 00 0 -0 -a- 0 2 0 2 -1 -1 0 -a- -0 0 sin 90° 00 = 25 1 60° 2 135° 3 150° 4 180° 6 240° 6 330° =~I -2 00 2 0 20 ~ 1 00 -2 -0 20 a-1 _20 _;1 -: J x J! - a r-(i=I :: 0 cot 90° = y - (i =aO - 7 270° 8 315° 25 Exponents of trigonometric functions .- When the trigonometric functions are to be raised to powers, they are written sin2 8, cos3 8, tan4 8, etc., instead of... RELATIONS BETWEEN PLANE AND SPHERICAL TRIGONOMETRY OM and MP Then, by definition, y cos (J = ~~, and, if OP is equal to 1, OM = cos (J, and MP = V OP2 - UM2=VI - cos2 (J ~; sin (J = FIG 35 MP eos 0 IVI - sin' VI VI / 'O =-= - sin' OI "n - ~in-IJ see 0 I cae (J VI lv/l 1 VI - I VI VI - 1 VI - 1 1 + tanie ~ cot 0 0 vsee' 0 - 1 llill fJ - + tan' 0 VI VI + tan' 0 + cot ,- 0 !V"""" 0 0 tan 0 ~ 1 cse 0 -= = vese'... cos -1 V2 5 eos sec .-1 5 2' 9 sin sec-1 H 2 sin sin-1 ~1~ 3 tan sec-1 2 4 cos ese-1 3 Prove the relations 6 tan sin-1 H 10 7 sin eos-1 O 11 8 sin tan-1 0 12 in Exercises 13 to 22 18 cas sin-1 a U, = ::f:V'I - u,2 cse eot-1 l sin cos-1 t cos sec-1 5 13 Sill eus'l = :+:~/ I - n' 14 sin tan-1 a = ::f: a v 1 + a' 19 eos tan-1 a = ::f: ~ V 1 + a' 15 sin cot-1 a = :1: ~ V 1 + a' 16 sin sec-1 a = + V a' -. .. cos4 1 - 2 cos2 to sin40 + cos4 0 to 1 - 2 sin2 0 cos2 O 2 sm see 4 (1 - '"' 1 covers (J)2 + ~ 2cos ~ + s~n -" I-sm J ~1=S~n sec = l+sm / 40 PLANE AND SPHERICAL TRIGONOMETRY 34 Identities.When two expressions... 1 - sm 8 sin fJ Vsec2 4 >- 1 (1 - sin2 4 -' = tan 4>tan fJ see 4>v / 1 - sin2 fJ cos.8 _1 - sin 8 2 tan 8 1 - sm 0 cas 8 = (1 - tan 4»2 sec 2 4> + 2 sin 4>cas 4>= 1 19 s~n 8 + s~n 4> s~n csc 4> + csc O sm 8 - sm 4>= esc 4 >- csc 8 (1 + sin 4» /1 - s~n4> /1 20 + ["J1+sm4> ~ 2cos ~ + s~n -" I-sm J ~1=S~n sec = l+sm r J/ 42 PLANE AND SPHERICAL TRIGONOMETRY RELATIONS 21 (tan' 6 + l)eot' . PLANE AND SPHERICAL TRIGONOMETRY BOOKS BY C. I. PALMER (Published by McGraw-Hill Book Company, Inc.) PALMER'S Practical Mathematics' Part I-Arithmetic with Applications Part II-Algebra. + + + + + + II . . + - - - - + III - - + + - - IV. """"""""""'" - + - - + - 'il 18 PLANE AND SPHERICAL TRIGONOMETRY . ( . . ) ordinate. in: -4 0°; -1 47°; -2 95°; -4 56°; -1 048°. 3. Layoff the following pairs of angles, using the same initial side for each pair: 170° and -1 90°; -4 0° and 320°; 150° and -2 10°. 4 PLANE AND SPHERICAL TRIGONOMETRY 4.