PLANE AND SPHERICAL TRIGONOMETRY PLANE AND SPHERICAL BOOKS BY TRIGONOMETRY C. I. PALMER (Published by McGraw-Hill Book Company, Inc.) PALMER'S Practical Mathematics' Part I-Arithmetic with Applications BY Part II-Algebra with Applications CLAUDE IRWIN PALMER Part III-Geometry with Applications Late Professor of Mathematics and Dean of Students, Armour Institute of Part IV-Trigonometry and Logarithms Technology; Author of a Series of Mathematics Texts PALMER'S AND Practical Mathematics for Home Study PALMER'S CHARLES WILBER LEIGH Practical Calculus for Home Study Professor Emeritus of Analytic Mechanics, Armour Institute of Technology, Author of Practical 1\1echanics 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 FOURTH EDITION PALMER, TAYLOR, AND FARNUM'S Plane and Solid Geometry NINTH IMPRESSION , 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. McGRAW-HILL BOOK COMPANY, INC. NEW YORK AND LONDON 1934 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 COPYRIGHT, 1914, 1916, 1925, 1934, BY THE and W. A. Spencer for their contribution of many new identities MCGRAW-HILL BOOK COMPANY, INC. and equations and also expresses thanks to Mr. Clark Palmer, PRINTED IN THE UNITED STATES OF AMERICA son of the late Dean Palmer, for assisting in checking answers to problems and in proofreading and for offering many constructive All rights reserved. This book, or criticisms. parts thereof, may not be reproduced in any form without permission of CHARLES WILBER LEIGH. CHICAGO, the publishers. June, 1934. THE MAPLE PRESS COMPANY, YORK, PA. v l 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 CONTENTS PAGE such angles. The tables are printed in an unshaded type, and the PREFACETOTHEFOURTHEDITION. .. V arrangement on the pages has received careful study. The authors take this opportunity to express their indebted- PREFACETOTHEFIRST EDITION. vii ness to Prof. D. F. Campbell of the Armour Institute of Tech- CHAPTER I nology, Prof. N. C. Riggs of the Carnegie Institute of Technology, INTRODUCTION and Prof. W. B. Carver of Cornell University, who have read ART. the work in manuscript and proof and have made many valuable 1. Introductory remarks. 1 suggestions and criticisms. 2. Angles, definitions. 2 3. Quadrants. 3 THE AUTHORS. 4. Graphical addition and subtraction of angles. 3 CHICAGO, 5. Angle measurement. 4 September, 1914. 6. The radian. 5 7. Relations between radian and degree. 6 8. Relations between angle, arc, and radius. 8 9. Area of circular sector. 10 10. General angles. 12 11. Directed lines and segments. 13 12. Rectangular coordinates. 14 13. Polar coordinates. 15 CHAPTER II TRIGONOMETRIC FUNCTIONS OF ONE ANGLE 14. Functions of an angle. 17 15. Trigonometric ratios. 17 16. Correspondence between angles and trigonometric ratios. 18 17. Signs of the trigonometric functions. 19 18. Calculation from measurements. 20 19. Calculations from geometric relations. 21 20. Trigonometric functions of 30°. .. ............ 21 21. Trigonometric functions of 45°. 22 22. Trigonometric functions of 120° . 22 23. Trigonometric functions of 0° . 23 24. Trigonometric functions cf 90°. 23 25. Exponents of trigonometric functions. 25 26. Given the function of an angle, to construct the angle. 26 27. Trigonometric functions applied to right triangles. 28 28. Relations between the functions of complementary angles. 30 29. Given the function of an angle in any quadrant, to construct the angle. 31 ix l x r CONTENTS CONTENTS xi ART. CHAPTER III 62. Summary. PAGE 85 RELATIONS BETWEEN TRIGONOMETRIC FUNCTIONS 63. Simple harmonic motion. ART. 64. Inverse functions. 86 30. Fundamental relations between the functions of an angle. PAGE 65. Graph of y sin-l x, or y 87 31. To express one function in terms of each of the other functions. 34 = = arc sin x . .. 87 32. To express all the functions of an angle in terms of one functioI) of 36 """ the angle, by means of a triangle. CHAPTER VII 33. Transformation of trigonometric 37 expressions. PRACTICAL APPLICATIONS AND RELATED PROBLEMS 34. Identities. .. .. .. .. 38 35. Inverse trigonometric 40 66. Accuracy. 90 functions. 36. Trigonometric equations. 42 67. Tests of accuracy. 91 . 43 68. Orthogonal projection. 92 69. Vectors. 93 CHAPTER IV 70. Distance and dip of the horizon. " """. 95 71. Areas of sector and segment. RIGHT TRIANGLES . 99 37. General statement. 72. Widening of pavements on curves . 73. Reflection of a ray of light. 97 38. Solution of a triangle. 47 . 102 . 74. Refraction of a ray of light. 39. The graphical solution. 47 102 40. The solution of right triangles by 48 75. Relation between sin e, e, and tan e," for small'" angles.""'" . 103 computation. 41. Steps in the solution. 48 76. Side opposite small angle given. 105 77. Lengths of long sides given.. 42. Remark on logarithms. 49 105 43. Solution of right triangles by logarithmic functions. 54 44. Definitions. 54 CHAPTER VIII 56 FUNCTIONS INVOLVING MORE THAN ONE ANGLE CHAPTER V 78. Addition and subtraction formulas. 108 FUNCTIONS OF LARGE ANGLES 79. Derivation of formulas for sine and cosine of the sum of two angles 108 46. Functions of !71'- e in terms of functions of e. 80. Derivation of the formulas for sine and cosine of the difference of 47. FUilptioni' of: 62 two angles. 100 + e in tnJJJ.j vi iUlldiuns of u. 48. Functions of 71'- e in terms of functions of e . 63 01. .Pruof of the addition formulas for other values of the angles. 110 49. Functions of 71'+ e in terms of functions of e . 63 82. Proof of the subtraction formulas for other values of the angles. 110 50. Functions of ~71'- e in terms of functions of e. 64 83. Formulas for the tangents of the sum and the difference of two 51. Functions of !71'+ ein terms of functions of e. 65 angles. .113 52. Functions of - e or 271'- e in terms of functions of e. 65 84. Functions of an angle in terms of functions of half the angle. 114 53. Functions of an angle greater than 271'. 66 85. Functions of an angle in terms of functions of twice the angle. 117 54. Summary of the reduction formulas. 67 86. Sum and difference of two like' trigonometric functions as a 55. Solution of trigonometric equations. 67 product. 119 71 87. To change the product of functions of angles to a sum. 122 88. Important trigonometric series. 123 CHAPTER VI GRAPHICAL REPRESENTATION OF TRIGONOMETRIC CHAPTER IX FUNCTIONS 56. Line representation of the trigonometric functions. OBLIQUE TRIANGLES . 89. General statement. 130 57. Changes in the value of the sine and cosine as the angle" increases 76 from 0 to 3600. 90. Law of sines.