PLANE AND SPHERICAL 
 
 TRIGONOMETRY 
 
 BY 
 
 GEORGE N. BAUER, PH.D. 
 
 N 
 
 AND 
 
 W. E. BROOKE, C.E.,M.A. 
 
 UNIVERSITY OF MINNESOTA 
 
 SECOND REVISED EDITION 
 
 D. C. HEATH & CO., PUBLISHERS 
 BOSTON NEW YORK CHICAGO 
 
PHYSIOS 
 
 COPYRIGHT, 1907 AND 1917, 
 BY D. C. HEATH & Co. 
 
 H7 
 
PREFACE 
 
 THE plan and scope of this work may be indicated briefly 
 by the following characteristic features : 
 
 Directed lines and Cartesian coordinates are introduced 
 as a working basis. 
 
 Each subject, when first introduced, is treated in a gen- 
 eral manner and is presented as fully as the character of 
 the work demands. This avoids the necessity of treating 
 the same subjects several times for the purpose of modify- 
 ing and extending certain conceptions. 
 
 Statements in the form of theorems and problems are 
 used freely to indicate the aim of various articles and to 
 define the data clearly. 
 
 Inverse functions are treated more fully than is customary. 
 
 The general principles governing the solution of triangles, 
 the solution of trigonometric equations, and the proof of 
 identities are carefully presented. 
 
 . Special attention is given to the arrangement of compu- 
 tations. 
 
 The sine and cosine series are obtained from De Moivre's 
 Theorem, thus completing the line of development which 
 leads to the calculation of the trigonometric functions. 
 
 The work on spherical trigonometry contains the devel- 
 opment of all the formulas that are generally used in 
 practical astronomy. 
 
 The right spherical triangle is treated from two points of 
 view : as a special case of the oblique triangle, and directly 
 
 iii 
 
iv PREFACE 
 
 from geometric figures. The work is so arranged that 
 either view may be presented independently. 
 
 The solutions of the oblique spherical triangle by means 
 of auxiliary quantities, characteristic of astronomy, are 
 included as interesting mathematical problems and as 
 preparation for astronomical work. 
 
 PREFACE TO THE SECOND REVISED EDITION 
 
 The second edition embodies such modifications, rear- 
 rangements, and additions as have been suggested by experi- 
 ence in the classroom. 
 
 Due to numerous requests tables have been added. 
 Simple three-place tables have been included. By their use 
 the numerical work is reduced to a minimum, thus leaving 
 the student free to give more attention to the principles in- 
 volved in the solution of a triangle. Accordingly a few tri- 
 angles, suitable for three-place table or slide rule computa- 
 tion, have been introduced at the beginning of the list of 
 problems under the right triangle and also under the oblique 
 triangle. 
 
 In order to give a wider range to computational work four- 
 place tables as well as five-place tables have been included. 
 
 G. N. B. 
 
 W. E. B. 
 
 MINNEAPOLIS, MINNESOTA, 
 June, 1917. 
 
CONTENTS 
 
 PLANE TRIGONOMETRY 
 
 CHAPTER I 
 RECTANGULAR COORDINATES AND ANGLES 
 
 ABT. PAOB 
 
 1. Introduction 1 
 
 RECTANGULAR COORDINATES 
 
 2. Directed lines .1 
 
 3. Lines of reference 2 
 
 4. Quadrants 2 
 
 5. Coordinates of a point 2 
 
 6. Signs of coordinates . . 3 
 
 7. Exercises 3 
 
 ANGLES 
 
 8. Magnitude of angles 4 
 
 9. Direction of rotation. Positive and negative angles . . 4 
 
 10. Initial and terminal lines 6 
 
 11. Sign of terminal line 5 
 
 12. Algebraic sum of two angles 6 
 
 13. Measurement of angles 6 
 
 14. Circular or radian measure 7 
 
 15. Value of radian 7 
 
 16. Relation between degree and radian 7 
 
 17. Relation between angle, radius, and arc . . . .9 
 
 18. Linear and angular velocity 9 
 
 19. Examples 10 
 
 v 
 
CONTENTS 
 
 CHAPTER II 
 TRIGONOMETRIC FUNCTIONS 
 
 ART. PAGE 
 
 20. Trigonometric functions introduced 12 
 
 21. Definitions of the trigonometric functions .... 12 
 
 22. Values of the trigonometric functions of 30, 45, and 60 . 13 
 
 23. Values of the trigonometric functions of 120, 135, and 150 15 
 
 24. Signs of the trigonometric ratios 17 
 
 25. Trigonometric functions are single valued .... 18 
 
 26. A given value of a trigonometric function determines an 
 
 infinite number of angles 19 
 
 27. Examples . 20 
 
 CHAPTER III 
 
 RIGHT TRIANGLES 
 
 28. Statement of problem 22 
 
 29. Application of the definitions of the trigonometric functions 
 
 to the right triangle 22 
 
 30. Trigonometric tables . . 23 
 
 31. Formulas used in the solution of right triangles ... 24 
 
 32. Selection of formulas 25 
 
 33. Check formulas 25 
 
 34. Suggestions on solving a triangle 25 
 
 35. Illustrative examples 26 
 
 36. Examples .......... 29 
 
 37 . Oblique triangles 30 
 
 38. Applications 31 
 
 CHAPTER, IV 
 
 VARIATIONS OF THE TRIGONOMETRIC FUNCTIONS 
 REDUCTION OF FUNCTIONS OF n90 
 
 39. Values of functions at 0, 90, 180, 270, and 360 . . 34 
 
 40. Variation of the functions 35 
 
 41. Graphical representation 39 
 
 42. Periodicity of the trigonometric functions .... 40 
 
 43. Examples 41 
 
CONTENTS vii 
 
 ABT PAGE 
 
 44. Use of formulas 42 
 
 45. Functions of a in terms of functions of a . . . .42 
 
 46. Functions of 90 -f a in terms of functions of a . .43 
 
 47. Functions of 90 a in terms of functions of a . .44 
 
 48. Functions of 180 a in terms of functions of a . . .45 
 
 49. Laws of reduction . . .46 
 
 50. Examples 46 
 
 CHAPTER V 
 FUNDAMENTAL KELATIONS. LINE VALUES 
 
 51. General statement 47 
 
 FUNDAMENTAL RELATIONS 
 
 52. Development of formulas 47 
 
 53. The use of exponents 48 
 
 54. Trigonometric identities 49 
 
 55. Trigonometric equations 49 
 
 56. Examples .51 
 
 LINE VALUES 
 
 57. Representation of the trigonometric functions by lines . 53 
 
 58. Variations of the trigonometric functions as shown by line 
 
 values 55 
 
 59. Fundamental relations by line values 66 
 
 60. Examples 57 
 
 CHAPTER VI 
 
 FUNCTIONS OF THE SUM OF TWO ANGLES 
 DOUBLE ANGLES. HALF ANGLES 
 
 61. Statement of problem 61 
 
 62. The sine of the sum of two acute angles expressed in terms 
 
 of the sines and cosines of the angles .... 61 
 
 63. The cosine of the sum of two acute angles .... 62 
 
 64. Importance of formulas ........ 62 
 
 65. Generalization of formulas 63 
 
 66. Tangent of the sum of two angles 65 
 
 67. Cotangent of the sum of two angles 65 
 
 68. Addition formulas . 65 
 
viii CONTENTS 
 
 ART. PAGB 
 
 69. Sine, cosine, tangent, and cotangent of the difference of two 
 
 angles 66 
 
 70. Exercises .66 
 
 71. Double angles 67 
 
 72. Half angles 68 
 
 73. Sum and difference of two sines and of two cosines . . 69 
 
 74. Equations and identities 70 
 
 75. Examples . .71 
 
 CHAPTER VII 
 INVERSE FUNCTIONS 
 
 76. Statement of problem 74 
 
 77. Fundamental idea of an inverse function .... 74 
 
 78. Multiple values of an inverse function . . . 75 
 
 79. Principal values .77 
 
 80. Interpretation of sin sin- 1 a and sin- 1 sin a . . . .77 
 
 81. Application of the fundamental relations to angles expressed 
 
 as inverse functions 78 
 
 82. The value of any function of an inverse function ... 79 
 
 83. Some inverse functions expressed in terms of other inverse 
 
 functions .81 
 
 84. Relations between inverse functions derived from the formulas 
 
 for double angles, half angles, and the addition formulas 82 
 86. Examples 82 
 
 CHAPTER VIII 
 OBLIQUE TRIANGLE 
 
 86. General statement 84 
 
 87. Law of sines 84 
 
 88. Law of tangents 86 
 
 89. Cyclic interchange of letters 85 
 
 90. Law of cosines 85 
 
 91. Sine of half angle in terms of sides of triangle ... 86 
 
 92. Cosine of half angle in terms of sides of triangle ... 87 
 
 93. Tangent of half angle in terms of sides 88 
 
 94. Area of plane triangle in terms of two sides and included 
 
 angle 89 
 
 95. Area of triangle in terms of a side and two adjacent angles . 89 
 
CONTENTS 
 
 96. Area in terms of sides 90 
 
 97. Formulas for solving an oblique triangle .... 90 
 
 98. Check formulas 91 
 
 99. Illustrative problems 92 
 
 100. The ambiguous case 95 
 
 101. Examples 98 
 
 102. MISCELLANEOUS EXERCISES 102 
 
 CHAPTER IX 
 
 DE MOIVRE'S THEOREM WITH APPLICATIONS 
 
 103. Introduction to chapter 113 
 
 104. Geometric representation of a complex number . . . 113 
 
 105. De Moivre's theorem 114 
 
 106. Geometric interpretation 115 
 
 107. Applications of De Moivre's theorem 116 
 
 108. Cube roots of unity 116 
 
 109. Fifth roots of unity . . . ... . .117 
 
 110. Square root of a complex number 118 
 
 111. Any root of a complex number ...... 119 
 
 112. Sin n a and cos n a. expressed in terms of sin a and cos a . 120 
 
 113. Comparison of the values of sin a, a, and tan a, a being 
 
 any acute angle 121 
 
 114. Value of sm a for small values of a 121 
 
 a 
 
 115. Sine and cosine series 122 
 
 116. Examples 124 
 
 SPHERICAL TRIGONOMETRY 
 CHAPTER X 
 
 FUNDAMENTAL FORMULAS 
 
 117. The spherical triangle 127 
 
 118. Law of sines 128 
 
 119. Law of cosines 129 
 
 120. Law of cosines extended 130 
 
 121. Relation between one side and three angles . . . 131 
 
 122. The sine-cosine law 131 
 
CONTENTS 
 
 123. Relation between two sides and three angles . . . 132 
 
 124. Relation between two sides and two angles . . . .132 
 126. Formulas independent of the radius of the sphere . . 133 
 
 CHAPTER XI 
 
 SPHERICAL RIGHT TRIANGLE 
 
 126. Definition of spherical right triangle 134 
 
 127. Formulas for the solution of right triangles .... 134 
 
 128. Direct geometric derivation of formulas .... 134 
 
 129. Sufficiency of formulas 137 
 
 130. Comparison of formulas of plane and spherical right 
 
 triangles 137 
 
 131. Napier's rules 138 
 
 132. Side and angle opposite terminate in same quadrant . . 139 
 
 133. Two sides determine quadrant of a third .... 139 
 
 134. Two parts determine a triangle 139 
 
 135. The quadrant of any required part 140 
 
 136. Check formula 140 
 
 137. Solution of a right triangle 141 
 
 138. Two solutions 141 
 
 139. Examples 143 
 
 140. Quadrantal triangles 143 
 
 141. Isosceles triangles 143 
 
 142. Examples 144 
 
 CHAPTER XII 
 
 OBLIQUE SPHERICAL TRIANGLE 
 
 143. Statement of aim of chapter 145 
 
 GENERAL SOLUTION 
 
 144. Angles determined from the three sides .... 145 
 
 145. Sides found from the three angles 146 
 
 146. Delambre's or Gauss's formulas . . . . . .147 
 
 147. Napier's analogies 147 
 
 148. Formulas collected . . 148 
 
 149. All formulas excepting law of sines determine quadrant . 148 
 
CONTENTS xi 
 
 ART. *AGE 
 
 150. Theorem to determine quadrant ...... 149 
 
 151. Second theorem to determine quadrant .... 149 
 
 152. Third theorem to determine quadrant ..... 150 
 
 153. Illustrative examples ........ 150 
 
 154. Two solutions ......... 153 
 
 155. Area of spherical triangle ....... 154 
 
 156. Examples .......... 154 
 
 SOLUTION WHEN ONLY ONE PART is REQUIRED 
 
 157. Statement of problem ....... .156 
 
 158. From two sides and the included angle, to find any one of 
 
 the remaining parts . . . . . ' . . 156 
 
 169. Two parts required ........ 158 
 
 160. Problems .......... 159 
 
 161. Each unknown part found from two angles and the included 
 
 side .......... 160 
 
 162. Each unknown part found from two sides and an angle 
 
 opposite one of them ....... 161 
 
 163. Each unknown part found from two angles and a side 
 
 opposite one of them ....... 163 
 
 164. The general triangle ........ 164 
 
 ANSWERS ........... 165 
 
PLANE TRIGONOMETRY 
 
 CHAPTER I 
 RECTANGULAR COORDINATES AND ANGLES 
 
 1. Introduction. The word trigonometry is derived from 
 the two Greek words for triangle (r/otytuvov) and measure- 
 ment (/Aerpta). Originally trigonometry was concerned 
 chiefly with the solution of triangles. At present this is 
 but one part of the subject. 
 
 Certain preliminary considerations, concerning directed 
 lines and angles, are neqessary before introducing the fun- 
 damental definitions of trigonometry. 
 
 RECTANGULAR COORDINATES 
 
 2. Directed lines. A positive and a negative direction 
 may be assigned arbitrarily to every line. 
 
 If the direction from A to C is positive, the opposite 
 direction from C to A is negative. 
 
 If we let the order of the let- ^ 
 
 ters indicate the direction in A c 
 
 which a segment is measured, it is evident that AC and CA 
 represent the same segment measured in opposite direc- 
 tions; hence 
 
 AC = - CA or -AC = CA. 
 
 Also, if B be a third point on the line, the segments AB 
 and BA are opposite in sign; likewise the segments BO 
 and CB. Hence 
 
 AB=-BA or -AB = BA, 
 and BC= - CB or - BC= CB. 
 
 1 
 
2 TRIGONOMETRY 
 
 Then for all positions of A, B, and C on a line it follows 
 that 
 
 Thus for the following figures : 
 % K_i_f Ti P Q 
 
 5 
 
 8 
 
 
 12 8 + f 4^ 12 
 
 -12 
 
 A -8 
 
 C 
 
 B 
 
 
 
 4-_6 + (4-l(r)-4 
 
 C - 
 
 B 
 
 
 
 4 
 
 \ A 
 
 X' 
 
 3. Lines of reference. Two directed lines, perpendicular 
 to each other, may be taken as lines of reference or axes. 
 They are usually designated by X'X and F'F and are 
 called the X-axis and the F-axis respectively. 
 
 The X-axis is positive from left 
 to right and negative from right 
 to left. 
 
 The F-axis is positive upward 
 _j_ and negative downward. 
 X The point of intersection of the 
 
 axes is the origin. The origin 
 serves as a convenient starting 
 point from which to measure dis- 
 tances. 
 
 4. Quadrants. The axes produced divide the plane into 
 four parts called quadrants. The quadrants are designated 
 by number. The first quadrant is indicated by XOF, the 
 second by FOX', the third by X'OF', and the fourth by 
 F'OX. 
 
 5. Coordinates of a point. From any given point P x in 
 the plane, draw a line parallel to the F-axis intersecting 
 the X-axis in some point A. 
 
 Then the distance from the origin to the point of inter- 
 section, or OA, is the abscissa of the given point P. 
 
RECTANGULAR COORDINATES AND ANGLES 3 
 
 X'C 
 
 The distance from the 
 point of intersection to 
 the given point, or AP lf 
 is the ordinate of the 
 given point P x . 
 
 The abscissa and the 
 ordinate of the point P 2 
 are OC and CP 2 respec- 
 tively. 
 
 The abscissa and the 
 ordinate of the point P 3 
 are OC and CP 3 respectively. 
 
 The abscissa and the ordinate of the point P 4 are OA and 
 AP 4 respectively. 
 
 6. Signs of coordinates. Abscissas are positive when 
 measured from the origin to the right, and negative when 
 measured from the origin to the left. 
 
 Ordinates are positive when measured upward from the 
 .X-axis and negative when measured downward. 
 
 Thus OA, APv and CP 2 are positive; 0(7, CP 3 , and AP 4 
 are negative. 
 
 7. EXERCISES 
 
 1. Locate the point whose abscissa is 4 and whose ordi- 
 nate is 7. This point is designated (4, 7). 
 
 2. Locate the point whose abscissa is 2 and whose ordi- 
 nate is 5, i.e. the point (2, 5). 
 
 3. Locate the points (- 3, 4), (- 6, - 3), (5, - 1), (0, 4), 
 (- 7, 0), and (- 8, 10). 
 
 4. Locate the points (1J, 3), (- |, 0), (m, n), (a?, y), (a, 0). 
 
 5. What is the locus of the points whose abscissas are 6 ? 
 
 6. What is the locus of the points whose ordinates are 3 ? 
 
 7. What is the locus of the points whose abscissas are 
 twice their ordinates ? 
 
TRIGONOMETRY 
 
 ANGLES 
 
 8. Magnitude of angles. In elementary geometry the 
 angles considered are usually less than two right angles ; 
 
 but in trigonometry it is necessary 
 to introduce angles of any magni- 
 tude, positive or negative. 
 
 To extend the conception of an 
 angle, suppose a line to revolve in 
 a fixed plane about a fixed point 0, 
 from the initial position OX to the 
 successive positions OY } OX', and 
 OP, generating a*, an angle greater 
 than two right angles. 
 If the line continues to revolve, 
 making more than one complete rev- 
 olution, it generates an angle a which 
 is greater than four right angles. 
 Evidently by continuing the rotation 
 an angle of any magnitude may be 
 generated. Thus the size of the angle a depends upon the 
 amount of rotation of OP and is designated by an arc. 
 
 9. Direction of rotation. Positive and negative angles. As 
 a positive and negative direction may be assigned arbitra- 
 rily to a line, so a positive and nega- 
 tive sense of generation may be assigned 
 arbitrarily to an angle. 
 
 The direction of rotation indicated 
 by the arrows in the figures of Art. 8 
 is the positive direction ; the opposite 
 direction, indicated by the arrow in the adjoining figure, is 
 the negative direction. 
 
 * Angles will usually be designated by Greek letters : 
 
 a ... Alpha 5 ... Delta 
 
 /3 . . . Beta 6 . . . Theta 
 
 7 ... Gamma ... Phi 
 
RECTANGULAR COORDINATES AND ANGLES 5 
 
 A positive angle is an angle generated by a line rotating 
 in the positive direction. 
 
 A negative angle is an angle generated by a line rotating 
 in the negative direction. 
 
 Negative angles, like positive angles, may have unlimited 
 magnitude. 
 
 10. Initial and terminal lines. The fixed line from which 
 both positive and negative angles are measured is the initial 
 line. It usually coincides with that part of the X-axis 
 lying to the right of the origin. Thus, in the last three 
 figures, OX is the initial line. 
 
 The final position of the revolving line, marking the ter- 
 mination of the angle, is the terminal line. Thus, in the 
 last three figures, OP is the terminal line. 
 
 Two or more unequal angles may P N 
 have the same terminal line. Thus 
 the positive angle a and the nega- 
 tive angles ft and y have the same 
 terminal line. 
 
 Angles having the same terminal 
 line, and the same initial line, are called coterminal angles. 
 
 11. Sign of terminal line. The terminal line, drawn from 
 
 Q f the origin in the direction of the ex- 
 
 / tremity of the measuring arc, is posi- 
 
 / tive ; the terminal line produced, 
 
 drawn from the origin in the oppo- 
 X site direction, is negative. 
 
 Thus the terminal line OP, of the 
 , angle a, is positive, and the terminal 
 
 line produced, OQ, is negative. As 
 the terminal line OP revolves it retains its positive sign. 
 
 12. The algebraic sum of two angles. To construct the 
 algebraic sum of two angles, conceive OP to rotate from the 
 
6 
 
 TRIGONOMETRY 
 
 position OX } through the angle a, to OP'; then from this 
 
 P f 
 
 X 
 
 a positive 
 R positive 
 
 O X 
 
 a positive 
 R negative. 
 
 a negative 
 positive 
 
 position let it rotate through the angle fi, whether positive 
 or negative, to OP. Then XOP is the desired angle a + ft. 
 
 13. Measurement of angles. Various units may be em- 
 ployed in the measurement of angles. In elementary geom- 
 etry the right angle is frequently used. Two other units in 
 common use are the degree and the radian. The degree 
 is generally used in practical problems involving numerical 
 computations, while the radian is essential in many theoret- 
 ical considerations. 
 
 The degree is defined as one ninetieth of a right angle. 
 The degree is divided into sixty equal parts called min- 
 utes. The minute is divided into sixty equal parts called 
 seconds. 
 
 Then 60 seconds (60") = 1 minute. 
 
 60 minutes (60') = 1 degree. 
 90 degrees (90) = 1 right angle. 
 
 The angle 26 degrees, 39 minutes, and 57 seconds is 
 written 26 39' 57". 
 
RECTANGULAR COORDINATES AND ANGLES 7 
 
 14. Circular or radian measure. A radian is an angle of 
 such magnitude that, if placed with its vertex at the center 
 of any circle, it will intercept an arc equal 
 
 in length to the radius of the circle. 
 
 Thus, if the arc XP is equal to the 
 radius OX, the angle XOP is a radian, 
 or Z. XOP = l r , r being used to designate 
 radian. 
 
 15. Value of the radian. Since, in the same circle or in 
 equal circles, angles at the center are proportional to their 
 intercepted arcs, it follows that 
 
 /.XOP 
 ZXOY 
 
 one radian 
 
 or 
 
 one right angle i (2 irr) 
 
 2 
 
 Therefore, one radian = one right angle. 
 
 7T 
 
 or TT radians = 180. 
 
 It is clear that the value of the radian is independent of 
 the radius, depending only upon the constant TT and the 
 right angle, and hence is an invariable unit. 
 
 16. Relation between degree and radian. From the pre- 
 ceding article it follows that 
 
 i-=^, a) 
 
 ' 
 
 Also, from equation (1), 
 
 r jf_ (3) 
 
 180' 
 or 1 = 0.01745''. (4) 
 
 Equations (1) and (2) are used to convert radians into 
 
8 TRIGONOMETRY 
 
 degrees, and equations (3) and (4) are used to convert 
 degrees into radians. Thus, 
 
 fromd), 
 
 o Y TT 
 
 from (2), 4 r = 4 (57 17' 44") = 229 10' 56", 
 
 fro m( 3), 2 0= 
 
 from (4), 3 = 3 (0.01745') = 0.05235 r . 
 
 Using equation (3) to convert degrees into radians intro- 
 duces TT into the numerical value of the angle. Thus TT 
 becomes associated with radian measure. Since the radian 
 is the angular unit with which TT is commonly associated, 
 no ambiguity arises by omitting to state with each angle, 
 expressed in terms of TT, that the radian is the unit. 
 
 Thus, 90 = - radians, 180 = TT radians, 29 = radians 
 are usually written 
 
 90 = -, 180 = TT, 29 = 
 
 2 loO 
 
 It must be especially noted that when no unit is specified 
 the radian is always understood. The constant TT is always 
 equal to 3.1416, and can never equal 180, but TT radians 
 are equal to 180. 
 
 17. Relation between angle, radius, and arc. It is evident 
 from the definition of a radian that if 
 any arc of a circle AB be divided by 
 
 ^^ the radius, the quotient indicates the 
 
 number of radians contained in the 
 central angle subtended by the given 
 arc, hence 
 
 radius 
 where angle AOB is expressed in radians. Representing 
 
RECTANGULAR COORDINATES AND ANGLES 9 
 
 the length of the arc AB by a, the radius by r, and the 
 angle AOB by 0, the relation above may be written 
 
 or 
 
 a=r9 (1) 
 
 It should be especially noted that the angle is expressed 
 in terms of radians. 
 
 In the above figure 9 is approximately equal to 21 radians. 
 
 Equation (1) expresses the relation between a, r, and 0, 
 and determines any one of these elements when the other 
 two are known. 
 
 PROBLEM. What is the length of the arc subtended by a 
 central angle of 112 in a circle whose radius is 15 feet ? 
 
 Solution. Reducing the angle to radians it is seen that 
 
 112 =?^- r = 1.95'". 
 45 
 
 Hence = 1.95. 
 
 Substituting in Equation (1) we have 
 
 a = 15 x 1.95 = 29.3 
 Hence the length of the arc is 29.3 ft. 
 
 18. Linear and angular velocity. Equation (1) of Art. 17 
 leads directly to the relation between linear velocity and 
 angular velocity in uniform motion in a circle. Suppose a 
 point P moves along the circumference of a circle at a 
 constant velocity v, describing an arc a in time t ; then a/t 
 is called the linear velocity and is represented by v. During 
 the same time t, the angle is generated ; and 6/t is called 
 the angular velocity, and is represented by the Greek letter 
 o> (omega). 
 
10 TRIGONOMETRY 
 
 Dividing Equation (1), Art. 17, by t gives 
 = ^or^ = r> 
 
 t t 
 
 i.e. the linear velocity is equal to r times the angular velocity. 
 
 While in general angular velocity may be expressed in 
 any units whatsoever, in the equation v = rco the angular 
 velocity must be expressed in radians per unit of time. 
 
 PROBLEM 1. If a point moves 26 feet in the arc of a 
 circle of radius 7 feet in 3 seconds, what is its angular 
 velocity ? 
 
 Solution. The linear velocity of the point is %- feet per 
 second. 
 
 Substituting in equation (1) we have 
 
 = 7 a, 
 .-. w = |f = 1.238 radians per second. 
 
 PROBLEM 2. A flywheel makes 200 revolutions per 
 minute. Show that its angular velocity is 72,000 per 
 
 minute or radians per second. 
 
 If the radius of the flywheel is 3 feet, show that the 
 velocity of a point on the rim is 42.84 miles per hour. 
 
 19. EXAMPLES 
 
 1. Construct the following angles : 30, 45, 60, 135, 
 300, - 60, - 90, - 390, - 420. 
 
 2. Construct approximately the following angles : 2 
 radians, 3|- radians, ^ radian, 4 radians, 9 radians. 
 
 3. Construct the following angles : 
 
 7T 7T 7T 5 7T 5 TT 
 
 2' ~3' 4' 1 ~T~' T' 
 
 4. Eeduce the following angles to radian or circular 
 measure : 10, 30, 45, 60, 135, 225, - 270, - 12, 
 - 18, 24 15', - 612 19' 25". 
 
RECTANGULAR COORDINATES AND ANGLES 11 
 
 5. Reduce to degree measure : 2 radians, 5 radians, 
 3 radians, ^ radian, ^ radian, b radians. 
 
 6. Reduce to degrees : 
 
 TT 3_7r _5jr 3.1416 2 TT + 1 2 
 
 3' 4 ' " 3 ' 7T "' TT' 2 ' 7T + 3* 
 
 7. If an arc of 30 ft. subtends an angle of 4 radians, 
 find the radius of the circle. 
 
 8. In a circle whose radius is 5, the length of an intercepted 
 arc is 12. Find the angle (a) in radians, (6) in degrees. 
 
 9. If two angles of a plane triangle are respectively equal 
 to 1 radian and ^ radian, express the third angle in degrees. 
 
 10. In a circle whose radius is 12 ft., find the length of 
 the arc intercepted by a central angle of 16. 
 
 11. Find the angle between the tangents to a circle at 
 two points whose distance apart measured on the arc of the 
 circle is 378 ft., the radius of the circle being 900 ft. 
 
 12. An automobile whose wheels are 34 inches in 
 diameter travels at the rate of 25 miles per hour. How 
 many revolutions per minute does a wheel make ? What is 
 its angular velocity in radians per second ? 
 
 13. Assuming the earth's orbit to be a circle of radius 
 92,000,000 miles, what is the velocity of the earth in its 
 path in miles per second ? 
 
 14. The rotor of a steam turbine is two feet in diameter 
 and makes 25,000 revolutions per minute. The blades of 
 the turbine, situated on the circumference of the rotor, 
 have one-half the velocity of the steam which drives them. 
 What is the velocity of the steam in feet per second ? 
 
 15. A belt travels around two pulleys whose diameters 
 are 3 feet and 10 inches respectively. The larger pulley 
 makes 80 revolutions per minute. Find the angular 
 velocity of the smaller pulley in radians per second, also 
 the speed of the belt in feet per minute. 
 
CHAPTER II 
 
 TRIGONOMETRIC FUNCTIONS 
 
 20. The trigonometric functions, upon which trigonom- 
 etry is based, are functions of an angle. 
 
 These functions are the sine, cosine, tangent, cotangent, 
 secant, cosecant, versed sine, and coversed sine of an angle. 
 
 For any angle a they are written sin a, cos a, tan a, cot a, 
 sec a, esc a, vers a, and covers a. 
 
 21. Definitions of the trigonometric functions. Let OX 
 and OP be the initial and terminal lines respectively of any 
 angle a. 
 
 Y 
 
 
 
 AX 
 
 X'A 
 
 X A 
 
 X' 
 
 Let P be any point on the terminal line, 
 
 OP or r the distance from the origin to the point P, 
 OA or x the abscissa of the point P, and 
 AP or y the ordinate of the point P. 
 12 
 
TRIGONOMETRIC ^UNCTIONS 13 
 
 It should be noted that OP, OA, and AP are directed 
 
 lines and hence 
 
 r = OP and not PO 
 
 x = OA and not AO 
 y = AP and not PA. 
 
 Then the trigonometric functions are defined as follows : 
 
 sin a = - = 
 
 ordinate 
 
 r distance 
 
 r distance 
 
 Y ordinate 
 tan a = ^ = : 
 x abscissa 
 
 = * = abscissa 
 
 y ordinate 
 = /: = distance 
 
 x abscissa 
 
 A* distance 
 esc a = - = 
 
 / ordinate 
 
 vers a = 1 cos a 
 covers a = 1 sin a 
 
 It will be observed that each of the first six functions is 
 defined as a ratio between two line segments. These are 
 the fundamental trigonometric functions, and the ratios de- 
 fining them are called the trigonometric ratios. 
 
 The first six functions are called trigonometric ratios. 
 The expression trigonometric functions is more general and 
 embraces the versed sine, coversed sine, and the trigono- 
 metric ratios. It is evident, from the definitions, that the 
 values of the trigonometric functions are abstract numbers. 
 
 22. Values of the trigonometric functions of 30, 45, and 
 
 60 6 . A few concrete illustrations serve to show the nature 
 of the trigonometric functions, to fix ideas, and to prepare 
 the way for more general considerations. 
 
14 
 
 TRIGONOMETRY 
 
 Functions of 30. Let OPF be an equilateral triangle 
 having its sides equal to 2 units. Place the triangle with 
 a vertex at the origin so that OX bisects the angle POP. 
 Then, by geometry, the angle A OP =30, the ordinate 
 AP= 1, and the abscissa OA = -\/3. 
 
 Hence, applying definitions, 
 
 sin30 = i=.500 
 P 
 
 cos 30 = =.866 
 
 X' 
 
 o 
 
 V3 jA X 
 
 \j 
 F 
 
 V3 3 
 cot 30 = = V 
 
 1.732 
 
 vers30=l-iV3=.134 sec 30 =-^z = |V3 = 1.155 
 
 covers 30= 1 - - = i = .500 esc 30 = | = 2.000 
 
 2i 2i J- 
 
 PROBLEM. Find the values of the trigonometric functions 
 of 30, as above, taking OP= 1. 
 
 Functions of 45. Let OAP be 
 a right-angled isosceles triangle 
 
 having its sides OA and AP each 
 
 equal to 1. Then Z. AOP = 45 and X ' 
 the distance OP = V2. 
 
 Hence, applying definitions, 
 
 sin 45 = = - V2 
 
 V2 2 
 
 cos 45 = = lV2 
 
 V2 2 
 
 1 
 
 1 A X 
 
 sec 45=-^ = 
 
 esc 45 = ^ = 
 
 vers45 = l-i- 
 
 cot45 = = 
 
 covers 45 
 
 1-V2 
 
TRIGONOMETRIC FUNCTIONS 
 
 15 
 
 PROBLEM. Find the values 
 of the trigonometric functions 
 of 45, taking OA = 3. 
 
 Functions of 60. Let OPF y 
 
 an equilateral triangle having 
 each side equal to 2, be placed 
 as in the figure. Then the 
 abscissa and ordinate of P 
 are 1 and V5, respectively. 
 Hence, applying definitions, 
 
 sin 60 = 
 
 1 A F X 
 
 cos 60 = - 
 
 2 
 
 V3 
 
 2~2 
 cot 60 = -L = | V3 covers 60 = 1 - - V3 
 
 PROBLEM. Find the values of the functions of 60, as 
 above, taking OP= 4 a. 
 
 The values of the sines and cosines of 30, 45, and 60 
 are used frequently and should be memorized. The follow- 
 ing table may be found helpful : 
 
 sin 30 = i VI = cos 60 
 
 2 
 
 23. Values of the trigonometric functions of 120, 135, and 
 
 150. By using the magnitudes of the figures of Art. 22 
 and properly placing them with respect to the axes, the 
 
16 
 
 TRIGONOMETRY 
 
 values of the trigonometric functions of various angles may 
 be obtained. 
 
 Functions of 120. From the figure and definitions it is 
 evident that 
 
 V3 
 
 X' A -1 
 
 120 
 
 sin 120 = -^ 
 
 -: 
 
 2 2 
 
 cos 120 = = -1 
 
 covers 120 = 1 
 
 V3 
 
 cotl20 = -=i = --V3 
 
 = - sec 120 = =-2 
 2 1 
 
 esc 120= 7==? A/3 
 
 Functions of 135. From the figure and definitions it is 
 evident that 
 
 V2 2" P 
 
 1 1 V2 
 
 :=l = _lV2 i V 
 
 V2 2 
 
 
 
 1 X' A -1 
 _^ ~ 
 
 "l"""" 1 Y 
 
 X. 
 
 sec 135 = = - V2 
 esc 135 = -^?= V2 
 
 covers 135 = l- 
 
TRIGONOMETRIC FUNCTIONS 
 
 IT 
 
 Functions of 150. From the figure and definitions it is 
 evident that 
 
 sin 150 = - 
 
 cos 150 = 
 tan 150 = 
 
 cot 150 = 
 sec 150 = 
 
 
 -V3 
 
 -V3 : 
 
 -V3 
 
 X A -Y3 
 
 3 V 
 V3 
 
 ?V3 
 3 
 
 J50" 
 
 csc 150 = = 
 
 covers 150 = l- = 
 
 w w 
 
 PROBLEM. Find the values of the trigonometric functions 
 of 210, 225, 240, 300, 315, and 330. 
 
 24. Signs of the trigonometric ratios. The signs of the 
 trigonometric ratios of any angle depend upon the signs 
 of the ordinate, the abscissa, and the distance of any point 
 on its terminal line. As the terminal line passes from 
 one quadrant to another, there is always a change of 
 sign in either the abscissa or the ordinate of any point 
 on that line, but the distance remains positive. When a 
 coordinate changes its sign, every trigonometric ratio de- 
 pendent upon it must also change its sign. 
 
 The following table is constructed by taking account of 
 the signs of the abscissa x and of the ordinate y, and re- 
 membering that the distance r is always positive. It gives 
 the sign of each trigonometric ratio of an angle terminating 
 in any quadrant. 
 
18 
 
 TRIGONOMETRY 
 
 
 1st Quadrant 
 
 2d Quadrant 
 
 3d Quadrant 
 
 4th Quadrant 
 
 
 -t- 
 
 + . 
 
 
 
 
 
 
 + -"* 
 
 + ~ 
 
 + 
 
 + 
 
 cos a 
 
 = + 
 
 T = 
 
 = 
 
 jH 
 
 tan a 
 
 H 
 
 _ 
 
 3=4- 
 
 T = 
 
 cot a 
 
 |^ 
 
 + ~ 
 
 3= + 
 
 4- _ 
 
 sec a 
 
 i^ 
 
 + _ 
 
 _ 
 
 1^ 
 
 pep /y 
 
 +. 
 
 + . 
 
 + 
 
 + 
 
 
 +- 
 
 Hh 
 
 
 
 
 
 A 2 A 
 
 25. Theorem. For every given angle there is one and only 
 one value of each trigonometric function. 
 
 The theorem is demonstrated 
 for the sine of an angle. The 
 same method is applicable to 
 each of the remaining functions. 
 - Let a be any angle. Eefer- 
 ring to Art. 21, it is clear that 
 if it be possible to obtain two 
 or more values for sin a they 
 must be obtained by taking dif- 
 ferent points on the terminal line. 
 
 Let Pj and P 2 be any two points on the terminal line. 
 Then by definition 
 
 X f 
 
 
 
 But since the right triangles OA^ and OA 2 P 3 are similar, 
 
TRIGONOMETRIC FUNCTIONS 
 
 19 
 
 and have their corresponding sides in the same direction, it 
 follows that 
 
 OP, OP 2 
 
 Hence the value of sin a is independent of the position of 
 the point chosen on the terminal line, but depends solely 
 upon the position of the terminal line, i.e. upon the angle. 
 
 The above theorem may be stated as follows : The trigo- 
 nometric functions are single-valued functions of the angle. 
 
 26. Theorem. Every given value of a trigonometric func- 
 tion determines an unlimited or infinite number of positive, 
 and negative angles, among which there are always two positive 
 angles less than 360. 
 
 The theorem is demonstrated for a given tangent. A 
 similar method is applicable to the remaining functions. 
 
 Let tan a = where m 
 n 
 
 and n are positive numbers. 
 Then 
 
 -f- m m 
 
 tan a 
 
 n 
 
 Locate the point P l5 whose 
 abscissa is n, and whose 
 ordinate is m. This point 
 and the origin determine the terminal line of the angle a 1} 
 
 whose tangent is 
 n 
 
 Likewise the point P 2 is located by using m and n as 
 ordinate and abscissa respectively. Drawing the terminal 
 line OP 2 , a second angle a 2 is found, which also has the 
 given tangent. 
 
 The angles a, and <*2> are evidently less than 360, and 
 
 have the given tangent There is an unlimited num- 
 ber of positive and negative angles coterminal with a, and 
 2, all of which have the given tangent. Hence the theorem. 
 
20 
 
 TRIGONOMETRY 
 
 27. EXAMPLES 
 
 In what quadrants does a terminate when 
 
 1. sin a = i 3. tan a = 5. 5. sin a = . 
 
 L <j 
 
 2. cos a = i. 4. cot a = 8. "^ 6. cos a = 4. 
 
 o 5 
 
 7. sin a is positive and cos is negative. 
 
 8. tan ct is positive and cos a is negative. 
 
 9. cosec a is negative and cos a is negative. 
 
 10. tan a is negative and sin a is positive. 
 
 11. cos a is negative and sin is negative. 
 
 Give the signs of the trigonometric functions of the fol- 
 lowing angles : 
 
 12. 750. 
 
 i*=. 
 
 U 14. 560. 
 
 ^ 15. 5|7T. 
 
 16. -15. 
 
 17. - 
 
 18. -470. 
 10 77r 
 
 "T 
 
 Find the negative angles, numerically less than 360, that 
 are coterminal with the following angles : 
 
 20. ^ 
 
 3 
 
 21. *. 
 
 22. 300. 
 
 23. 
 
 24. 5. 
 
 25. -495 
 
 26. Construct the positive angles, less than 360, for which 
 the sine is equal to J- , and find the values of the other func- 
 tions of both angles. 
 
 SOLUTION. Determine the points 
 whose ordinates are 2 and whose 
 distances are 5, as follows : 
 
 With as center and a radius 5, 
 describe a circle. Through a point 
 on the Y"-axis, 2 units above the 
 origin, draw a line parallel to the 
 X-axis, intersecting the circle in 
 the two required points PI and P 2 . 
 Draw the terminal lines OPi and 
 OP, giving the angles i and #2, f* 
 which we have 
 
TRIGONOMETRIC FUNCTIONS 21 
 
 2 2 
 
 sin a,i = - and sin 2 = - 
 
 5 5 
 
 -V2T 
 
 COS i = - COS 2 = - 
 
 tan 0! = - = 
 
 \/21 21 -V21 21 
 
 5 5 
 
 CSC i = - CSC 2 = ~ 
 
 27. Construct the positive angles, less than 360, for 
 which the cosine is f, and find the values of the other func- 
 tions of both angles. 
 
 Find the values of the functions of all angles less than 
 360 determined by 
 
 ^ 28. tana = -f ^32. sin = -}. 
 
 29. cot a = If 33. sin<* = ^. 
 
 30. esc a = 3J* 34. cos a = A. 
 -31. cota = -5. 
 
 35. Given tan a = - 4, find the value of sm a cos a 
 
 36. Given sec = 6; find the value of 
 
 cot a 
 sin 2 a + cos 2 a 
 
 cos a 
 
 37. Given sin a = .3, find the value of tan a sec a cos a. 
 
 38. Given esc a = 8 and tan ft = 3, find the value of 
 sin a cos ft -f cos a sin ft. 
 
 39. Given tan = i and cos/3 = ^-VS, a terminating 
 in the first quadrant and /3 in the second, show that the angle 
 between the terminal lines of a and (3 is a right angle. 
 
CHAPTER III 
 
 RIGHT TRIANGLES 
 
 \ 
 
 28. In every triangle there are six elements or parts. 
 These are the three sides and the three angles. 
 
 When three elements are given, one of which is a side, 
 the other three elements can be determined. 
 
 The solution of a triangle is the process of determining 
 the unknown parts from the given parts. 
 
 In the present chapter it will be shown how the trigono- 
 metric functions can be employed to solve the right triangle. 
 
 29. Applications of definitions of the trigonometric func- 
 tions to the right triangle. Let ABC be any right tri- 
 angle. Place the triangle in the first quadrant with the 
 vertex of one acute angle coinciding with the origin, and 
 one side, not the hypotenuse, coinciding with the X-axis, as 
 in the figure. 
 
 Y 
 
 B 
 
 X' 
 
 Then, by definitions, 
 
 sin a = - = 
 
 side opposite 
 hypotenuse 
 side adjacent 
 
 cos a = - = 
 
 c hypotenuse 
 
 22 
 
RIGHT TRIANGLES 23 
 
 = g = side opposite 
 b side adjacent 
 
 6 side adjacent 
 cot a = - = 
 
 a side opposite 
 
 == c = hypotenuse 
 6 side adjacent 
 
 a side opposite 
 
 It is seen that the functions of any acute angle of a right 
 triangle are expressed in terms of the side adjacent, the 
 side opposite, and the hypotenuse, without reference to the 
 coordinate axes. 
 
 It follows that 
 
 sin/3 = - cot0 = - 
 
 C 
 
 cos/? = - sec/3 = - 
 
 c a 
 
 30. Trigonometric tables. In the preceding chapter the 
 trigonometric functions of 30, 45, and 60 were calculated. 
 By processes too complicated to introduce here, tables have 
 been computed, giving the values of the trigonometric func- 
 tions of acute angles. These tables generally contain two 
 parts. In one part the values of the functions, called 
 natural functions, are given ; in the other part the logarithms 
 of the trigonometric functions, called logarithmic functions, 
 are given. 
 
 When an angle is given its trigonometric functions can 
 be taken from the tables, and vice versa. The functions of 
 known angles thus become known numbers and can be used 
 in problems of computation. 
 
 Approximate values of the trigonometric functions can be 
 obtained by graphical methods. 
 
24 TRIGONOMETRY 
 
 PROBLEM. Measure the distance, abscissas and ordinates 
 of the points a, 6, c, , j. From these measurements 
 compute to two figures the sine, cosine, and tangent of 0, 
 10, 20, . , 90. By arranging the results in tabular 
 form, a two place table is constructed. 
 
 70 C 
 
 60< 
 
 ,IO e 
 
 This table may be used to solve Examples 1 to 6, Art. 36. 
 On account of its extreme simplicity the use of this table 
 allows the attention to be focused upon the fundamental 
 processes involved in the solution of triangles. 
 
 l n o 
 
RIGHT TRIANGLES 25 
 
 A 
 31. Formulas used in the solution of right triangles. A 
 
 right triangle can always be solved when, in addition to 
 the right angle y, two independent parts are given. The 
 formulas usually employed are : 
 
 c a 2 + 6 2 = c 2 
 
 cos a = - cos 8 = - 
 
 c c 
 
 tona=- tan/? = - 
 
 o a 
 
 When the computations are made without logarithms, 
 the formulas involving the cotangent, secant, and cosecant 
 may sometimes be used advantageously. 
 
 32. Selection of formulas. 
 
 (a) If an angle and a side are given, it is always possible 
 to find the unknown parts directly from the given parts 
 without the use of the formula a 2 -f- 6 2 = c 2 . To find the 
 unknown side, select that formula which contains the given 
 parts and the desired unknown side. The unknown angle 
 can always be found from a. -f /? = 90. 
 
 (6) If two sides are given, the third side would naturally 
 be found by the use of a 2 + b 2 = c 2 , but in practice it is 
 generally preferable first to compute an angle by the use of 
 a formula involving the given sides and an angle. To find 
 the third side, select a formula containing one of the given 
 sides, the angle already computed, and the required side. 
 
 33. Check formulas. In all computations it is necessary 
 constantly to guard against numerical errors. However 
 carefully the computations are made, errors may still occur, 
 and therefore computed parts should be checked by means 
 of check formulas. Any formula which was not used in 
 the solution of the triangle may be used for this purpose. 
 
 For the right triangle the formula 
 
 may conveniently be used, and in general it is sufficient. 
 
26 TRIGONOMETRY 
 
 34. Suggestions on solving a triangle. Make a careful 
 free-hand construction of the required triangle, and write 
 down an estimate of the values of the unknown parts. 
 Large errors will be detected readily, without the use of 
 check formulas, when the computed parts are compared 
 with the estimates. 
 
 Before entering the tables, and before making any com- 
 putations, select all the formulas to be used, solve the 
 formulas for the required parts, and make an outline in 
 which a place is provided for every number to be used in 
 the computation. This will often lessen the actual work, 
 for frequently several required numbers are found on the 
 same page of the table. 
 
 The arrangement of the work is of considerable impor- 
 tance in every extended computation. 
 
 35. Illustrative examples. 
 
 1. In a right triangle, given b = 14, a = 35, to find a, c, 
 and ft. ^ 
 
 SOLUTION. Approximate construction. S^<\ 
 
 Estimate a = 9, c = 17. %,''' \ a 
 
 By natural functions 
 
 a-- cosa = - 
 
 C 
 7) tan flf ' r ft Q0 ct. 
 
 6=14 
 
 Check 
 c 2 = a 2 + & 2 
 a 2 = 96.10 
 6 2 = 196.0 
 
 cos a 
 a = 14 x. 7002 c = 14 -*- .8192 j8 = 56 
 a = 9.803 c = 17.09 
 
 By logarithms 
 
 292.1 
 c 2 = (17.09)2 = 292.1 
 
 a = 6tan a c = = 90 - 
 
 b 
 a 
 
 log 6 
 log tan a 
 
 log a 
 a 
 
 14 log b 
 
 35 log cos a 
 
 logo 
 
 c 
 
c 
 
 6 
 c-b 
 
 c+ b 
 
 RIGHT TRIANGLES 
 
 Check 
 
 6) 
 
 log (c - 6) 
 
 log (c + 6) 
 
 2 log a 
 
 log a 
 
 Filling in the above outline, the completed work appears as follows ; 
 
 b nno 
 
 a = b tan a 
 
 b 
 a 
 log 6 
 log tan a 
 log a 
 a 
 
 14 
 35 
 1.14613 
 9.84523 
 
 -10 
 
 0.99136 
 9.8030 
 
 c = 
 
 cos a 
 
 log b 
 
 1.14613 
 
 0= 55 
 
 log cos a 
 
 9.91336 - 10 
 
 
 logc 
 
 1.23277 
 
 
 c 
 
 17.091 
 
 
 Check 
 
 a i - C 2 _ 52 _ ( C _ 5) ( C + 5^ 
 
 c 
 
 b 
 c-b 
 
 c-f & 
 
 17.091 
 14. 
 
 log(c-&) 
 log(c + &) 
 
 0.49010 
 1.49263 
 
 3.091 
 31.091 
 
 2 log a 
 log a 
 
 1.98273 
 0.99136 
 
 Since the check formula gives the same value for log a as that found 
 in the solution, the computation is in all probability correct. 
 
 2. In a right triangle, given c = 6.275, /3 = 1847 f , to find 
 a, &, and a. 
 
 SOLUTION. Approximate construction. 
 Estimated a = 5, b = 2. 
 
 By natural functions Check 
 
 a = c cos j8 b = c sin c 2 = a 2 + 6 2 
 
 a = 6.275 x .9468 b = 6.275 x .3220 a 2 = 35.30 
 
 a = 5.941 b = 2.021 6 2 = 4.084 
 
 a = 90 - ft 39.384 
 
 a = 71 13' c 2 = (6.275) 2 = 39.38 
 
28 
 
 TRIGONOMETRY 
 
 By logarithms 
 a = c cos /3 
 
 b = csin/3 
 
 a = 90 - /a 
 
 c 
 ft 
 
 logc 
 log cos /3 
 log a 
 a 
 
 6.275 
 18 47' 
 0.79761 
 9.97623 - 10 
 
 logc 
 log sin /3 
 log 6 
 6 
 
 Check 
 2 = (c - 6) 
 
 0.79761 
 9.50784 
 
 a = 71 13' 
 -10 
 
 0.30545 
 2.0205 
 
 (c+6) 
 
 0.77384 
 5.9407 
 
 
 . c 6.275 
 6 2.0205 
 
 log (c - 6) 
 log (c + 6) 
 
 0.62885 
 0.91884 
 
 c - & 4.2545 
 c + 6 8.2955 
 
 2 log a 
 log a 
 
 1.54769 
 0.77384 
 
 3. Given a = .064873, b = .12574, to 
 find a, /?, and c. 
 
 SOLUTION. Approximate construction. 
 Estimate a 30, /3 - 60, c = .15. 
 
 sin a 
 
 
 a 
 
 .064873 
 
 
 6 
 
 .12574 
 
 log a 
 log 6 
 
 8.81206 - 10 
 9.09948 - 10 
 
 log tan a 
 a 
 
 9.71258 - 10 
 27 17' 23" 
 
 
 /3 62 42' 37" 
 
 c 
 
 .14149 
 
 b 
 c-6 
 
 .12574 
 
 .01575 
 
 c + 6 
 
 .26723 
 
 log a 
 
 log sin- a 
 
 logc 
 
 c 
 
 8.81206 - 10 
 9.66133 - 10 
 
 9.15073 - 10 
 .14149 
 
 Check 
 
 log (c - 6) 
 
 log (c + 6) 
 
 2 log a 
 
 log a 
 
 8.19728-10 
 9.42689 - 10 
 
 17.62417 - 20 
 8.81208 - 10 
 

 RIGHT TRIANGLES 
 
 36. EXAMPLES 
 
 Solve the following right triangles, y being the right 
 angle. It is recommended that the first ten problems be 
 solved by the use of a three-place table of natural func- 
 tions, or by means of a slide rule. 
 
 1. a = 6 +A. c = .0091 
 a = 20 a = .0029 
 
 2. c = 2.5 7. 6=371 
 a = 35 a = 43 
 
 3. 6 = .84 8. c = 7.72 
 ft = 75 6 = 6.87 
 
 4. a = 25 9. a=18 
 
 6 = 60 c = .0938 
 
 5. c = 82 10. /?=4930' 
 a = 37 c = 12.47 
 
 The following problems should be solved by the use of 
 a four-place * or a five-place table. 
 
 11. a = 1870 16. c = 12.145 
 
 a = 19 55' a = 9.321 
 
 <**-12. c=.3194 17. 6 = 78.545 
 
 a = 25 41' = 31 41' 6" 
 
 13. 6 = .9292 18. 6=3.4572 
 
 /3 = 32 43' ft = 57 57' 57" 
 
 14. a = .00006 19. c = 20.082 
 6 = .000019 6 = 16.174 
 
 15. c = 1200.7 20. a = 78 0' 3" 
 a = 885.6 a = 271.82 
 
 * Whenever sides are given to four significant figures and angles to 
 minutes. 
 
30 TRIGONOMETRY 
 
 a = 5987.2 j6. /? = 83 15' 6" 
 
 ft= 88 53' 2" c = 7000 
 
 22. c = . 09008 27. a =66 6' 18" 
 a = .07654 c = 8070.6 
 
 23. a = 46 39' 50" 28. a = 978.45 
 a = 26.434 6 = 1067.2 
 
 24. a = 30.008 -^29. a = 5280 
 b = 29.924 6 = 5608 
 
 25. a = 111.45 /A 30 - = 17 26' 34" 
 b = 121.69 c = 46.474 
 
 31. Solve the isosceles triangle, one of the equal sides 
 being 690.13, and one of the base angles being 15 20' 25". 
 
 32. Solve the isosceles triangle whose altitude is 606.6, 
 one of the equal sides being 955.7. 
 
 33. Solve the isosceles triangle whose base is 2558, and 
 whose vertical angle is 104 0' 46". 
 
 34. Solve the isosceles triangle whose base is 161.4, and 
 whose altitude is 204.4. 
 
 35. Find the length of a side of a regular octagon in- 
 scribed in a circle whose radius is 49. 
 
 37. Oblique triangles. When any three independent parts 
 of an oblique triangle are given it can be solved by means 
 of right triangles. The oblique triangle may be divided 
 into two right triangles by drawing a perpendicular from 
 one vertex to the opposite side, or the opposite side pro- 
 duced. 
 
 When one of the given parts is an angle, the perpendicu- 
 lar must be selected so that one of the resulting right 
 triangles will contain two of the given parts. 
 
 In case the three sides are given, a second part of one of 
 
RIGHT TRIANGLES 31 
 
 the right triangles can be obtained by equating the expres- 
 sions for the length of the perpendicular obtained from each 
 right triangle. 
 
 Thus a 2 - a 2 = 6 2 - (c - a) 2 , 
 
 from which a? = 
 
 2c 
 
 38. APPLICATIONS 
 
 1. To find the distance from B to (7, two points on op- 
 posite sides of a river, a line BA, 200 feet long, was laid off 
 at right angles to the line BC, and the angle BAC was 
 measured and found to equal 55 29'. What was the re- 
 quired distance ? 
 
 2. A railway is inclined 4 23 '20" to the horizontal. 
 How many feet does it rise per mile, measured along the 
 horizontal ? 
 
 3. From the top of a tower 120 feet high the angle of 
 depression of an object in the horizontal plane of the base 
 of the tower is 24 27'. How far is the object from the foot 
 of the tower ? How far from the observer ? 
 
 Given any point A and a second point B at a greater elevation 
 than A. 
 
 The angle of elevation of B from A is the 
 angle that the line AB makes with its orthogonal 
 projection upon the horizontal plane through A. 
 _ In the figure it is the angle a. 
 A The angle of depression of A from B is the 
 
 angle that the line BA makes with its orthogonal projection upon the 
 horizontal plane through B. In the figure it is the angle /3. It is clear 
 that a = 0. 
 
 4. Find the length of one side of a regular pentagon 
 inscribed in a circle whose radius is 18.24 feet. 
 
 5. Find the perimeter of a regular polygon of n sides 
 inscribed in a circle whose radius is r. 
 
32 TRIGONOMETRY 
 
 6. Find the perimeter of a regular pentagon circum- 
 scribed about a circle whose radius is 18.24 feet. 
 
 7. Find the perimeter of a regular polygon of n sides 
 circumscribed about a circle whose radius is r. 
 
 8. Find the length of a chord subtending a central angle 
 of 63 14' 20" in a circle whose radius is 124.93 feet. 
 
 9. A straight road, PR, makes an angle of 19 27' 30" 
 with another straight road, PS. Having given PR = 640 
 feet, find US, the perpendicular distance from R to PS. 
 
 /"* I 
 
 ACUAt a certain point the angle of elevation of a moun- 
 tain is 34 28'. At a second point 500 feet farther away, 
 the angle of elevation is 31 12'. Find the height of the 
 mountain above the table-land. 
 
 11. One side of the square base of a right pyramid is 15 
 inches, and the altitude is 20 inches. Find the slant height 
 and the edge. Find the inclination of a face to the base of 
 the pyramid. 
 
 12. How far can you see from an elevation of 2000 feet, 
 assuming the earth to be a sphere with a radius of 3960 
 miles ? 
 
 13. Two forces of 95.75 pounds and 120.25 pounds, at 
 right angles to each other, act at a point ; find the magnitude 
 of their resultant and the angle it makes with the greater 
 force. 
 
 14. Find the velocity of a point, whose latitude is 
 44 30' 20", due to the rotation of the earth. Assume the 
 radius of the earth to be 3960 miles. 
 
 15. Find the length of a belt running around two pulleys 
 whose radii are 12 inches and 4 inches, respectively, the 
 distance between the centers of the pulleys being 6 feet. 
 
 16. A diagonal of a cube and a diagonal of a face of a 
 cube intersect at a vertex. Find the angle between them. 
 
RIGHT TRIANGLES 33 
 
 17. A force of 2000 pounds applied at the origin makes 
 an angle of 33 25' with the positive X-axis. Find its 
 components along the X and Y axes respectively. 
 
 18. A force of 185 pounds applied at the origin makes an 
 angle of 82 12' with the positive X-axis. Another force of 
 327 pounds applied at the origin makes an angle of 11 32' 
 with the positive X-axis. Find the X and Y components 
 of each of these forces. Add the X-components and also 
 the y-components and then find the resultant in magnitude 
 and direction. 
 
 19. A cylindrical tank, whose axis is horizontal, is 8 feet 
 in diameter and 12 feet long. The tank is partly filled with 
 water, so that the depth of the water at the deepest point is 
 3 feet. How many gallons of water are in the tank, there 
 being 7-J- gallons in a cubic foot ? 
 
 20. A man who can paddle his canoe 5 miles per hour in 
 still water paddles at his usual rate directly across a river 
 one-half mile wide. If the river 'flows 4 miles per hour, 
 where will the canoe land and what is its speed in the water ? 
 
 * ' *= 
 
 
 
 
-A &-o 
 CHAPTER IV 
 
 O-^v 
 
 VARIATIONS OF THE TRIGONOMETRIC FUNCTIONS 
 REDUCTION OF FUNCTIONS OF /i90a 
 
 39. In the study of the variations of the trigonometric 
 functions, their values at 0, 90, ISO , 270, and 360 are of 
 special importance. 
 
 Values of functions of 0. The terminal line of coin- 
 cides with the positive X-axis. For a point on this line, at 
 a distance r from the origin, we have 
 
 Y 
 
 
 
 
 x=r, 
 
 y=( 
 
 ) 
 
 
 
 Hence 
 
 
 sin 
 
 = 
 
 
 
 = 
 
 cotO 
 
 _ r *_ 
 
 00 
 
 
 P 
 
 
 r 
 
 
 
 0~ 
 
 
 X' 
 
 X 
 cos 
 
 = 
 
 r 
 
 = 1 
 
 secO 
 
 _ r _ 
 
 1 
 
 
 
 
 r 
 
 
 
 r 
 
 
 
 tan 
 
 = 
 
 
 
 = 
 
 cscO 
 
 _ r _ 
 
 00 
 
 Y 
 
 
 
 r 
 
 
 
 
 
 
 Values of functions of 90. The terminal line of 90 coin- 
 cides with the positive F-axis, therefore 
 
 X- 
 
 = U, y = r. 
 
 Y 
 
 
 Hence 
 
 
 
 P 
 
 sin 90 = - = 1 
 
 cot90 = - = 
 
 
 
 r 
 
 r 
 
 
 
 cos 90 = - = 
 
 r X' 
 
 sec 90 =. - = oo 
 
 
 
 X 
 
 r 
 
 
 
 
 
 tan90 = - = oo 
 
 csc90 = - = l 
 
 i 
 
 
 
 
 r 
 
 Y 
 
 * For the 
 
 interpretation of - see any College 
 
 Algebra. 
 
 
 34 
 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 35 
 
 Values of functions of 180. The terminal line of 180 co 
 incides with the negative X-axis, 
 therefore 
 
 x= r, 
 
 Hence 
 
 sin 180= -- =0 
 r 
 
 cos 180 = = - 1 
 
 X' 
 
 tan 180 = = 
 r 
 
 cot 180 = - = oo 
 
 sec 180 = = -1 
 r 
 
 esc 180= =00 
 
 Values of functions of 270. The terminal line of 270 
 coincides with the negative Y-axis, therefore 
 
 x = 0, y = r 
 Hence 
 
 sin 270 =nr= -1 cot 270 ==0 
 r r 
 
 cos 270= - =0 
 r 
 
 sec 270= =00 
 
 tan 270 = =00 esc 270= =-1 
 r 
 
 The values of the functions of 360 are identical with the 
 values of the functions of 0, since these angles are 
 coterminal. 
 
 40. Variation of the functions. It has been shown that 
 the trigonometric functions are functions of an angle. As 
 the angle varies, the trigonometric functions depending upon 
 the angle also vary. 
 
36 
 
 TRIGONOMETRY 
 
 Let the line OP, of fixed 
 length r, revolve about 
 from the initial position OX. 
 Then the angle XOP or a 
 increases from to 360. 
 The variations of the trigo- 
 nometric functions can be 
 traced by observing the 
 changes in the abscissa and 
 ordinate of the point P. 
 
 a. As the angle a increases from to 90, 
 
 y increases from to r, 
 and x decreases from r to 0. 
 Hence 
 
 sin a or - is positive and increases from to 1 
 
 fm 
 
 cos a or - is positive and decreases from 1 to 
 tan a or - is positive and increases from to oo 
 cot a or - is positive and decreases from co to 
 
 y 
 
 T 
 
 sec a or - is positive and increases from 1 to oo 
 esc a or - is positive and decreases from oo to 1 
 
 y 
 
 As the angle a increases through 90, x passes through 
 zero, changing from a positive number to a negative num- 
 ber. Then, immediately before a becomes 90, cos a or - 
 
 r 
 
 is a very small positive number ; while immediately after 
 a has passed 90, cos a is a very small negative number. 
 This may be expressed by saying that cos a passes through 
 zero and changes sign as a passes through 90. 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 37 
 
 Likewise, immediately before a becomes 90, tan a or 
 
 x 
 
 is a very large positive number ; while immediately after 
 a has passed 90, tan a is a very large negative number. 
 It has been seen that tan 90 = oo . Hence tan a passes 
 through oo and changes sign as a passes through 90. 
 Hence we may say that tan 90 = GO , choosing the posi- 
 tive sign when associating 90 with the first quadrant and 
 the negative sign when associating 90 with the second 
 quadrant. 
 
 Similarly, whenever a trigonometric function passes throiigh 
 oo it changes sign. 
 
 b. As the angle a increases from 90 to 180, 
 
 y decreases from r to 0, 
 and x decreases from to r. 
 Hence 
 
 sin a or ^ is positive and decreases from 1 to 
 r 
 
 cos a or - is negative and decreases from to 1 
 
 tan a or ^ is negative and increases from oo to 
 x 
 
 cot a or - is negative and decreases from to GO 
 
 y 
 
 sec a or - is negative and increases from oo to 1 
 x 
 
 esc a or is positive and increases from 1 to + oo 
 
 y 
 
 c. As the angle a increases through 180, y passes through 
 zero, changing from a positive number to a negative number. 
 As the angle a increases from 180 to 270, 
 
 y decreases from to r, 
 and a increases from r to 0. 
 Hence 
 
 sin a or ^ is negative and decreases from to 1 
 
 T 
 
38 TRIGONOMETRY 
 
 cos a or - is negative and increases from 1 to 
 
 T 
 
 tan a or - is positive and increases from to -f oo 
 
 cot a or - is positive and decreases from + oo to 
 
 sec a or - is negative and decreases from 1 to oo 
 x 
 
 esc a or - is negative and increases from oo to 1 
 
 d. As the angle a increases through 270, x passes through 
 zero, changing from a negative number to a positive 
 number. 
 
 As the angle a increases from 270 to 360, 
 
 y increases from r to 0, 
 and x increases from to r. 
 Hence 
 
 sin a or - is negative and increases from 1 to 
 
 flf* 
 
 cos a or - is positive and increases from to 1 
 tan or - is negative and increases from oo to 
 
 cot a or - is negative and decreases from to oo 
 sec a or - is positive and decreases from + oo to 1 
 
 7* 
 
 esc a or - is negative and decreases from 1 to oo 
 The above results are presented in tabular form. 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 39 
 
 ( 
 
 1st QUADRANT | 
 
 P 9C 
 
 2d QUADRANT 
 18 
 
 8d QUADRANT 
 
 3 27 
 
 4th QUADRANT 
 360 
 
 sin =- 
 
 + inc. -f 1 
 
 + 1 dec. + 
 
 - dec. - 1 
 
 - 1 inc. - 
 
 X 
 
 cos a=- 
 r 
 
 + 1 dec. + 
 
 -Odec. - 1 
 
 - 1 inc. - 
 
 + inc. + 1 
 
 V 
 
 tan a=- 
 
 X 
 
 -f inc. +00 
 
 oo inc. 
 
 + inc. +00 
 
 ao inc. 
 
 cot=* 
 
 + 00 dec. + 
 
 dec. oo 
 
 +00 dec. + 
 
 - dec. -oo 
 
 sec =- 
 
 X 
 
 + 1 inc. +00 
 
 oo inc. 1 
 
 1 dec. oo 
 
 + 00 dec. + 1 
 
 esc a - 
 
 y 
 
 + oo dec. + 1 
 
 + 1 inc. +00 
 
 oo inc. 1 
 
 1 dec. oo 
 
 It is thus seen that the sine and cosine can never be 
 greater than -+- 1 nor less than 1, while the secant and 
 cosecant have no values between -f 1 and 1, but have 
 values ranging from +- 1 to +00 and from 1 to oo . 
 
 The tangent and cotangent may 
 have any value from +- oo to oo . 
 
 41, Graphical representation. A 
 graphical representation of the trigo- 
 nometric functions is effected by first 
 locating points using the different 
 values of the angle as abscissas and 
 the corresponding function-values as 
 ordinates, and then drawing a smooth 
 curve through these points taken in 
 the order of increasing angles. 
 
 The values of the functions of 
 the angles previously calculated are 
 sufficient to determine an approxi- 
 mate graph. For greater accuracy 
 the values of the functions may be 
 taken from the table of natural 
 functions. 
 
 a 
 
 since 
 
 a 
 
 tan 
 
 
 
 
 
 
 
 
 
 7T 
 
 1 
 
 7T 
 
 1 
 
 6~ 
 
 2 
 
 6 
 
 gV3 
 
 it 
 
 -A/~ 
 
 IT 
 
 1 
 
 4 
 
 2 
 
 4 
 
 
 7T 
 3" 
 
 |V3 
 
 7T 
 3 
 
 V3' 
 
 7T 
 
 1 
 
 IT 
 
 00 
 
 2 
 
 
 2 
 
 
 27T 
 
 1 _ 
 
 2* 
 
 V3 
 
 3 
 
 2 
 
 3 
 
 
 37T 
 
 1 - 
 
 Sir 
 
 -1 
 
 4 
 
 2 
 
 4 
 
 
 67T 
 
 1 
 
 STT 
 
 V3 
 
 6 
 
 2 
 
 6 
 
 3 
 
 7T 
 
 
 
 IT 
 
 
 
 ?7T 
 
 1 
 
 77T 
 
 1 
 
 6 
 
 2 
 
 "6~ 
 
 gV3 
 
 etc. 
 
 etc. 
 
40 
 
 TRIGONOMETRY 
 
 Sine Curve y = sin x 
 
 Tangent Curve ytanx 
 
 The sine and tangent curves illustrate the truth of the 
 theorems of Arts. 25 and 26. 
 
 o 
 
 Thus for any given value of the angle, as ^, there is but 
 
 4 
 
 one value of each function, the curves showing the sine and 
 tangent to be 1 V2 and 1 respectively. 
 
 But for any given value of a trigonometric function, as 
 sin a, = ^, there are an unlimited number of angles, as 
 
 , Z, 2J, 2f, etc. . 
 
 Also for tan a = V^we see that a may have the values 
 I HTT, 21 TT, 31 TT, etc. 
 
 42. Periodicity of the trigonometric functions. From the 
 sine curve it is readily seen that the sine is when the 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 41 
 
 angle is 0, and that it increases to 1 as the angle increases to 
 
 ?-. At this point the sine begins to decrease and has the 
 
 2 
 
 value when the angle is TT, and finally reaches the value 
 
 o 
 
 1 when the angle is - ; then the sine again begins to in- 
 
 2i 
 
 crease and has the value when the angle has the value 27r. 
 
 When the angle increases from 2 TT to 4 TT, the sine repeats 
 its values of the interval from to 2?r. If the angle were 
 increased indefinitely, the sine would repeat its values for 
 each interval of 2 ?r. For this reason the sine is called a 
 periodic function, and 2 TT is its period. 
 
 A study of the tangent curve shows that the tangent has 
 the same values between and TT that it has between ?r and 
 27T or between 2?r and 3?r. Hence the tangent is also a 
 periodic function having the period- TT. 
 
 43. EXAMPLES 
 
 1. Plot y = cos x and give the period of the cosine. 
 
 2. Plot y = cot x and give the period of the cotangent. 
 
 3. Plot y = sec x and give the period of the secant. 
 
 4. Plot y = esc x and give the period of the cosecant. 
 
 5. Plot y = sin x + cos x. 
 
 6. Plot y = x + sin x. 
 
TRIGONOMETRY 
 
 REDUCTION OF FUNCTIONS OF n 90 a 
 
 44. The formulas to be developed in this section enable 
 us to express any function of any angle in terms of a func- 
 tion of an angle differing from the given angle by any 
 multiple of 90. They may be used to express any func- 
 tion of any angle in terms of a function of an angle less 
 than 90 or less than 45. 
 
 45. Functions of a in terms of functions of a(n = 0). 
 Let XOP be any positive angle and XOP' a numerically 
 equal negative angle. 
 
 Then taking OP' = OP we have, for each figure, 
 x f = x, y' = - y, r' = r, 
 
 where #, #, r and #', y', r' are associated with P and P f 
 respectively. 
 
 Then in each quadrant 
 
 sn a = ~ f = _ _ sn a 
 
 x f x 
 cos ()= = - = cos a 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 43 
 
 tan ( a) = ^- = ^ = tan a 
 x x 
 
 X f X 
 
 cot () = = = cot a 
 
 y' -y 
 
 r' r 
 
 sec ( a) - sec a 
 x' x 
 
 esc ( a) = = -^- = esc a 
 2/' -2/ 
 
 46. Functions of 90 + a in terms of functions of a(n = I), 
 Let XOP be any positive angle a and XOP' the angle 
 90 + a. 
 
 Then taking OP' = OP we have, for each figure, 
 
 where a?, ?/, r and #', #', r' are associated with P and P r 
 respectively. 
 
 X' 
 
 x 
 
 Similar figures can be constructed for the other quadrants. 
 Then in each quadrant 
 
 sin (90 + ) = ' = - = cos a 
 
 - 
 r' 
 
 = - snce 
 
 tan (90 + a) = = = - cot 
 v -y 
 
44 
 
 TRIGONOMETRY 
 
 cot (90 + )=- = ^ = - tan a 
 
 y' 
 
 sec (90 + )== = = - esc 
 a; y 
 
 esc (90 + )= = - = sec a 
 ' 
 
 47. Functions of 90 a in terms of functions of a (n = 1). 
 Let XOP be any positive angle and XOP' the angle 
 90 - a. 
 
 Then taking OP' = OP we have, for each figure, 
 
 x = 
 
 y 
 
 x, r f = r. 
 
 Y 
 
 Y 
 
 
 
 
 D' 
 
 
 
 7 
 
 y P 
 
 
 
 && 
 
 >. . v 
 
 T> 
 
 .X' 
 
 x ' x x X' J 
 
 aj^V 
 
 / Y 
 
 
 y \/< 
 
 ^ 
 
 $* 
 
 
 p' r y 
 
 / 
 
 
 Y 
 
 \ 
 
 / 
 
 D ' 
 
 Y 
 
 
 Similar figures can be constructed for the other quadrants. 
 Then in each quadrant 
 
 sin (90 - a) = = - = cos a 
 
 cos (90 -) = - = % = sin a 
 r' r 
 
 tan(90-<x) = ^ = - 
 x' y 
 
 cot (90 - a) = - = ^ = tan a 
 
 y % 
 
 sec (90 - a) = = - = esc a 
 x' y 
 
 esc (90 a) = - = - = sec a 
 
 y 1 ^ 
 
VARIATIONS OF TRIGONOMETRIC FUNCTIONS 45 
 
 48. Functions of 180 a in terms of functions of a(n=2). 
 Let XOP be any positive angle and XOP' the angle 
 180 - a. 
 
 Then taking OP' = OP we have, for each figure, 
 
 ' = -, y'=y, r' = r. 
 
 P'Y 
 
 P 
 
 Y 
 
 
 IY 
 
 r 
 
 
 
 
 \i 
 
 Bg. 
 
 2/ 
 
 
 
 \ 
 
 fij 
 
 
 a?' X5 
 
 "^ a? 
 
 X ** vj 
 
 x X ' 2/' 
 
 
 ^S-J^ x 
 
 
 P 
 
 p 
 
 r 
 
 
 | 
 
 
 Y 
 
 
 Y 
 
 
 Similar figures can be constructed for the other quadrants. 
 Then in each quadrant 
 
 sin (180 - ) = ^ = ^ = sin 
 
 T T 
 
 cos (180 - a) = -,= = - cosa 
 
 tan (180 -) = ^ = -^- = - tana 
 
 x' x 
 
 cot (180 - )= - = -^^ = - cot a 
 2/' y 
 
 sec (180 -) = - = -^- = - sec a 
 x' x 
 
 esc (180 - a) = - = -= csca 
 
 y 1 y 
 
 49. Laws of reduction. The method of the last four 
 articles may be applied to any angle of the form n 90 a 
 to obtain formulas of reduction for all positive and negative 
 integral values of w, a being any angle. ' A complete inves- 
 tigation would show the following laws to be true : 
 
I 
 
 46 TRIGONOMETRY 
 
 When n is even, any function of n 90 a is numerically 
 equal to the same function of a. ; when n is odd, any function 
 ofn 90 a is numerically equal to the cofunction of a. 
 
 If the function of n 90 a is positive, a being considered 
 acute, the members of the equation have like signs ; if the 
 function of n 90 a. is negative, a being considered acute, 
 the members of the equation have unlike signs. The results 
 thus obtained are valid for all positive and negative values 
 of a. 
 
 50. EXAMPLES 
 
 Express as functions of a positive angle less than 90 : 
 
 1. sin 130. 
 
 SUGGESTION. 130 = 90 + 40, or 130 = 180 - 60. 
 
 2. cos 170. 5. cos (-20). 
 
 3. tan 110. 6. tan (-80). 
 
 4. cot 160. 7. sin (-120). 
 
 Express as functions of : 
 
 8. sin (810 -0). 12. tan (0 - 180). 
 
 9. tan(360-0). 13. sec (- 180 - 6). 
 
 10. cot (270 + 0). 14. esc (- 630 + 0). 
 
 11. sin (9 - 90). 15. cos (990 - 0). 
 
 Express each of the trigonometric functions of the follow- 
 ing angles as functions of a without using the laws of 
 reduction. 
 
 16. 180 + a. 18. 270 + a. 
 
 17. 270 -a. 19. 360 -a. 
 
CHAPTER V 
 FUNDAMENTAL RELATIONS. LINE VALUES 
 
 51. In the present chapter eight fundamental relations 
 between the trigonometric functions are developed. It is 
 then shown that each trigonometric function may be repre- 
 sented geometrically by a single line, giving the so-called 
 line values. This gives a second view of the trigonometric 
 functions. The line values offer a simple method of demon- 
 strating the fundamental relations, and of developing the 
 properties already derived by means of the trigonometric 
 ratios. In fact the line values might serve as the funda- 
 mental definitions of the trigonometric functions and thus 
 trigonometry could be based upon these values. Inciden- 
 tally the line values suggest the origin of some of the terms 
 used to designate the several trigonometric functions. 
 
 . FUNDAMENTAL RELATIONS 
 
 52. Certain relations of fundamental importance exist 
 between the trigonometric functions of an angle. These 
 will now be developed. 
 
 From definitions, 
 
 sin a = -, 
 r 
 
 csca=-. .-. sina = - . (1) 
 
 y csc a 
 
 x 
 cos a = -. 
 
 seca=- .-. cosa= - (2) 
 
 x sec a 
 
 47 
 
48 TRIGONOMETRY 
 
 tan a = ^ , 
 x 
 
 cot a = -. .-. tan a = 
 
 cot a 
 
 sin a = -j 
 
 cos a 
 
 tan a = . .-. = tan a. (4) 
 
 x cos a 
 
 cota=-, and - cot a. (5) 
 
 sin a 
 
 From geometry, 2/ 2 + x 2 = r 2 . 
 
 Dividing successively by r 2 , # 2 , and y z , 
 
 2? + 1 2 = 1, .-. sin 2 a + cos 2 a = 1. (6) 
 
 + 1=^, ... tan 2 a + 1 = sec 2 a. (7) 
 
 a 2 x 2 
 
 ^ + 1=^, /. cot 2 a + 1 = esc 2 a. (8) 
 
 These eight formulas are called the fundamental relations. 
 They are used frequently, and should be memorized. In 
 doing this, it is well to have the method of derivation 
 clearly in mind. 
 
 53. The use of exponents. In affecting trigonometric 
 functions with exponents, the exponent is usually placed 
 immediately after the function. Thus, in the formulas of 
 the preceding article, sin 2 a is identical in meaning with 
 (sin a) 2 , and tan 2 a is identical' with (tan a) 2 . 
 
 An exception to the above is made for the exponent 1, 
 in which case it is always necessary to make use of paren- 
 
FUNDAMENTAL RELATIONS 49 
 
 theses. Thus, (cos a)" 1 is used to express , while 
 
 cos a 
 
 cos" 1 a has an entirely different meaning, as will be ex- 
 plained later. 
 
 54. Trigonometric identities. An identical equation, or 
 an identity, is an equation which is satisfied for all values 
 of the unknown quantities. The eight fundamental rela- 
 tions are trigonometric identities. There are many other 
 identities depending upon these. 
 
 The truth of an identity can be established in two ways : 
 
 First. Begin with one of the fundamental relations, and 
 produce the given identity by means of the fundamental 
 relations and algebraic principles. 
 
 Second. Begin with the given identity and transform it 
 to one of the fundamental relations, or reduce one member 
 of the equation to the other by means of the fundamental 
 relations and algebraic principles. 
 
 The choice of the trigonometric transformations neces- 
 sary to effect the reductions is often suggested by the 
 functions involved in the given problem. When no trans- 
 formation is thus suggested, the problem may generally be 
 simplified by expressing each of the functions in terms 
 of sines and cosines, and making use of the relation 
 sin 2 a 4- cos 2 a = 1. 
 
 Avoid the use of radicals whenever possible. 
 
 55. Trigonometric equations. A conditional equation is 
 an equation which is not satisfied for all values of the 
 unknown quantity, but is satisfied only for particular values. 
 
 (a) Trigonometric equations involving different trigono- 
 metric functions of the same angle may often be solved by 
 simplifying the equation by the use of the fundamental 
 relations. Thus s i n a _ CO s a 
 
 may be written in a = 1 or tan a = 1 
 
 cos a 
 
 /.a = 45 or 225. 
 
50 TRIGONOMETRY 
 
 (5) Some equations may conveniently be solved by trans- 
 posing all of the terms to the first member of the equation, 
 factoring, and then placing each factor equal to zero. Thus 
 
 2 sin 2 a + V3 cos a = 2 V3 sin a cos a -f sin a 
 may be written 
 
 sin a (2 sin a 1) -f- V3 cos a (1 2 sin a) = 
 or 
 
 (sin a V3 cos a) (2 sin a 1) = 0. 
 
 This equation is satisfied if either 
 
 sin a V3 cos a = or 2 sin a 1 = 
 whence tan=V3 or sina = i. 
 
 /.a = 60, 240 or a =30, 150. 
 
 (c) When no other method suggests itself, the equation 
 may be transformed, by the use of the fundamental rela- 
 tions, into an equation containing only one function. The 
 equation thus obtained may be solved algebraically for the 
 function involved, from which the values of the angle may 
 
 be obtained. Thus 
 
 2 
 
 10 cos 3 a tan a 9 cos 2 a 10 sin a + 9 = 
 
 esc a 
 
 10 sin (1 - sin 2 a)- 9(1 - sin 2 a) 12 sin + 9 = 
 10 sin a 10 sin 3 a + 9 sin 2 a 12 sin a = 
 .-. sin a = or 10 sin 2 a 9 sin a -f 2 = 
 from which sin a = 0, f , or . 
 
 .-. a = 0, 23 35', 30, 150, 156 25', or 180. 
 A trigonometric equation is considered completely solved 
 when every positive angle less than 360 which satisfies it 
 has been determined. All other angles coterminal with 
 these angles also satisfy the equation. 
 
 In the solution of trigonometric equations extraneous 
 roots may occur. These may be detected by substitution in 
 the original equation. 
 
FUNDAMENTAL RELATIONS 51 
 
 56. EXAMPLES 
 
 Prove the following trigonometric identities : 
 
 1. sin 6 = cos tan 0. co t A 
 
 3. cos 6 = 
 
 2. tan = sin sec 0. esc 
 
 4. (1 sin #)(! + sin x) = cos 2 x. 
 
 5. (sec x tan #)(sec x + tan a?) = 1. 
 
 6. (sin x -f- cos #) 2 = 2 sin a; cos cc -}- 1. 
 
 7. 
 
 cot a; 
 
 8. cos 2 a tan 2 + sin 2 a. cot 2 = 1. 
 
 9. Given cos a. = -| , a being in the fourth quadrant, find 
 the values of the 1 remaining trigonometric functions. 
 
 SOLUTION. Since sin 2 a + cos 2 a = 1, 
 
 sin 2 a = 1 - cos 2 a =1 - |f = |f. 
 .*. sin a = \/|f = $\/l4. 
 sin a 
 
 Also tan a = 
 
 cot a = 
 
 esc a = 
 
 sin a fv/14 2\/14 
 
 Compare with the method of examples 26 and 27, Art. 27. 
 
 10. Given tan y 4, y being in the third quadrant, find 
 the values of the remaining trigonometric functions. 
 SOLUTION. Since sec 2 y = I -f tan 2 y, 
 
 sec y = Vl -f 16 = \/17. 
 
 Also cot y = = - ; 
 
 tan y 4 
 
 esc 2 y = 1 + cot 2 y = 1 + ^ = f ; 
 esc y = 
 

 52 I/ TRIGONOMETRY 
 
 sin 3, = -!- = 1 = -AVl7; 
 
 cos y = = I = - T V VI7. 
 secy _vi 7 
 
 11. Given sin = -|, terminating in the second quadrant, 
 find the values of the remaining trigonometric functions. 
 
 12. Given cot = T 7 ^, terminating in the first quadrant, 
 find the values of the remaining trigonometric functions. 
 
 13. Given sec = 2, being in the third quadrant, 
 find the values of the remaining trigonometric functions. 
 
 14. Given tan a = -, find the remaining functions of- 
 the angles less than 360 which satisfy the equation. 
 
 Solve the following trigonometric equations for angles 
 less than 90: 
 
 15. 5 sin x + 8 = 3(4 sin x). 
 
 16. sinw cos w = 0. 
 
 17. 2 cos 2 x 3 cos x 4- 1 = 0. 
 
 18. (tan 6 - l)(tan - V3) = 0. 
 
 19. Solve sin x = cot x for sin x. 
 
 Some of the following examples are identities and others 
 are equations. Establish the identities and solve the equa- 
 tions for angles less than 180. 
 
 20. sin 4 x = 1 2 cos 2 x + cos 4 x. 
 
 21. sin 4 x = 2 6 cos 2 x + cos 4 x. 
 
 22. (V + l)L+ = sin 
 
 tan 6 
 
 23 2 Bin! + costf VS 
 
 tan $ cot $ 2 
 
 24. tan u + cot u = sec u esc u. 
 
 25. 2 sin 2 y esc ?/ + 3 esc y = 1. 
 
FUNDAMENTAL RELATIONS 53 
 
 Prove the following trigonometric identities : 
 
 26. tan* + cot 2 = sec 2 esc 2 6-2. . 
 
 27. sec 2 ft + cos 2 (3 = tan 2 ft sin 2 ft-}- 2. 
 
 28. esc 2 y sec 2 y = cos 2 y esc 2 y sin 2 y sec 2 y. 
 
 rtrt . sin a 4- tan a 
 
 29. sin a tan ce = 
 
 cot + csc <* 
 
 30. cot 2 a cos 2 a = cot 2 a cos 2 a. 
 
 31. 
 
 sin cos 
 
 32. (sin 4- cos 0) 2 4- (sin - cos 0) 2 = 2. 
 
 33. sin 2 x 4- vers 2 a? = 2(1 cos x). 
 
 34. sec 2 a csc 2 a _(l~tan 2 a) 2 = 4 
 
 tan 2 a 
 
 Solve the following trigonometric equations for angles 
 less than 180: 
 
 35. 2 cos 2 a 3 sin a = 0. 
 
 36. sec 2 a 4- cot 2 a = 4, 
 
 37. 1 4- tan 2 a; 4 cos 2 a; = 0. 
 
 38. tan x 4- cot x = 2. 
 
 39. 2 sin 2 x 4- 3 cos x == 0. 
 
 40. V3 cos x 4- sin # = V2. 
 
 41. csc 2 a: 4 sin 2 a; = 0. 
 
 42. cot aj 4- csc 2 x = 3. 
 
 LINE VALUES 
 
 57. Representation of the trigonometric functions by lines. 
 
 The trigonometric functions of an angle have been studied 
 solely from the standpoint of ratios ; but each function can 
 also be represented in magnitude and sign by a single line. 
 
54 
 
 TRIGONOMETRY 
 
 Let LOP be any angle. Take OP=l. Draw the ordi- 
 nate and abscissa of the point P. About as the center 
 construct a circle with OP as a radius. Draw the tangents 
 to this circle at L and F, the beginning of the first and second 
 quadrants respectively, and produce them to intersect the 
 terminal line, or the terminal line produced, in M and G. 
 
 Then 
 
 y AP AP 
 
 sin = - = = 
 
 OP 
 
 AP 
 
 x OA OA KA 
 
 cos a = - = = = OA 
 
 r OP 1 
 
 x OA OL 
 = x = OA = FG ==FG 
 
 OP 
 
 x OA OL 
 
FUNDAMENTAL RELATIONS 55 
 
 or 
 
 Thus the trigonometric functions are represented by the 
 segments AP, OA, LM, FG, OM and OG. It is evident 
 that these segments represent the trigonometric functions 
 in magnitude. They also represent the trigonometric func- 
 tions in sign. In accordance with the conventions of Arts. 
 6 and 11, the segments AP and LM are positive when drawn 
 upward from the X-axis and negative when drawn down- 
 ward ; the segments OA and FG are positive when drawn 
 from the F-axis to the right and negative when drawn to 
 the left; the segments OM and OG are positive when they 
 coincide with the terminal line of the angle and negative 
 when they coincide with the terminal line produced. 
 
 It is evident that the sign of each trigonometric function 
 is determined by the segment representing the function, 
 since in each quadrant the segment is positive whenever the 
 ratio defining the function is positive, and negative whenever 
 the ratio is negative. Thus in each quadrant LM has the 
 
 same sign as the ratio ?, and similarly for the other func- 
 x 
 
 tions. Therefore the segments represent the trigonometric 
 functions in sign as well as in magnitude. 
 
 The segments which represent the trigonometric functions 
 are called the line values of the functions. 
 
 PROBLEM. Represent vers a and covers a by line values. 
 
 58. Variations of the trigonometric functions as shown by 
 line values. The line values of the trigonometric functions 
 give a simple method of tracing the variations in the func- 
 tions as the angle varies from to 360. Thus as the point 
 P, in the figures of Art. 57, describes the circle of radius 
 unity, the changes in the segments AP, OA, LM, etc., 
 represent the variations in the sine, cosine, tangent, etc., 
 respectively. 
 
56 
 
 TRIGONOMETRY 
 
 Numerous figures, with three or four values of the angle in 
 each quadrant, serve to suggest these variations. 
 
 59. Fundamental relations by line values. By the use of 
 line values the fundamental relations of Art. 52 may be 
 simply obtained and easily memorized. 
 
 F cot a -, 
 
 Thus 
 
 From the triangle OAP, 
 
 AP 2 +OA 2 =OP 2 or sin 2 a + cos 2 a = 1. 
 From the triangle OLM, 
 
 or tan 2 a + 1 = sec 2 a. 
 or cot 2 a+l = csc 2 a. 
 
 From the triangle OGF, 
 
 From definitions, 
 
 AP sin a 
 
 tan a = = 
 
 OA cos a 
 
 OA cos a 
 
 cot a 
 
 AP sin a 
 
 sec = - = 
 
 OA cos a 
 
 esca-O?.- 1 
 
 AP sin a 
 
 or 
 or 
 or 
 or 
 
 tana = 
 cot a = 
 seca = 
 csca = 
 
 sin a 
 cos a 
 cos a 
 sin a 
 
 cos a 
 
 1 
 sin a 
 
FUNDAMENTAL RELATIONS 57 
 
 co_= = __ or 
 
 AP LM tan a tan a 
 
 The process of derivation of these formulas applies 
 equally well to an angle of the third or fourth quadrant ; 
 hence the formulas are true for all values of a. 
 
 To memorize these formulas most easily, form a clear 
 mental picture of the figure for the first quadrant and read 
 each formula directly from the figure, applying the ratio 
 definitions and the Pythagorean theorem. 
 
 60. EXAMPLES 
 
 1. Construct the line values of an angle terminating in 
 the third quadrant. 
 
 2. Construct the line values of an angle terminating in 
 the fourth quadrant. 
 
 3. Obtain the fundamental relations for an angle in the 
 third quadrant, by the use of line values. 
 
 4. Obtain the fundamental relations for an angle termi- 
 nating in the fourth quadrant, by the use of line values. 
 
 5. Deduce the relations between the functions of 90 + x 
 and the functions of x by means of line values. 
 
 SUGGESTION. Construct two figures, one for the angle 90 + x and 
 the other for the angle x. 
 
 6. Deduce the relations between the functions of 180 x 
 and the functions of x by means of line values. 
 
 7. Deduce the relations between the functions of 90 x 
 and the functions of 270 + x by means of line values. 
 
 By means of line values, trace the variations in the follow- 
 ing functions as a varies from to 360 : 
 
 8. sin a. 11. cot . 
 
 9. cos . 12. sec a. 
 10. tan a. 13. esc a. 
 
58 TRIGONOMETRY 
 
 Prove the following trigonometric identities : 
 
 14. = sin x cos x. 
 
 cot x + tan x 
 
 15. cos 2 a (1 + cot 2 a) = cot 2 a. 
 
 16. sin cot 2 = esc - sin 0. 
 
 17. tan a tan sin 2 a = sin a cos a. 
 
 18. cos a tan 2 a = sec a cos a. 
 
 jo cos sin _ sin 3 a -f- cos 3 a 
 tan cot a sin cos a 
 
 on sin 3 a + cos 3 .< 
 
 20. = 1 sin a cos a. 
 
 sin a , cos a 
 
 O i sin x . tan x . sec x 2 cot a? 4- 1 
 
 1 
 
 22. 
 
 23. 
 
 04 
 
 cosa; cotx csca; cot 2 re 
 
 1 cos x _ sec a; 1 
 1 -|- cos a? sec x + 1 
 
 1 cos x sin 2 a; 
 
 1 + cos x (1 + cos a:) 
 1 cos # (1 cos # 2 
 
 1 -j- cos a? sin'" a; 
 
 25. 2 sin ?/ cos 2 y + (2 cos 2 ?/ 1) sin y = 3 sin y 4 sin 3 y. 
 
 26. cos 4 a sin 4 a = cos 2 a sin 2 a. 
 
 27. sec 2 a + cosec 2 a = sec 2 a esc 2 a. 
 
 no tan a; -f- tan y _ sec x -+- sec y 
 \ sec a; sec ?/ tan a; tan y 
 
 29. tan 4 w cot 4 u = (tan 2 u + cot 2 w)(sec 2 w esc 2 w). 
 
 on tan a; -f tan ?/ 
 
 30. - -i- - ? = tan x tan y. 
 
 cot x + cot y 
 
 01 tana? tan y 
 
 31. - - 2 == tan x tan w. 
 cot x cot ?/ 
 
FUNDAMENTAL RELATIONS 59 
 
 on tan x -f cot y tan x 
 64. = 
 
 cot x -f- tan y tan y 
 
 33. Express sin 2 a; cos 2 a; in terms of tan x. 
 
 34. Express vers x covers x in terms of sin x. 
 
 35. Express cot 2 a? -f- csc 2 # in terms of cos x. 
 
 yp-pQ /V 
 
 36. Express in terms of tan x. 
 
 covers x 
 
 37. Express tan 2 x + sec 2 x in terms of cot x. 
 
 38. Express '(4 sin 3 - 3 sin 0)(2 cos 2 0-1) 
 
 -f- (3 cos 4 cos 3 6) (2 sin 6 cos 0) in 
 terms of sin 0. 
 
 39. Given sin = a, find the remaining functions of 0. 
 
 40. Given cot = 6, find the remaining functions of 0. 
 
 41. Express each trigonometric function of in terms of 
 cos 0. 
 
 42. Express each trigonometric function of in terms of 
 tan0. 
 
 43. Simplify (a + 6) sin 30 P - 6 cos 60 + a tan 180. 
 
 44. Simplify I sin (270 - a;) + m cos (180 + a?) 
 
 + w sin (90 - a;). 
 
 45. Simplify a sin 135 +(a 6) cos 225 + & cos 315. 
 
 46. Simplify tan (-120) + cot 150-tan 210+ cot 240. 
 
 Find the positive values of 0, less than 180, that satisfy 
 the following equations : 
 
 47. sin0 cos = 0. 
 
 48. sin + cos = 0. 
 
 49. sin0(2sin0-l)(2cos0-l) = 0. 
 
 50. sin + cos cot = 2. 
 
60 TRIGONOMETRY 
 
 51. sin0cos0 = |. 
 
 52. 2 sin cos <9 + sin 0-2 cos0 = l. 
 
 53. 2 cos + sec 6 = 3. 
 
 54. sec (9 + tan = 2. 
 
 55. 2 sin + 5 cos 9 = 2. 
 
 56. 1 + sin 2 = 3 sin cos 0. 
 
 57. tan 4 0-4 tan 2 0-5 = 0. 
 
 58. What negative angles, numerically less than 180, 
 satisfy the equation sin 2 x -f- sin x = cos 2 x 1 ? 
 
 59. Given 9 cos 2 w + 9 sin ^ = 11, find tan u. 
 
 60. Given tan 2 x 5 sec a; + 7 = 0, find sin x. 
 
 Find the positive values of 0, less than 360, that satisfy 
 the following equations : 
 
 61. sin = - cos 285. 
 
 62. tan = cot (-144). 
 
 63. cos ( - 0) = sin 190. 
 
 64. sin0 = sin 50. 
 
 65. If sin 122 = k, prove that tan 32 = ~ V/J 
 
 66. If cot 255 = a, prove that cos 345 = 
 
 67. If cos(- 100)= k, prove that tan 80 = - 
 
 k 
 
 Solve the following equations for any trigonometric func- 
 tion of a. 
 
 68. 2 sin a + esc a = 1. 71. tan 2 a -f 4 sec a = 5. 
 
 69. esc a cot a = j. 72. tan a + cot a = 2. 
 
 70. 2 sin a + cos 2 a = 1. 
 
 73. Given sin ^ = k sin v and tan u = Z tan v, find sin w 
 and sinv. 
 
CHAPTER VI 
 
 FUNCTIONS OF THE SUM OF TWO ANGLES 
 DOUBLE ANGLES. HALF ANGLES 
 
 61. We have thus far considered the properties and rela- 
 tions of the functions of a single angle. We have also 
 shown that the functions of the angles n 90 a depend 
 upon the functions of the angle a. 
 
 In the present chapter we consider the relations between 
 the functions of the sum of two independent angles and the 
 functions of the separate angles, and develops some related 
 formulas. 
 
 -* 62. The sine of the sum of two acute angles expressed in 
 terms of the sines and cosines of the angles. 
 
 B X 
 
 Let a and /? be any given acute angles. 
 
 Construct the angle (a -f- /?), Art. 12. 
 
 Then (a + ft} may be acute as in Fig. 1, or obtuse as in 
 Fig. 2. 
 
 From any point P, in the terminal line of the angle 
 (a 4- /?), draw PA perpendicular to OX, and PQ perpendicu- 
 lar to OS. Through Q draw RQT parallel to OX, and QB 
 perpendicular to OX. 
 
 61 
 
62 TRIGONOMETRY 
 
 Then the angle TQP is equal to 90 + a. 
 By definition 
 
 OP OP OP OP 
 
 BQ.OQ.RP QP 
 OQ' OP QP' OP 
 
 and =s 
 
 therefore sin (a + P) = sin a cos p 4- cos a sin p. (1) 
 
 PROBLEM 1. Show that formula (1) is true when either 
 
 angle is 90. 
 
 PROBLEM 2. Show that formula (1) is true when each 
 
 angle is 90. 
 
 63. The cosine of the sum of two acute angles expressed in 
 terms of the sines and cosines of the angles. ' Referring to the 
 figures of the previous article, we have by definition 
 
 OB OQ QR QP 
 OQ' OP QP' OP 
 
 But = cos , = cos ft 
 
 =cos (90 +) = - sin a, and = sinft 
 
 therefore cos (a -f p) =cos a cos p sin a sin p. *" (1) 
 
 PROBLEM 1. Show that the formula (1) is true when 
 
 either angle is 90. 
 
 PROBLEM 2. Show that formula (1) is true when each 
 
 angle is 90. 
 
 64. The formulas developed in the last two articles are 
 of the utmost importance, since many other formulas are 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 63 
 
 derived from them. They form the basis of the present 
 chapter. In developing these formulas a and ft were con- 
 sidered positive acute angles, but the formulas are true for 
 all values of a and ft } as may be shown either geometrically 
 or analytically. 
 
 It is shown geometrically by constructing figures in 
 which a and (3 are of any magnitude, and following the 
 method of proof given above for acute angles. 
 
 We present the analytic demonstration as the more satis- 
 factory. 
 
 65. To prove that 
 
 sin ( -|- ft) = sin a cos ft 4- cos a sin ft (1) 
 
 and cos (a -f- ft) = cos a cos ft sin a sin ft (2) 
 
 are true for all values of a and ft. 
 
 First. To show that a can be replaced by a', where a' = 
 a + 90. 
 
 Let a' = a -f 90 where a is acute. 
 
 As a varies from to 90, a' varies from 90 to 180. 
 
 Also sin a' = sin (a + 90) = cos a, (3) 
 
 cos a' = cos (a + 90) = sin a. (4) 
 
 Now sin (' + /?)= sin (90 + a + )= cos(a -f ). 
 
 Since a and /? are both acute, cos (a + ft) may be ex- 
 panded by (2), hence 
 
 sin (a' + ft) = cos (a + ft) = cos a cos ft sin a sin ft. 
 Therefore, by (3) and (4), 
 
 sin (a' + ft) = sin a' cos ft + cos ' sin ft. (5) 
 
 Similarly, 
 
 cos(' -f ft) = cos (90 + a + ft) = - sin ( -f /8) 
 = sin a cos /? cos a sin /?. 
 
64 TRIGONOMETRY 
 
 Therefore, by (3) and (4), 
 cos (' + /?) = cos a' cos ft sin a' sin j3. (6) 
 
 Formulas (5) and (6) show that one angle, a, in (1) and 
 (2) may be extended to 180. 
 
 Second. To show that a can be replaced by a" where a" = 
 a + 18CP- 
 
 Let a" = a + 180. 
 
 As a varies from to 180, a!' varies from 180 to 360. 
 Also sin a" = sin (a + 180) = sin a, (7) 
 
 cos a" = cos (a + 180)= cos a. (8) 
 
 Now sin (a" -f 0)= sin (180 + a + /?)= - sin (a + 0). 
 
 Since a is less than 180 and ft acute, sin ( 4- /?) may be 
 expanded by (1), as extended in the first part of the proof, 
 hence 
 
 sin (a" + /?) = sin (a + ft) = sin a cos ft cos a sin ft. 
 
 Therefore, by (7) and (8), 
 
 sin (a" + ft) = sin a!' cos ft -f cos a" sin ft. (9) 
 
 Similarly, 
 
 cos (a" 4- /?)= cos (180 4- a + 0) = - cos (a + 0) 
 
 = cos a cos /? + sin a sin /3. 
 Therefore, by (7) and (8), 
 
 cos (a" -f ft} = cos a" cos sin a" sin /?. (10) 
 
 Formulas (9) and (10) show that one angle, a, of (1) and 
 (2) may be extended to 360. 
 
 Third. In a similar manner it may be shown that (1) and 
 (2) are true when ft varies from to 360, a having any 
 value from to 360, from which it easily follows that (1) 
 and (2) are true for all positive values of a and ft. 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 65 
 
 Fourth. Formulas (1) and (2) may be shown to be true 
 when either a or ft or both are negative, by the use of the 
 substitutions 
 
 a' = a-n 360 
 
 hence (1) and (2) are true for all positive and negative 
 values of a and (3. 
 
 66. To find the tangent of the sum of any two given angles 
 in terms of the tangents of the given angles. 
 Let a and ft be the given angles. 
 
 Then tanQ + 8) = sin ( " + ffl = sina cos ft + cos sin ^ 
 cos (a + ft) cos a cos ft sin a sin ft 
 
 Dividing numerator and denominator of the last fraction 
 by cos a cos ft and simplifying, we have 
 
 1 tan a tan p 
 
 67. To find the cotangent of the sum of any two given 
 angles in terms of the cotangents of the given angles. 
 Let a and ft be the given angles. 
 
 Then cot ( + /?) = cos < + ^ = cos cos /3 - sin sin/?. 
 
 sm (a -h ft) sm a cos ft + cos a sm ft 
 
 Dividing numerator and denominator of the last fraction 
 by sin a sin ft and simplifying we have 
 
 cot p+ cot a 
 
 68. Addition formulas. Collecting the formulas for the 
 sum of two angles for convenience of reference, we have 
 
 sin (a -h ft) = sin a cos ft + cos a sin ft (1) 
 
 cos (a + ft) = cos a cos ft sin a sin /3 (2) 
 
66 TRIGONOMETRY 
 
 69. To find the sine, cosine, tangent, and cotangent of the 
 difference of two given angles. 
 
 Since the formulas of Art. 68 are true for all values of ot 
 and ft they are true when ft is substituted for ft. 
 
 Hence 
 
 sin (a ft) = sin a cos ( ft) + cos a sin ( ft) 
 cos (a ft)= cos a cos ( ft) sin a sin ( ft) 
 
 tan (ct K\ - tang + 
 ~^- 
 
 cot(-0)+cota 
 But remembering that 
 
 sin ( ft) = sin ft, cos ( ft) = cos ft 
 tan (- ft) = - tan ft, and cot (-) = - cot ft 
 these formulas become U 
 
 sin (a ft) = sin a cos ft cos sin /? (1) Ir. 
 
 cos (a ft)= cos a cos + sin a sin ft (2) 
 
 tan (a - ft) = an <*~ ?L/L (3) 
 
 1 4- tan 
 
 ' cot/3-cot 
 
 70. EXERCISES 
 
 1. Find sin 75. 
 
 SOLUTION. 75 = sin (45 + 30) = sin 46 cos 30 + cos 45 sin 30 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 67 
 
 2. Find cos 75. 7. Find sin (90 - ). 
 
 3. Find sin 15. 8. Find tan (90 + a). 
 
 4. Find cos 15. 9. Find cos (180 - a). 
 
 5. Find tan 75. 10. Find sec 15. 
 
 6. Find cot 15. 11. Find esc (90 + a). 
 
 71. Double angles. To find the sine, cosine, tangent, and 
 cotangent of twice a given angle in terms of the functions of 
 the given angle. 
 
 From the formulas of Art. 68, letting /3 = a, we have, 
 after reduction, 
 
 sin 2 a = 2 sin a cos a (1) 
 
 cos 2 a . = cos 2 a sin 2 a (2) 
 
 = l-2sin 2 a (3) 
 
 = 2 cos 2 a - 1 (4) 
 
 rt 2 tan a /K -, 
 
 tan 2 a = (5) 
 
 1 - tan 2 a 
 
 cot2a = cot2a - 1 . (6) 
 
 2 cot a 
 
 To express clearly the real significance of these formulas, 
 we may state them from two points of view. 
 
 If a be the angle under consideration, the formula 
 
 sin 2 a = 2 sin a cos a 
 
 may be stated : The sine of twice an angle is equal to twice 
 the sine of the angle times the cosine of the angle. 
 
 If 2 a be the angle under consideration, the same formula 
 may be stated : TJie sine of an angle is equal to twice the 
 sine of half the angle times the cosine of half the angle. 
 
 It then follows that 
 
 sin a = 2 sin J a cos | a. (7) 
 
 Similarly, if a be the. angle under consideration, the 
 formula cos 2 a. = cos 2 a sin 2 a 
 
68 TRIGONOMETRY 
 
 may be stated : The cosine of twice an angle is equal to the 
 square of the cosine of the angle minus the square of the sine 
 of the angle. 
 
 If 2 a be the angle under consideration, the same formula 
 may be stated : The cosine of an angle is equal to the square 
 of the cosine of half the angle minus the square of the sine of 
 half the angle. 
 
 It then follows that 
 
 cos a = cos 2 1 a sin 2 ^ a. (8) 
 
 Similarly, 
 from cos 2 a = 1 2 sin 2 we have 
 
 cosa = l -2sin 2 ia, (9) 
 
 from cos 2 a = 2 cos 2 a 1 we have 
 
 l, (10) 
 
 from tan 2 a = 2 tan " we have 
 1 tan 2 a 
 
 2 tan i a 
 
 from cot 2 a = cot2 a ~ l we have 
 2 cot a 
 
 cota^****- 1 . (12) 
 
 2 cot |a 
 
 72. Half angles. To find the sine, cosine, tangent, and co- 
 tangent of one half a given angle in terms of functions of the 
 given angle. 
 
 From formula (9), Art. 71, we have 
 cos = 1 2 sin 2 ~j 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 69 
 
 
 From formula (10), Art. 71, we have 
 
 cos a = 2 cos 2 - 1, 
 
 2 
 
 or COS^ = A/- "~ (2) 
 
 & A 
 
 From (1) and (2), 
 
 ^^ (3) 
 
 2 ^ 1 + cos a 
 and cot 
 
 2 
 
 In each, of these formulas the positive or negative sign 
 
 is chosen to agree with the sign of the function of ^ , depend- 
 
 2i 
 
 ing on the quadrant in which - lies. 
 
 2 
 
 These formulas may be considered from two view points. 
 For example, formula (1) may be stated : The sine of half 
 an angle is equal to the square root of the fraction whose 
 numerator is one minus the cosine of the angle, and whose 
 denominator is two; or, Tlie sine of an angle is equal to the 
 square root of the fraction whose numerator is one minus the 
 cosine of twice the angle, and whose denominator is two. 
 
 73. To find the sum and difference of the sines of any two 
 angles, also the sum and difference of the cosines of any two 
 angles. 
 
 From the formulas of Arts. 68 and 69 we have, by ad- 
 dition and subtraction, 
 
 sin (a + ft) + sin (a ft) = 2 sin a cos (3 
 sin (a -f- ft) sin (a (3) = 2 cos a sin ft 
 cos (a + ft) + cos (a ft) = 2 cos a cos ft 
 cos (a + ft) cos (a ft) = 2 sin a sin ft. 
 
70 TRIGONOMETRY 
 
 Let a -J- ft = y and a ft = d. Solving for a and /?, we 
 have a = (y 4- 3) and ft = 1 (y d). Substituting these 
 values in the above equation, it follows that 
 
 sin Y + sin d = 2 sin (y + d) cos 1 (Y - 3), (1) 
 
 sin Y- sin 3 = 2 cos 1 (y + 3) sin | (>- 3), (2) 
 
 cos Y 4- cos 3 = 2 cos J (Y 4- 3) cos J (? 3), (3) 
 
 cos Y cos d = 2 sin $(i + d) sin -J (<y 3). (4) 
 
 Equations (1), (2), (3), and (4) may be read from two 
 view points. 
 
 Eegarding y and 3 as the given angles, (1) may be 
 stated : The sum of the sines of two angles is equal to twice 
 the product of the sine of half the sum of the given angles into 
 the cosine of half the difference of the given angles. 
 
 Thus, sin 6 x -f- sin 4 x = 2 sin 5 x cos x. 
 
 Eegarding i(y + 3) and -J(y 3) as the given angles, it is 
 clear that their sum is y and their difference 3. Then, by 
 reading the second member first, equation (1) may be stated : 
 Twice the sine of any angle times the cosine of any other angle 
 is equal to the sine of the sum of the angles plus the sine of the 
 difference of the angles. 
 
 Thus, 2 sin 20 cos 5 = sin 25 + sin 15. 
 
 74. Equations and identities. The formulas developed in 
 the present chapter are true for all values of the angles 
 involved ; hence they are trigonometric identities. By the 
 use of these identities many others may be established. 
 The remarks of Art. 54, concerning the use of the funda- 
 mental relations in establishing identities, apply here. 
 
 The identities of the present chapter are also useful in 
 solving trigonometric equations. By their aid an equation 
 involving functions of multiple angles may be transformed 
 into an equation containing functions of a single angle 
 (see Ex. 33, Art. 75). This transformed equation can then 
 be solved as indicated in Art. 55. Frequently equations 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 71 
 
 may be much simplified by reducing sums or differences of 
 sines and cosines to products by the relations of Art. 73 
 (see Ex. 41, Art. 75). 
 
 75. EXAMPLES 
 
 1. Find sin 221, cos 221, tan 22J, and cot 22. 
 
 2. Given cosa = J; find sin 2 a, cos 2 a, tan2, and 
 cot 2 a. 
 
 3. Given tan a = 3 ; find sin 1 a, cos % a, tan % a, and 
 cot \ a. 
 
 Prove the following identities : 
 
 4. (cos a + sin a) (cos a sin a) = cos 2 . 
 
 5. (sin a + cos a) 2 = 1 -f sin 2 a. 
 
 6. sin 3 a = 3 sin a 4 sin 3 a. 
 
 SUGGESTION : sin 3 a = sin (2 a + a). 
 
 7. cos3a = 4cos 3 a 3 cos a. 
 
 8. tan3 = 3tan "- tan3 ". 
 
 l-3tan 2 a 
 
 1 + COS X 
 
 , f. sin x 
 10. cot i x = 
 
 1 cos x 
 
 - - 1 + tan i x _ 1 4-sin^ 
 
 1 tan i x ~ cos x 
 
 - ft 1 cos 2 a 
 
 12. tan 2 a=- 
 
 1 + cos 2 
 
 13. sec2= sec tt 
 
 2 sec 2 
 
 14. 4 sin 2 1 a cos 2 i a = 1 cos 2 . 
 
 15. sin4 + sin2 = 2sin 3acos a. (See Art. 73.) 
 
 16. cos 6 a cos 2a= 2 sin 4 a sin 2 a. 
 
72 TRIGONOMETRY 
 
 17. sin (45 + ) sin (45 - a) = V2 sin a. 
 
 18. sin (30 + a) + sin (30 - a) = cos a. 
 
 19. Express cos 4 a sin 3 a as the difference of two sines. 
 
 20. Express cos 5 a cos a as the sum of two cosines. 
 
 Prove the following identities : 
 
 21. cos 2 a cos 2 /? = sin (a + ft) sin (a ft). 
 sin 2 a + sin a 3 a 
 
 00 
 22. 
 
 - --^ 
 
 cos 2 a -j- cos a 2 
 
 cos /? cos a 
 24. 
 
 1 + tan 2 ! 
 
 25. sin |- a; + 2 sin 2 ^a? cos J # cot # = sin x cos -|-a/*. 
 
 26. 2 tanicsca; = l4-tan 2 ^aj. 
 
 27. cos (a -f- ft) cos /? cos ( + y) cos y = 
 
 sin (a -f y) sin y sin (a -f- /?) sin ft. 
 
 28. cos2a = 8sin 4 -i-a-8sin 2 a + l. 
 
 29. cos 2 a - cos a = 2 (4 sin 4 ! a - 3 sin 2 ! a). 
 
 30. (1 cos 2 a) cot 2 ! a = cos 2 a + 4 cos + 3. 
 
 31. 2sin 2 |-0cot0 csc20 + csc0 = cot2 + sin0. 
 
 32. Given tan a; = a tan 1 a? ; find tan ^ a; in terms of a. 
 
 Solve the following equations for all positive values of a 
 less than 180: 
 
 33. sin2a-4cos 2 ia-sina + 3 = 0. 
 
 SOLUTION. 2 sin a cos a - 4 / 1 + cos <*\ - sin a + 3 = 0, 
 
 or 2 sin a cos a 2 cos a sin + 1 = 0, 
 
 or (2 cos a 1) (sin a 1) =0, 
 
 cos a = ^ or sin a = 1. 
 and a = 60 or 90. 
 
FUNCTIONS OF THE SUM OF TWO ANGLES 73 
 
 34. sin 2 a cos = 0. 
 
 35. sin 2 + sin = 0. 
 36. 
 
 37. 
 
 38. cos 3 a + 2 cos 2 8 cos 2 a + 2 sin 2 a 4- 5 cos a = 0. 
 
 39. cos 2 a H- 2 sin 2 ! a 1 = 0. 
 40 sin2a + cos2a + sin a = l. 
 
 41. sin 5 a sin3cc + sin a = 0. 
 
 SOLUTION. 2 sin 3 a cos 2 a sin 3 a = 0. Art. 73. 
 or sin3a(2cos2a-l) = 0. 
 
 sin 3 a = 0, or cos 2a = i. 
 3 a = 0, 180, 360, and 2 a = 60, 300. 
 a = 0, 60, 120, 30, 150. 
 
 42. cos 3 a + sin 2 a cos a = 0. 
 
 43. sin 3 a -+- sin 2 -|- sin a = 0. 
 
 44. cos 3 a sin 2 a + cos a = 0. 
 
 45. sin 3 a + cos 3 a sin a cos a '= 0. 
 
 46. sin a; + sin 2 # + sin 3 x = 1 + cos x -f- cos 2 #. 
 
 47. ^-^=20032*. 
 1 + tan x 
 
 Prove the following identities : 
 
 48. tan (30 + a) tan (30 - <*) = 2 cos 2 a ~ 1 . 
 
 ' 2 cos 2 a + 1 
 
 49. tan a -J- cot a = 2 esc 2 a. 
 
 50. sin 80 = sin 40 + sin 20. 
 
 51. 1 + sintt-cosft^^ 
 1 + sin a + cos a 2 
 
 52. cos 2 a - sin (30 + a) sin (30 - a) = f . 
 
 cos -h cos 
 
CHAPTER VII 
 INVERSE FUNCTIONS 
 
 76. Inverse trigonometric functions are closely related to 
 the trigonometric functions previously studied. 
 
 After introducing the fundamental idea of an inverse 
 function, it will be shown that they lead to new relations, 
 closely allied to the relations already developed. 
 
 77. Fundamental idea of an inverse function. From the 
 equation sin a = u, it is evident that a is an angle whose 
 
 sine is it. The statement a is an 
 angle whose sine is u is abbreviated 
 into 
 7 a, = sin" 1 u. 
 
 The equation u = sin a expresses 
 u in terms of a. 
 
 The equation a = sin" 1 u expresses a in terms of u. 
 
 We thus have two methods of expressing the relation 
 between an angle and its sine. 
 
 The symbol sin" 1 ?* is the inverse sine of u. It may be 
 read the inverse sine of u, the anti sine of u, arc sine u, or an 
 angle whose sine is u. The last form should be used until 
 the conception of an inverse function is perfectly clear. 
 
 Corresponding to each direct function there is an inverse 
 function. Thus, 
 
 a = sin" 1 u corresponds to sin a = u, 
 a = cos"" 1 u corresponds to cos a = u, 
 a tan" 1 u corresponds to tan a = u, etc. 
 
 CAUTION. Since sin" 1 u is an inverse function, the \ 
 
 74 
 
INVERSE FUNCTIONS 75 
 
 cannot be an exponent. It is. simply part of a symbol for an 
 inverse function. When a function is affected by 1 as an 
 exponent, it must be written with parentheses, thus (sin a;) 
 
 -i 
 
 78. Multiple values of an inverse function. It has been 
 shown that the trigonometric functions are single-valued 
 functions. Thus, if a is given, there is only one value for 
 sin a. See Arts. 25 and 41. 
 
 On the contrary, the inverse functions are multiple-valued 
 functions. Thus, if u is given there are an infinite number 
 of values for sin" 1 ^. See Arts. 26 and 41. A few ex- 
 amples will illustrate this property of inverse functions, 
 and show that all the values of any given inverse function 
 may be combined in a single expression. 
 
 To find all the values of tan" 1 
 
 V3 
 
 tan- 1 -l r = 30 
 
 V3 
 
 = 180 + 30 
 = 360 + 30 
 
 - _ 180 -f 30, or - 150 
 = _ 360 -|- 30, or 330 
 = _540 ^30, or -510 
 
 All these angles may be expressed by n 180 + 30, n 
 being any positive or negative integer. Hence 
 _ 1 i 
 
 VB = 
 
 In general, if a is an angle whose tangent is u, it may be 
 shown that tan -i u = n >n + Q.. 
 
76 TRIGONOMETRY 
 
 To find all the values o/cos" 1 ^. 
 
 cos- 1 ^ 60 
 
 = -60 
 
 = 360 + 60 
 
 = 360 - 60 
 
 = 720 + 60 
 
 = 720 - 60 
 
 . = _ 360 + 60 
 = -360 -60 
 
 = _ 720 + 60 
 = _ 720 - 60 
 
 All these angles may be expressed by n 360 60, n 
 being any positive or negative integer. Hence 
 
 o 
 
 In general, if a is an angle whose cosine is u, it may be 
 shown that cos -i u = 2 nir a.. 
 
 To find all the values o/sin- 1 !. 
 
 sin- 1 i= 30 
 
 = 180 - 30 
 
 = 360 + 30 
 
 = 540 - 30 
 
 = 720 + 30 
 
 = 900 - 30 
 
 - 180 - 30 
 
 - 360 + 30 
 
 - 540 - 30 
 : - 720 + 30 
 
INVERSE FUNCTIONS 77 
 
 All these angles may be expressed by n 180 + ( - 1)"30, 
 n being any positive or negative integer. Hence 
 
 In general, if a is an angle whose sine is w, it may be 
 own that sin-' U = / nr +(-!)" a. 
 
 In a similar manner it may be shown that 
 
 sec" 1 u = 2 nir a, 
 csc~ l u = nir + ( l) n a. 
 
 79. Principal values. The smallest numerical value of an 
 inverse function is called its principal value, preference be- 
 ing given to positive angles in case of ambiguity. 
 
 The principal values of the inverse sine and the inverse 
 
 cosecant lie between and - ; of the inverse cosine and 
 
 2 2 
 
 the inverse secant, between and ?r ; of the inverse tangent 
 and the inverse cotangent, between - and - 
 
 The principal values of an inverse function are sometimes 
 distinguished from the general values by the use of a capital 
 letter. 
 
 Thus Sin- 1 -^-, 
 
 while sin- 1 1 = mr + ( - l) w - 
 
 6 
 
 80. To interpret sin sin~ l u and sin~ l sin a. 
 
 The expression sin sin" 1 !* is read: the sine of the angle 
 whose sine is u. This sine is evidently u, hence 
 
 sin sin' 1 u = u. 
 
78 TRIGONOMETRY 
 
 The expression sin" 1 sin a is read : the angle whose sine 
 is the sine of a. This angle is evidently a, hence 
 
 sin" 1 sin a == a, 
 or more generally, 
 
 sin" 1 sin a = nir -\- ( l) n . 
 
 Similar relations exist between any direct function and 
 the corresponding inverse function. 
 
 Thus cos cos" 1 u = u ; 
 
 cos" 1 cos a = a, or cos" 1 cos a = 2 n-n a ; 
 
 tan tan" 1 u = u ; 
 
 tan" 1 tan a = a, or tan" 1 tan = nir + a, etc. 
 
 81. Application of the fundamental relations to angles ex- 
 pressed as inverse functions. The fundamental relations, 
 being true for all angles, must necessarily be true when 
 the angles are expressed as inverse functions. 
 
 Thus, letting a = tan" 1 u in the identity 
 
 sin 2 a + cos 2 a = 1, 
 
 we have (sin tan" 1 u) 2 -f- (cos tan" 1 u) 2 = 1. 
 
 Similarly (sin esc" 1 u) 2 -f- (cos esc" 1 u) 2 = 1 ; 
 
 (tan sec" 1 u) 2 + 1 = (sec sec" 1 u) 2 j 
 
 sin cos" 1 u = 
 
 esc cos" 1 u 
 
 By expressing the angle of the fundamental relations as 
 an inverse function, we may develop relations between the 
 inverse functions. 
 
 82. Given an angle, expressed as an inverse function of u, 
 to find the value of any function of the angle in terms of u. 
 
 By the application of one or more of the fundamental 
 relations, it is always possible to solve the stated problem. 
 
INVERSE FUNCTIONS 
 
 79 
 
 Several illustrations are given below. The method em- 
 ployed can be readily applied to the other functions. 
 
 1. To find the value of tan cos" 1 u in terms of u. 
 
 If tan cos" 1 u is expressed in terms of the cosine of 
 cos" 1 u, the problem is solved, since cos cos" 1 u = u. 
 
 This may be done as follows : 
 
 tan cos" 1 u 
 
 sin cos" 1 u _ Vl (cos cos" 1 u) 2 __ Vl u 2 
 
 COS COS" 1 U U U 
 
 This result may be obtained geometrically. Since u is 
 given, it is evident that cos" 1 ^ represents, among others, 
 two positive angles, a and o, each less than 360. 
 
 Let us assume u positive and let us construct these angles 
 defined by cos" 1 u. 
 
 Then from the figure and the definition of the tangent, 
 
 and 
 
 Since the tangent of any angle Co- 
 terminal with ! is -t- , and the tangent of any 
 
 angle coterminal with 0% is , and since cos" 1 ^ 
 
 u 
 
 represents all angles coterminal with either ai or #2, we have 
 
 tan cos" 1 u = 
 
 Vl - u* 
 
 2. To find the value of sec cot 1 u in terms of u. 
 
 To solve this problem it is only necessary to express 
 sec cot" 1 u in terms of cot cot" 1 u. 
 
80 
 
 TRIGONOMETRY 
 
 Thus 
 
 sec coir 1 u = Vl 4- (tan cot" 1 u) 2 = -v/1 4- 
 
 (cot cot" 1 w) 2 
 
 This result may be obtained geometrically. Construct 
 the angles given by cot" 1 u. Let us assume in this problem 
 that 16 is negative and hence that u is positive. If the 
 
 cotangent of an 
 ^angle is nega- 
 tive the angle 
 must terminate 
 in either the 
 second or fourth 
 quadrant. Since 
 OA and OB are 
 
 the terminal lines of otj and 2 respectively, and since the 
 terminal line is always positive, we have 
 OA = OB = 
 
 u u 
 
 or by considerations similar to those in the previous ex- 
 ample, we have 
 
 + Vl 4- u 2 Vl 4- w 2 
 
 sec cot" 1 16=- 
 
 16 U 
 
 3. To find the value of sin cos" 1 u in terms ofu. 
 
 We have sin cos" 1 u = Vl (cos cos" 1 16) 2 = Vl i* 2 . 
 
 4. To find the value of cot vers" 1 16 in terms of u. 
 We have 
 
 1-u 
 
 Vl (1 vers vers" 1 w) 2 
 1-u 
 
 ^/2u u* 
 
INVERSE FUNCTIONS 81 
 
 83. Some inverse functions expressed in terms of other in- 
 verse functions. 
 
 1. To express cos" 1 u in terms of an inverse tangent. 
 
 From tan cos" 1 u = , (Art. 82, prob. 1) by 
 
 u 
 
 taking the inverse tangent of each member (Art. 80), there 
 
 results yi _ ^2 
 
 cos" 1 u = tan" 1 
 
 u 
 
 2. To express the cot" 1 u in terms of an inverse secant. 
 
 From sec cot" 1 u = + M> (Art. 82, prob. 2) there 
 
 u 
 
 results 
 
 3. Similarly from sin cos" 1 u = Vl w 2 there results 
 
 cos" 1 u = sin" 1 ( Vl w 2 ). 
 
 4. Also from cot vers" 1 M = ~ there results 
 
 V2 u - u* 
 1-u 
 
 vers" 1 u = cot" 1 
 
 V2 u - u 2 
 
 By the method exemplified in Arts. 82 and 83 it is possi- 
 ble to express any inverse function in terms of any other 
 inverse function. 
 
 In applying the above formulas care must be exercised 
 in selecting the angles, since each inverse function repre- 
 sents an infinite number of angles and one member of 
 the equation may represent angles not represented by the 
 other. For example, in problem 1, if u be positive the 
 cos" 1 u represents angles terminating in the first and fourth 
 
 quadrants ; but tan" 1 -5^- -^- represents angles terminat- 
 ing in the second and third quadrants as well as angles 
 terminating in the first and fourth quadrants. 
 
82 TRIGONOMETRY 
 
 84. Some relations between inverse functions derived from 
 the formulas for double angles, half angles, and the addition 
 formulas. 
 
 The general method applicable to this class of problems 
 will be illustrated by a few examples. 
 
 1. To express cos (2 sec" 1 u) in terms of u. 
 
 cos (2 sec" 1 u) = 2 (cos sec" 1 u) z - 1 Art. 71, Eq. 4. 
 
 1=1-1. 
 
 (sec sec" 1 u) 2 u 2 
 
 From this relation it follows that, 
 
 (2 \ 
 1 ). 
 u 2 ) 
 
 2. To express tan (1 cos" 1 u) in terms of u. 
 
 lY^^L 
 
 COS 
 
 Art. 72, Eq. 3. 
 From this relation it follows that, 
 
 l , *. ,, _i_ /I if 
 
 2 
 
 3. To express sin (sin" 1 u + cos" 1 v) in terms of u and v. 
 sin (sin" 1 u + cos" 1 v) = sin sin" 1 u cos cos" 1 v 
 + cos sin" 1 u sin cos" 1 v 
 
 Art. 62, Eq. 1. 
 
 From this relation it follows that, 
 
 sin" 1 u + cos" 1 v = sin" 1 (uv Vl w 2 Vl v 2 ). 
 
 85. EXAMPLES 
 
 Find the value of each of the following : 
 
 1. sin-^VS. 3. tan" 1 !. 
 
 2. cos-^-iVS). 4. tan cot" 1 4. 
 
 5. sin cot" 1 4. 
 
INVERSE FUNCTIONS 83 
 
 Express the following in terms of u and v: 
 
 6. cos cot" 1 w. 17. cos (2 cos' 1 w). 
 
 7. sec cot" 1 u. 18. cos (2 skr 1 u). 
 
 8. esc cot" 1 u. 19. cos (2 tan" 1 u). 
 
 9. cos sin" 1 u. 20. sin (sin" 1 u -f sin" 1 v). 
 
 10. cos tan" 1 u. 21. cos (sin" 1 w 4- sin" 1 v). 
 
 11. cossec^w. 22. tan (tan" 1 u + cot" 1 v). 
 
 12. cos esc" 1 u. 23. tan (sec" 1 u + sec" 1 v). 
 
 13. sin (2 cos" 1 w). 24. cos (see" 1 w + osc" 1 v). 
 
 14. tan (2 tan- 1 w). 25. sin ( cos- 1 w) 
 
 15. tan (2 sec- 1 w). 26. cos (J cos' 1 u). 
 
 16. tan (2 cos- 1 w). 27. sin (i sec' 1 w). 
 
 28. sin (| tan- 1 w). 
 Find x in terms of a. 
 
 29. tan" 1 a; = cot" 1 a. 
 
 30. sin - 1 x = tan" 1 a. 
 
 31. cos" 1 x = 2 sin"" 1 a. 
 
 32. cos" 1 # = sin" 1 a + tan" 1 a. 
 
 33. sin" 1 x = -J- sec" 1 a. 
 Find a; in terms of a and &. 
 
 34. tan" 1 x = sin" 1 a + sin" 1 6. 
 
 35. cos" 1 x = sec" 1 a sec" 1 b. 
 
 36. sin" 1 x = 2 cos" 1 a -f- cos" 1 6. 
 
CHAPTER VIII 
 OBLIQUE TRIANGLE 
 
 86. In the present chapter we develop the formulas by 
 means of which a triangle may be completely solved when 
 any three independent parts are given. 
 
 87. Law of sines. In a plane triangle any two sides are to 
 each other as the sines of the opposite angles. 
 
 c c 
 
 Let a, b, c be the sides of a triangle and a, ft, y the angles 
 opposite these sides, respectively. 
 
 From the vertex of y draw h perpendicular to the side c, 
 or c produced. 
 
 Then, for each figure, sin a = - , 
 
 b 
 
 and 
 
 siny3=-< 
 a 
 
 Art. 21 
 Art. 29 
 
 Dividing the first equation by the second, we have 
 
 a _ sin a 
 
 b sin/? 
 
 In a similar manner, dropping a perpendicular from the 
 vertex of a, it is seen that 
 
 c sny 
 The last two equations may be written 
 
 a b c 
 
 sin a sin p sin -y 
 84 
 
OBLIQUE TRIANGLE 85 
 
 88. Law of tangents. The tangent of half the difference of 
 two angles is to the tangent of half their sum, as the difference 
 of the corresponding opposite sides is to their sum. 
 From the law of sines we have 
 a _ sin a 
 b sin/3 
 
 By division and composition, this becomes 
 a b _ sin a sin ft 
 a + b sin a + sin/?' 
 
 which reduces to 
 
 a- 6 2cos -( + sina ff) 
 
 a + 6 2 sin 1 (a + J3) cos | (a - ft 
 = cot i (a + J3) tan *- ( - J3) 
 
 tana + a + 
 
 Singly tanHP-V) ^_g, (2) 
 
 tani(p+Y> 6+* 
 and tanj^^ L o) = c--a > (3) 
 
 tan i (-y + a) c + a 
 
 89. Cyclic interchange of letters. Each formula pertaining 
 to the oblique triangle gives rise to two other formulas of 
 the same type by a cyclic interchange of 
 
 letters. A cyclic interchange of letters 
 
 may be accomplished by arranging the 
 
 letters around the circumference of a circle 
 
 as in the figure, and then replacing each 
 
 letter by the next in order as indicated by 
 
 the arrows. Thus by this cyclic interchange of letters 
 
 formula (1) of Art. 88 gives rise to formula (2) ; likewise 
 
 formula (2) gives rise to formula (3). 
 
 90. Law of cosines. The square of any side of a plane tri- 
 angle is equal to the sum of the squares of the other sides 
 minus twice the product of those sides into the cosine of the 
 included angle. 
 
86 
 
 TRIGONOMETRY 
 
 c D 
 
 B 
 
 DA B 
 
 For each figure 
 
 AB = AD + DB, or DB = AB- AD. Art. 2 
 But AB = c, ^4Z> = 6 cos a, hence DB = c b cos a. 
 Also D(7 = b sin a. 
 
 From the right triangle CDB we have 
 
 BC 2 = DC 2 -\- DB 2 
 
 Substituting values, this equation becomes 
 a 2 = (6 sin a) 2 + (c 6 cos a) 2 
 
 = 6 2 sin 2 a + c 2 2 fcccos a + 6 2 cos 2 a 
 = 6 2 (sin 2 a + cos 2 a) + c 2 2 6c cos a. 
 Hence a 2 = 6 2 + c 2 - 2 be cos a. 
 
 Similarly 6 2 = c 2 + a 2 
 and c == fl ~f~ o 
 
 91. To find the sine of half an angle of a plane triangle in 
 terms of the sides of the triangle. 
 
 From equation (1), Art. 72, 
 
 cos 
 
 whence 
 
 From the cosine law 
 
 . cos a. = 
 
 52 + c2 _ 
 2 be 
 
 (1) 
 
 (2) 
 
OBLIQUE TRIANGLE 87 
 
 
 
 From equations (1) and (2) 
 
 2 be 
 
 2 be 
 (a 6 
 
 2 be 
 Let a + b -f c = 2 s. (4) 
 
 Subtracting 2 a, 26, and 2 c from each member of (4) we 
 have respectively 
 
 -a + 6 + c = 2( S -a) 
 
 a -f- 6 c = 2 (s c). 
 Then equation (3) becomes 
 
 a 
 
 Similarly sin = -j- 
 
 ca 
 
 -82. 
 
 ^/in ter 
 
 and 6in? = + J(* -")(*-*). 
 
 2 * a0 
 
 In these formulas the positive sign is given to the radical, 
 since it is known that half an angle of any plane triangle is 
 less than 90. The same applies to the corresponding for- 
 mulas for the cosine and the tangent of half an angle. 
 
 To find the cosine of half an angle of a plane triangle 
 'n terms of the sides of the triangle. 
 
88 
 
 TRIGONOMETRY 
 
 From equation (2), Art. 72, we have 
 
 2cos 2 - = l + cosa. 
 2 
 
 Then 
 
 1 . fr 2 + c 2 - a 2 
 l+___ 
 
 (b + c) 2 - a 2 
 2 be 
 
 26c 
 
 and 
 Hence 
 
 Similarly 
 and 
 
 93. To find the tangent of half an angle of a plane tri- 
 s angle in terms of the sides of the triangle. 
 
 Since 
 
 Similarly 
 and 
 
 sin? 
 tan"- 2 - 
 
 
 (1) 
 (2) 
 (3} 
 
 HUI 
 
 A , a 
 
 tan a + % fe 
 
 _ )($ _ c ) 
 
 2 "" \ 
 
 5(5 - a) 
 
 tan P 4- A /( 5 
 
 - c)(s a) 
 
 2 * 
 
 s(s - b) 
 
 ,n Y - ^fe 
 
 -a)(s-b) t 
 
OBLIQUE TRIANGLE 
 Formula (1) may be written 
 
 89 
 
 Letting 
 
 Similarly 
 and 
 
 tan" 1 ,/(-)(*-&)(* - c >- 
 
 (4) 
 (5) 
 (6) 
 
 (7) 
 
 Ln 2 s-a\ 
 
 r _:,_.J(-)(*-W*-c) 
 
 * * 
 tan a - r 
 
 2 s -a 
 
 tan p - r 
 
 2~-6' 
 tan'' r 
 
 2 s-o 
 
 94. To find the area of a plane triangle in terms of two 
 sides and the included angle. 
 
 Let A be the area and h the altitude. Then in each figure 
 
 A = ^ ch, 
 
 and h = b sin a. 
 
 Therefore A = \ be sin a. 
 
 Similarly A = J ca sin 0, 
 
 and A = \ab sin -y. 
 
 95. To find the area of a plane triangle in terms of a side 
 and two adjacent angles. 
 
 From Art. 94, A = % be sin a. 
 
90 TRIGONOMETRY 
 
 From the sine law , _ csin/? 
 
 siny 
 . _ c 2 sin a sin J3 
 
 2 siny 
 Then since a + ft + y = 180, 
 
 M _c 2 sin a sin p 
 ~2sin(a + p)' 
 
 96. To find the area of a plane triangle in terms of the 
 three sides. 
 
 From Art. 94. 
 
 A = bc sin a. 
 
 Since sin a = 2 sin - cos - . 
 
 2 2' 
 
 A be sin - cos - 
 
 2 2 
 
 Substituting the values of sin ^ and cos ^ as found in 
 Arts. 91 and 92, we have, after reduction, 
 
 -a)( S -b)(s-c). 
 
 97. Formulas for solving an oblique triangle. The formu- 
 las developed in the present chapter are sufficient to solve a 
 plane triangle when three independent parts are given. 
 
 The law of sines 
 
 a b c 
 
 sin a sin ft sin y 
 
 is used when two of the given parts are an angle and the 
 opposite side. 
 
 The law of tangents 
 
 tan i (a- ft) = ^| tan 1 (a + 0) 
 
 a + 6 
 
 is used when two sides and the included angle are given. 
 
OBLIQUE TRIANGLE 91 
 
 If a, 6, and y are the given parts, \ (a + ft) is obtained from 
 the relation a + /? + y = 180. The formula then gives the 
 value of -J (a /3), which, united with the value of % (a 
 gives a and /?. 
 
 TTie ta;s of half angles 
 
 s (s a) s a 
 
 are used when the three sides are given. The last formula 
 is the most accurate, since the tangent varies more rapidly 
 than either the sine or the cosine. The formula involving 
 r is advantageous when all the angles are to be computed. 
 
 TJie law of cosines 
 
 may be used to determine the third side when two sides 
 and an angle are given. It may also be used to determine 
 an angle when the three sides are given. 
 
 This formula is used with natural functions, not being 
 adapted to logarithmic computation. 
 
 98. Check formulas. Any formula which was not used 
 in the solution of the triangle may be used as a check 
 formula. 
 
 The relation a + (3 + y = 180 cannot be used as a check 
 when a problem has been solved by the law of tangents, 
 since the law of tangents involves this relation. 
 
 When two equations from the sine law have been used to 
 find two elements of a triangle, the third equation from the 
 sine law cannot be used as a check, since the first two 
 equations involve the third. 
 
92 
 
 TRIGONOMETRY 
 
 99. Illustrative problems. 
 
 1. Given c = 127.32 
 
 a = 71 58' 22" 
 J3 =52 19' 40" 
 
 SOLUTION. Construction and estimates. 
 
 A 
 A\ 
 
 7 V 
 
 to find a 
 b 
 y . 
 
 a = 140 
 
 6 = 120 
 
 7 = 60 
 
 A 
 
 Outline 
 
 
 c sin a tan * fa ^ a 
 
 C%ecfc 
 _ 5 
 
 
 ian 2 ^ p; 
 sm 7 a 
 
 + 6 
 
 c 
 a 
 ft 
 
 a-6 
 
 a + 6 
 
 
 y 
 
 log a b ) 
 colog (a + 6) 
 
 
 log sin a 
 
 log tan (a + /3) 
 
 
 logc 
 colog sin 7 
 
 log tan (a /3) 
 
 
 log a 
 a 
 
 !>+ 
 
 
 , 
 
 ~ 
 
 sin 7 
 
 r, more compactly, 
 
 log sin /3 
 logc 
 colog sin 7 
 log& 
 6 
 
 
 
 log sin a 
 
 . w> j colog sin 7 
 ^ [ log sin /3 
 log a 
 log& 
 a 
 6 
 
OBLIQUE TRIANGLE 
 
 93 
 
 Filling in the above outline, the completed work appears as follows 
 
 c sin a 
 
 sin 7 
 
 
 c 
 a 
 
 V 
 
 127.32 c , , 
 71 58' 22" 
 52 19' 40" tan*Ca /3) a ~ & tan *fa + /3) 
 
 124 18' 2" a + 6 
 
 55 41' 58" a-6 
 
 24.57 
 268.55 
 62 9' 1" 
 
 log sin a 
 logc 
 colog sin 7 
 log a 
 a 
 
 9.97814-10 a + b 
 2.10490 ?( + ) 
 
 0.08297 log (a -6) 
 
 1.39041 
 7.57098 
 0.27708 
 
 2.16601 colog (a + 6) 
 146.56 log tan (a + /3) 
 
 3 sin/* logtanKa-0) 
 
 9.23847-10 
 9 49' 28" 
 62 9' 1" 
 71 58' 29" 
 52 19' 33" 
 
 Shl7 Ha -I-/9) 
 
 log sin ft 
 logc 
 colog sin 7 
 log 6 
 6 
 
 Or, in the 
 
 9.89846-10 ' r ' a 
 2.10490 ,3 
 0.08297 
 
 2.08633 
 121.99 
 
 more compact form, 
 
 (log sin a 9.97814-10 
 f logc 2.10490 
 "S)j colog sin 7 0.08297 
 -2 1 log sin ft 9.89846 
 
 log a 2.16601 
 log b 2.08633 
 a 146.56 
 b 121.99 
 
 2. Given 
 
 a = 1674.3 
 c = 1021.7 
 = 28 44' 39" 
 
 to find a 
 
 7 
 b. 
 
 Estimates 
 a = 120 
 7 = 30 
 6 = 900. 
 
94 
 
 TRIGONOMETRY 
 
 
 a -f c 
 
 T _ c sin /3 
 sin 7 
 
 a 
 c 
 
 /s 
 
 a c 
 a + c 
 a + 7 
 
 log (a - c) 
 
 colog (a + c) 
 
 log tan (a + 7) 
 
 log tan H-7) 
 
 1674.3 
 1021.7 
 28 44' 39" 
 
 652.6 
 2696.0 
 151 15' 21" 
 
 75 37' 40" 
 
 logo 
 
 log sin /3 
 
 colog sin 7 
 
 log 6 
 
 3.00932 
 
 9.68205-10 
 
 0.27268 
 
 2.96405 
 920.66 
 
 Check 
 
 2.81465 
 
 6.56928-10 
 
 0.59135 
 
 b = 
 
 sin a 
 
 9.97528-10 
 43 22' 12" 
 75 37' 40" 
 
 118 59' 52" 
 32 15' 28" 
 
 log a 
 
 log sin 
 
 colog sin a 
 
 log 6 
 
 3.22384 
 
 9.68205-10 
 
 0.05817 
 
 2.96406 
 
 3. Given 
 
 a = 1.4932 
 b = 2.8711 
 = 1.9005 
 
 to find 
 
 a 
 
 Estimates 
 
 ft 
 
 a= 25 
 /3 = 120 
 
 y- 
 
 7 = 35 
 
 r 
 
 tin 7 r 
 
 2 S - C 
 
 a 
 
 1.4932 
 
 6 
 
 2.8711 
 
 c 
 
 1.9005 
 
 2s 
 
 6.2648 
 
 s 
 
 3.1324 
 
 s a 
 
 1.6392 
 
 s- 6 
 
 0.2613 
 
 * c 
 
 1.2319 
 
 s 
 
 3.1324 
 
 log (s - a) 
 
 0.21463 
 
 log 0-6) 
 
 9.41714 - 10 
 
 log (s - c) 
 
 0.09058 
 
 colog s 
 
 9.50412 - 10 
 
 logr 2 
 
 19.22647 - 20 
 
 logr 
 
 9.61324 - 10 
 
 log tan J 
 
 9.39861 - 10 
 
 log tan | 
 
 0.19610 
 
 log tan J 
 
 9.52266 - 10 
 
 
 
 2 
 
 14 3' 26" 
 
 e 
 
 2 
 
 57 31' 2" 
 
 2 
 2 
 
 18 25' 34" 
 
 a 
 
 28 6' 52" 
 
 ft 
 
 115 2' 4" 
 
 7 
 
 36 51' 8" 
 
 CAecfc : a + + 7 
 
 180 00' 4" 
 
OBLIQUE TRIANGLE 
 
 95 
 
 4. Given 
 
 6 = .0060041 
 c = .0093284 
 = 44 47' 58" 
 
 to find 
 
 Estimates 
 j8= 30 
 a = 105 
 a = .012. 
 
 Check 
 
 sin/3 
 
 6 sin 7 
 
 tin "" ("ft "v} a c tan 1 (OL 1 V^ 
 
 c 
 
 a + c 
 
 b 
 c 
 7 
 
 .0060041 
 .0093284 
 44 47' 58" 
 
 a 
 c 
 a c 
 
 a + c 
 a + 7 
 K + 7) 
 
 .012574 
 .0093284 
 .0032456 
 .021902 
 153 1'48" 
 76 30' 54" 
 
 log& 
 colog c 
 log sin 7 
 log sin /3 
 
 a 
 a 
 
 7.77845 - 10 
 2.03019 
 9.84796 - 10 
 
 9.65660 - 10 
 26 58' 12" 
 71 46' 10" 
 108 13' 50" 
 
 csina 
 
 log(a-c) 
 colog (a + c 
 log tan K a + 7) 
 log tan ( - 7) 
 U-7) 
 
 7 
 
 7.51129 - 10 
 1.65952 
 0.62015 
 
 9.79096 - 10 
 31 42' 51 
 76 30' 54" 
 108 13' 45" 
 44 48' 3" 
 
 sin 7 
 
 logc 
 colog sin 7 
 log sin a 
 log a 
 a 
 
 7.96981 - 10 
 0.15204 
 9.97764-10 
 
 8.09949-10 
 0.012574 
 
 100. The ambiguous case. When two sides and an angle 
 opposite one of them are given, the triangle may admit of 
 no solution, of one solution, or of two solutions. 
 
 Let a, b, a be given. The formula 
 
 sin/?; 
 
 determines 
 
96 
 
 TRIGONOMETRY 
 
 If the calculated 
 sin ft is greater than 1, 
 there can be no solu- 
 tion. 
 
 a < b sin a 
 
 If the calculated 
 sin ft equals 1, ft = 
 90 and there is one 
 solution. 
 
 .5 1 a 
 
 a = 6 sin a 
 
 If the calculated 
 sin ft is less than 1, 
 two supplementary 
 values of ft are deter- 
 mined, giving two so- 
 lutions unless the 
 larger value of ft plus 
 a is equal to or greater 
 than 180. 
 
 a > b sin a 
 
 a<b 
 
 Given 
 
 6 = 420 
 c = 389.73 
 y =53 47' 20" 
 
 to find ft 
 a 
 a. 
 
 Estimates 
 
 /3 = 65 
 a =60 
 a = 390. 
 
 Ttoo solutions 
 
 /S' = 115 
 a' = 10 
 a' = 90. 
 
 a 
 
OBLIQUE TRIANGLE 
 
 97 
 
 Ch'jck of 1st solution 
 
 sin/3 = 
 
 6 sin 7 
 
 tan \ (a - 7) = 
 
 - - 
 a + c 
 
 b 
 c 
 y 
 
 420 
 389.73 
 53 47' 20" 
 
 a 
 c 
 a c 
 
 a + c 
 ft + 7 
 *( + 7) 
 
 440.61 
 389.73 
 50.88 
 830.34 
 119 35' 51' 
 59 47' 50" 
 
 log b 
 colog c 
 log sin 7 
 log sin p 
 ft 
 180 -p = p' 
 P + y 
 
 ' + 7 
 ft 
 ft' 
 
 2.62325 
 7.40924 - 10 
 9.90679 - 10 
 
 9.93928 - 10 
 60 24' 9" 
 119 35' 51" 
 114 11 '29" 
 173 22' 11" 
 65 48' 31" 
 6 37' 49" 
 
 log (a - c) 
 colog (a + c) 
 log tan ( + 7) 
 log tan (a - 7) 
 
 i(-7) 
 
 H+7) 
 
 ft 
 
 7 
 
 1.70655 
 7.08074- 10 
 0.23502 
 
 9.02231 - 10 
 6 0'34" 
 59 47' 50" 
 65 48' 24" 
 53 47' 16" 
 
 Check of 2d solution 
 
 tan \ (7 - a') = 
 
 c + a 
 
 tan | (7 + 
 
 c sin ft 
 
 c 
 a' 
 c a' 
 c + a' 
 7 + ft' 
 
 * (y + f ) 
 
 389.73 
 55.771 
 333.96 
 445.50 
 60 25' 9" 
 30 12' 34" 
 
 sin 7 
 , c sin a' 
 
 sin 7 
 
 log sin a 
 
 r logo 
 
 II colog sin 7 
 log sin a' 
 log a 
 log a' 
 a 
 a' 
 
 9.96008 - 10 
 2.59076 
 0.09321 
 9.06244 - 10 
 
 log (c - a') 
 colog (c + a') 
 log tan i (7 + a') 
 log tan (7 - a') 
 H7-ft') 
 
 K7 + 
 7 
 ft' 
 
 2.52370 
 7.35115 -10 
 9.76509-10 
 
 9.63994 - 10 
 23 34' 45" 
 30 12' 34" 
 53 47' 19" 
 6 37' 49" 
 
 2.64405 
 1.74641 
 440.61 
 55.771 
 
 FIRST SOLUTION 
 
 p = 60 24' 9" 
 ft = 65 48' 31" 
 a = 440.61 
 
 SECOND SOLUTION 
 
 j8' = 11935'51" 
 a*= 6 37' 49" 
 a' = 55.771 
 
98 TRIGONOMETRY 
 
 101. EXAMPLES 
 
 Solve the following triangles, using a three-place table or 
 a slide-rule. 
 
 1. a =26 2. a = 48 
 
 a = 53 ft 61 
 
 ft = 49 y = 69 
 
 3. a = 73 4. a = 69 
 
 6 = 55 6 = 64 
 
 5. a = 54 40' 6. a = 163 
 
 6 = 122 6 = 241 
 
 c = 110 y = 34 20' 
 
 7. a = 42 8. 6 = 115 
 
 6 = 28 c = 96 
 
 y = 72 a = 110 
 
 9. a = 51 10. a = 8.03 
 
 6 = 63 6 = 6.42 
 
 c = 70 c = 7.15 
 
 11. a = 20 12. y = 4350' 
 
 a = 15 a = .34 
 
 6 = 25 c = .30 
 
 Solve the following triangles : 
 
 ^13. 6 = 63.67 14. 6' =20.007 
 
 ft = 100 10' a = 40 27' 30" 
 
 = 400'10" y = 42 30' 15" 
 
 15. a = 238.61 . NsJ6. a = 8.0038 
 
 6 = 216.77 6 = 4.6259 
 
 c = 98.435 c = 4.3167 
 
 17. 6 = .76328 ^4. 8 - 6 = 85.249 
 
 c = 2.4359 c = 105.63 
 
 y = 120 46' 18" a = 50 40' 24" 
 
OBLIQUE TRIANGLE 
 
 99 
 
 19. a = 1.4562 
 c = .45296 
 /3 = 74 19' 38" 
 
 21. a = 2.1469 
 6 = 3.2824 
 j^= 4.0026 
 
 23. 6 = .06532 
 c = .01846 
 y = 8 0' 20" 
 
 25.; a = 764.38 
 
 = 143 18' 31" 
 13 = 13 34' 26" 
 
 20. a = 
 
 6 = 
 c = 
 
 22. c 
 
 a 
 a 
 
 24. 6 
 
 c 
 
 26. a 
 
 6 
 
 c 
 
 83.831 
 56.479 
 74.025 
 
 7.2693 
 .54871 
 5 41' 30" 
 
 10.246 
 18.075 
 33 30' 5" 
 
 962.27 
 637.34 
 655.80 
 
 Find the areas of the following triangles 
 
 28. a = 
 
 c = 
 
 30. a = 
 
 6 = 
 
 \32. c = 
 
 a = 
 
 ^54. 6 = 
 
 27. a = 15 
 
 6 = 20 
 c = 25 
 
 29. a = 20.46 
 6 = 19.72 
 c = 15.04 
 
 31. 6 = 3.46 
 c = 4.09 
 a = 56 10' 
 
 33. a = 48 
 
 v = 43 
 
 172 
 103 
 141 
 
 18.3 
 22.4 
 32 
 
 435.3 
 
 289.6 
 31 7' 
 
 10.34 
 
 83 22' 
 60 40' 
 
 35. The horizontal distance from a point on top of a 
 tower to a distant flagpole is 468 ft. The angle of elevation 
 of the top of the flagpole is 5 8' 30.". The angle of depres- 
 sion of the foot of the pole is 15 36'. Find the height of 
 the flagpole. 
 
100 TRIGONOMETRY 
 
 36. A tower is situated on a hill which inclines at an 
 angle of 23 19' 10" to the horizontal. The angle of eleva- 
 tion of the top of the tower, from a point on the hillside, 
 was measured and found to be 43 39' 50". At a point 75.5 
 ft. farther down, the angle of elevation was found to be 
 39 23' 20". How high is the tower ? 
 
 37. From a point 5 miles from one end of a lake and 
 3 miles from the other end, the lake subtends an angle of 
 47 34' 30". Find the length of the lake. 
 
 38. Find the altitudes of a triangle whose sides are 125.4, 
 230.6, and 179.8. 
 
 39. A flagpole stands on the summit of a hill. The hill 
 inclines 32 18' 20" to the horizontal. At a point 25.5 ft. 
 from the base of the flagpole, measured along the incline, 
 the angle subtended by the flagpole was found to be 41 24'. 
 Find the height of the flagpole. 
 
 Two observers, A and B, 
 stationed 4000 ft. apart, at the same 
 instant observe the angles BAG and 
 CBA to an automobile traveling on 
 a straight road. Three minutes 
 later they measure the angles DAB and ABD to the second 
 position of the automobile. 
 
 If 
 
 Z CBA = 32 8', 
 2. DAB = 40 12', 
 
 what is the rate of the automobile ? 
 
 41. A man wishes to measure the length of a lake from 
 his position on a hill top 185 ft. above the level of the lake. 
 He finds the angles of depression of the ends of the lake to be 
 6 18' and 2 30'. The angle subtended by the lake, formed 
 
OBLIQUE TRIANGLE 
 
 101 
 
 by the two lines of sight, is 66 27'. 
 the lake. 
 
 Find the length of 
 
 (42) The angles of elevation of a cloud, directly above a 
 straight road, from two points of the road on opposite sides 
 of the cloud, are 78 15' 20" and 59 47' 40". Find the height 
 of the cloud, the distance between the two points of observa- 
 tion being 5000 ft. 
 
 43. Two observers at A and 
 B. whose longitudes are the 
 same, simultaneously observe 
 the moon and find the angles 
 ZAM&vdi Z'BMto be 35 2' 20" 
 and 51 17' 10" respectively. 
 The latitude of A (Greenwich) 
 is N, 51 17' 15" and the lati- 
 tude of B (Cape of Good Hope) is S. 33 45' 16". The moon 
 is in the plane determined by A, E } and B. Find EM, the 
 distance from the center of the earth to the moon, the 
 radius of the earth being 3960 miles. 
 
 44. Show that twice the radius, 2 Jf2, of a 
 circle circumscribing a triangle is. given by 
 the equations 
 
 2 _ a _ b _ c 
 
 sin a sin ft sin y 
 SUGGESTION. ZBAC=DOC. 
 45. Find the radius of a circle in- 
 scribed in a triangle whose sides are 
 given. 
 
 SOLUTION. Representing the area of the 
 triangle by A, we have 
 
 A \ ; ar + br + J cr = rs. 
 
 But A = Vs(s - a) (-&)(*- c). Art. 96. 
 
 Therefore r =J(* - )(* ~ *)(* - <0. See Art. 93. 
 
 \ s 
 

 
 MISCELLANEOUS EXERCISES 
 
 102. 1. An angle of 4 radians, having its vertex at the 
 center of a circle, intercepts an arc of 7 inches; find the 
 radius of the circle. 
 
 2. Express the following in degrees : \ radians, -^ radi- 
 
 o o 
 
 STT ,, 
 
 ans, -j- radians. 
 
 3. Express the following in radian measure: 40, 55, 
 38, 52 16'. 
 
 4. Eeduce f radians to degrees. 
 
 5. Find the number of degrees in a central angle which 
 intercepts an arc of 5 feet in a circle whose radius is 8 feet. 
 
 6. An arc of 12 inches subtends a central angle of 50 ; 
 find the radius of the circle. 
 
 7. The number of minutes in an angle is 1\ times the 
 number of degrees in its supplement ; find the number of 
 radians in the angle. 
 
 8. An angle exceeds another by ^ radians, and their sum 
 
 is 160 ; express each angle in radian measure. 
 
 9. The angles of a triangle are to each other as 2 : 3 : 4 ; 
 express each angle in radian measure. 
 
 10. The angles of a triangle are in arithmetical progres- 
 sion, and the mean angle is twice the smallest ; express each 
 angle in radian measure. 
 
 102 
 
MISCELLANEOUS EXERCISES 103 
 
 11. The circumference of a circle is divided into 7 parts 
 in arithmetical progression, the greatest part being 10 times 
 the least ; express in radians the angle which each arc sub- 
 tends at the center. 
 
 12. An arc of 40 on a circle whose radius is 6 inches is 
 equal in length to an arc of 25 on another circle ; what is 
 the radius of the latter circle ? 
 
 Prove each of the following : 
 
 13. sin cos a = sin 3 a cos a -f cos 3 a sin a. 
 
 14. cot a esc a = . 
 
 15. 
 
 sec a cos a 
 1 + tan 2 a _ sin 2 a 
 1 + cot 2 a cos 2 a 
 
 1/S a , SID. ft COS ft 
 
 16. sm + - ^ = ___cos& 
 
 cot ft 1 1 tan ft 
 
 17. sin 6 a -\- cos 6 a = 1 3 sin 2 cos 2 a. 
 
 18. cot a tan cc == esc a sec a (1 2 sin 2 a). 
 
 19 1 ~ tan fi _ Got ft -I 
 1 + tan ~~ cot/2-f l' 
 
 20. sec 4 a tan 4 a = 2 sec 2 a 1. 
 
 21. tan 15 = 2 -V3. 
 
 22. sin 4 a; = 4 (cos 3 cc sin a? sin 3 x cos #). 
 
 23. cos 4 # = 4 cos 4 # 4- 4 sin 4 x 3. 
 
 24. sin 5 x = 16 sin 5 x 20 sin 3 x -f- 5 sin a?. 
 
 25. cos 5 a? = 16 cos 5 x 20 cos 3 x + 5 cos x. 
 
 26. sin (a + ft -f y) = sin a cos /? cos y 
 
 + cos a sin /? cos y 
 + cos a cos (3 sin y 
 sin a sin /? sin y. 
 
 27 tan(a-f-#+ x _ tan ot+ tan ^ + tan y tan tan ft tan y 
 1 tan /3 tariy tan y tan a tan a tan ft' 
 
104 TRIGONOMETRY 
 
 28. 2 sin (a - ft) cos a = sin (2 a - ) - sin/3. 
 
 29. cos2(*:= 1 
 
 V2 
 31. 
 
 (* ) V2 
 
 32. 1 
 
 4 1 - tan $ 
 
 33. tan a + tan ft = sin O + ) 
 
 cos a cos /3 
 
 34. cot + cot^= s -H^^l. 
 
 sin a sin /? 
 
 35. cos (a -f ft) cos (a ft)= cos 2 a sin 2 )8. 
 
 I. COS0 = 
 
 1 + tan 2 1 
 37. 
 
 cot tan ^ 
 
 38. tan + cot = 2 cosec 2 0. 
 
 39. C0s " + sin '* 
 cos a sin a 
 
 40. 
 
 cos (45 - 0) 
 41. sin (a + ft) cos (-/?) +cos (a+ft) sin (a-)8) = sin 2 a. 
 
 cos a -f- cos yS 
 
MISCELLANEOUS EXERCISES 105 
 
 43. tan 2 tan a = tan a sec 2 a. 
 
 44. sin = 2cos2 K 
 
 cot^a 
 
 45. cos 3 a cos a + sin 3 a sin a = cos 2 a. 
 
 4, 
 
 [sin [-] 4. cos [ ^-oTj = sin a + cos /?. 
 \ A ) \ * yj 
 
 47. 4 sin sin (60 - 0) sin (60 + 0) = sin 3 0. 
 48. 
 
 2sin0 + sin20 
 
 Solve each of the following equations for all values of 
 the unknown quantity less than 360. 
 
 49. tan 0= sin 0. 
 
 50. (3-4cos 2 a)cos2a=0. 
 
 51. sin a + cos a cot a = 2. 
 
 52. tan (45 + 0) = 3 tan (45 -0). 
 
 53. 5 sin = tan 0. 
 
 54. 1 -f sin 2 a = 3 sin a cos a. 
 
 55. 2 cos x + sec x = 3. 
 
 56. tan 4 2/-4tan 2 2/-h3 = 0. 
 
 57. sec J3 4- tan ft = 2. 
 
 58. 2 sin a + 5 cos a = 2. 
 
 59. sin 5 x -\- sin 3 x = 0. 
 
 60. cos 7 x cos a? = 0. 
 
 61. tan60 = l. 
 
 62. tan 20 tan = 1. 
 
 63. sin40 + sin20-f cos0 = 0. 
 
106 TRIGONOMETRY 
 
 64. sin20-2cos0 + 2sin0-2 = 0. 
 
 65. sin 2 + sin30 = 2sin0cos20 J. 
 
 66. tan20cot0-tan20 + cot0-l = 0. 
 
 68. sec 2 esc 2 + 4 = 4 esc 2 + sec 2 0. 
 
 69. Given tan (3 = u, find sin 2 /?. 
 
 70. Given tan = esc 2 0, find cos 0. 
 
 71. Given cos x = -|, find cos i #. 
 
 72. Given cos x = f , find tan i x and tan 2 a?. 
 
 73. Given tan 2x = m, find tan x. 
 
 74. If + p + y = 180, show that 
 
 tan a + tan /? + tan y = tan a tan /? tan y. 
 
 75. If a + ft + y = 180, show that 
 
 cot ^ + cot f + cot 2 = cot - cot cot 2 . 
 
 ^ ^ 2 222 
 
 76. If a + + y = 180, show that 
 
 sin a + sin ft + sin y = 4 cos cos ^ cos 2 
 
 222 
 
 Prove the following : 
 
 77. tan-'i + tan- 1 !^. 
 ^^S 4 54 
 
 78. tan" 1 - + tan" 1 
 
 6 
 
 79. tan- 1 k + tan- 1 1 = 
 
 1 kl 
 80. 
 
 81. 
 
MISCELLANEOUS EXERCISES 107 
 
 82. sin- 1 
 
 83. cos- 1 
 
 m + n * n 
 
 m _, / n 
 
 n 
 84. cot- 1 fe2 ~ 2 - = 2csc- 1 fc. 
 
 Solve the following for y: 
 85. 
 
 86. sm- 1 2/ + sin- 1 2;y = - 
 
 2 
 
 87. cot -1 3 y 
 
 88. tan' 1 y = sin" 1 a + cos" 1 b. 
 
 89. Prove that in a plane triangle, right-angled at y, 
 
 90. In a right triangle, c being the hypothenuse, prove 
 that 
 
 . 9 1 c b 9 1 
 
 l - 
 
 91. In an isosceles triangle in which a = 6, prove that 
 
 cos = ^; vers y = ^ 2 . 
 
 92. In a triangle in which y = 60, prove that 
 
 93. Show that the area of a regular polygon, inscribed in 
 
 i u j* . Tir 2 . 2?r 
 
 a circle whose radius is r, is -^- sin -- 
 
108 TRIGONOMETRY 
 
 94. Show that the area of a regular polygon, circum- 
 scribed about a circle whose radius is r, is nr 2 tan -. 
 
 
 95. Show that the area of a regular polygon of n sides is 
 -j- cot -i a being the length of a side. 
 
 96. Prove that the areas of an equilateral triangle and of 
 a regular hexagon, of equal perimeters, are to each other as 
 2:3. 
 
 97. A flagpole 125 ft. high, standing on a horizontal 
 plane, casts a shadow 250 ft. long ; find the altitude of the 
 sun. 
 
 98. From the top of a rock that rises vertically 325.6 ft. 
 out of the water, the angle of depression of a boat was found 
 to be 24 35' ; find the distance of the beat from the rock. 
 
 99. A balloon is directly above a station A. From a 
 second station B, in the same horizontal plane with A, the 
 angle of elevation of the balloon is 60 30'. If AB = 300 ft., 
 find the height of the balloon. 
 
 * 100. From a tower 64 ft. high the angles of depression 
 of two objects in the same horizontal line with the base of 
 the tower and on the same side of the tower, are measured 
 and found to be 28 14' and 42 47' respectively; find the 
 distance between the objects. 
 
 101. Find the area of a triangular field whose sides are 
 48 rods, 62 rods, and 74 rods. 
 
 *** 102. The earth subtends an angle of 17" at the sun ; find 
 the distance of the sun from the earth, the radius of the 
 earth being 3960 mi. 
 
 103. A vertical tower makes an angle of 113 12' with the 
 inclined plane on which it stands ; and at a distance of 89 ft. 
 8 in. from its base, measured down the plane, the angle sub- 
 tended by the tower is 23 26'. Find the height of the tower. 
 
MISCELLANEOUS EXERCISES 109 
 
 104. In a circle of radius 8, find the area of a sector with 
 an arc of 50. 
 
 105. In a circle of radius 12, find the area of a segment 
 with an arc of 132. 
 
 106. In a circle of radius r, find the area of a sector with 
 an arc of 1 radian. 
 
 107. The area of a square inscribed in a circle is 200 sq. ft. ; 
 find the area of an equilateral triangle inscribed in the same 
 circle. 
 
 '* 108. Find the area of a triangle of which two sides are 
 6 + V5 and 6 - V5, the included angle being 32 12'. 
 
 109. At what latitude is the radius of the circle of latitude 
 equal to 1 the radius of the earth ? 
 
 110. From the top of a lighthouse 85 feet high, standing 
 on a rock, the angle of depression of a ship was 3 38', and 
 at the bottom of the lighthouse the angle of depression was 
 2 43' ; find the horizontal distance of the vessel and the 
 height of the rock. 
 
 111. At a point directly south of a flagpole, in the hori- 
 zontal plane of its base, I observed its elevation, 45; then 
 going east 200 ft. its elevation was 35. Find the height 
 of the flagpole. 
 
 112. A castle and a monument stand on the same hori- 
 zontal plane. The angles of depression of the top and the 
 bottom of the monument viewed from the top of the castle 
 are 40 32' 18" and 80 17' 46", and the height of the castle 
 is 104f feet. Find the height of the monument. 
 
 113. At the distance a from the foot of a tower the angle 
 of elevation a of the top of the tower is the complement of 
 the angle of elevation of a flagstaff on top of the tower; 
 show that the length of the staff is 2 a cot 2 a. 
 
110 TRIGONOMETRY 
 
 114. A flagstaff a feet high is on a tower 3 a feet high ; 
 the observer's eye is on a level with the top of the staff, 
 and the staff and tower subtend equal angles. How far is 
 the observer from the top of. the staff? 
 
 115. Two towers on a horizontal plane are 120 feet apart. 
 A person standing successively at their bases observes that 
 the angular elevation of one is double that of the other; 
 but, when he is halfway between them, the elevations are 
 complementary. Find the heights of the towers. 
 
 116. An observer sailing north sees two lighthouses 8 
 miles apart, in a line due west ; after an hour's sailing one 
 lighthouse bears S. W., and the other S. S.W. Find the 
 ship's rate. 
 
 117. From the top of a house 42 ft. high, the angle of 
 elevation of the top of a pole is 14 26' 9"; at the bottom of 
 the house it is 23 21' 33" ; find the height of the pole. 
 
 118. From the top of a hill I observe that the angles of 
 depression of two successive milestones in the horizontal 
 plain below, in a straight line before me, are 14 20' 24" 
 and 5 31' 14". Find the height of the hill. 
 
 119. Along the bank of a river is measured a line 500 
 ft. in length ; the angles between this line and the lines 
 of sight from its extremities to an object on the opposite 
 bank are 53 1' 7" and 79 44' 55". Find the breadth of 
 the river. 
 
 120. A cape bears N. by E. (1 of 90 E. of K), as seen 
 from a ship. The ship sails N. W. 30 miles and then the 
 cape bears E. How far is it from the second point of 
 observation ? 
 
 121. Find the height of a precipice, its angles of eleva- 
 tion at two stations in a horizontal line with its base being 
 39 30' and 34 15' and the distance between the stations 
 being 145 ft. 
 
MISCELLANEOUS EXERCISES 111 
 
 122. From the summit of a hill, 360 ft. above a plain, 
 the angles of depression of the top and bottom of a tower 
 standing on the plain were 41 and 54 ; required the height 
 of the tower. 
 
 123. At one side of a canal is a flagstaff 21 feet high 
 fixed on the top of a wall 15 feet high; on the other side 
 of the canal, at a point on the ground directly opposite, the 
 flagstaff and the wall subtend equal angles. Find the width 
 of the canal. 
 
 124. From the top C of a cliff 600 feet high, the angle of 
 elevation of a balloon B was observed to be 47 22', and the 
 angle of depression of its shadow S upon the sea was 61 
 10' ; find the height of the balloon, the altitude of the sun 
 being 65 31' and J5, S, C being in the same vertical plane 
 and the sun being behind the observer. 
 
 125. At each extremity of a base AB = 758 yards, the 
 angles between the other extremity and two objects C and 
 D were observed, viz. CAB = 103 50' 41", DAB = 53 17' 
 24", DBA = S5 47' 30", and CBA = 6 13' 27"; find CD. 
 
 126. Show that if r be the radius of the earth, h the 
 height of the observer above the sea, and d the angle of 
 
 depression of the horizon ; then tan d = ^ - 
 
 127. A privateer lies 12.75 miles S. W. of a harbor, and 
 a merchantman leaves the harbor in a direction E. by S., at 
 the rate of 10 miles an hour ; on what course and at what 
 rate must the privateer sail in order to overtake the mer- 
 chantman in \\ hours ? 
 
 128. From the top of a hill the angles of depression of 
 two objects in the plain at its base were observed to be 45 
 and 30, and the horizontal angle between them is also 30; 
 find the height of the hill in terms of the distance a be- 
 tween the objects. 
 
112 TRIGONOMETRY 
 
 V-^129. The topmast, 120 feet above the water line of a 
 man-of-war coming into port at the rate of 10 miles an hour, 
 was first seen on the horizon at 8.45 A.M. by a person swim- 
 ming near the water's edge; and at 10.06 A.M. she cast 
 anchor. Find an approximate value for the radius of the 
 earth. 
 
 130. A railway curve which is a circular quadrant has 
 telegraph poles at its extremities and at equal distance 
 along the arc, the whole number of poles being 10. A per- 
 son in one of the extreme radii produced and at a distance 
 of 300 feet from its extremity, sees the third and sixth 
 poles in line. Find the radius of the curve. 
 
CHAPTER IX 
 
 DE MOIVRE'S THEOREM WITH APPLICATIONS 
 
 103. In the present chapter De Moivre's theorem is 
 introduced with some of its applications, including the 
 demonstration of the fundamental series by means of which 
 we may calculate the trigonometric tables. 
 
 A few preliminary considerations pertaining to the com- 
 plex number furnish the necessary basis. 
 
 104. Geometric representation of a complex number. In 
 algebra it is shown that every complex number can be re- 
 duced to the form a + b V 1 or a + ib where a and b are 
 any real numbers, and i=V 1. 
 
 Having given a complex number a + ib, we can construct 
 a point P whose coordinates are y 
 
 a and b. This point may be 
 considered the geometric repre- 
 sentation of the complex num- 
 ber. It is thus seen that to 
 every complex number there cor- 
 responds a point in the plane. 
 
 Conversely, to every point in 
 the plane there corresponds a 
 complex number, since the coor- 
 dinates of the point represent the two elements, a and 6, of 
 the complex number. 
 
 The line OP, instead of the point P, may be considered 
 the geometric representation of the complex number. 
 
 113 
 
 X'- 
 
114 TRIGONOMETRY 
 
 From the figure it is evident that 
 a = r cos a, 
 
 and b = r sin a. 
 
 Therefore a + ib = r cos a + ir sin a. 
 
 Hence any complex number, a -f ib, can be reduced to the 
 
 r(cos a + t" sin a). (1) 
 
 The form (1) includes all real and all pure imaginary 
 numbers as special cases. Letting a = or 180, (1) reduces 
 to r or r, which represents any real number since r repre- 
 sents any real positive number. Letting a = 90 or 270, 
 (1) reduces to ri, any pure imaginary number. 
 
 The angle XOP or a is the argument of the complex num- 
 ber a -f ib. It is determined by the equation 
 
 tana=*. 
 a 
 
 The distance OP or r is the modulus of the complex num- 
 ber a + ib. It is determined by the equation 
 
 r Va 2 -f- 6 2 . 
 105. To show that 
 
 [r (cos a + i sin )] n = r n (cos n + 1 sin wa), 
 
 91 6eift<7 a positive vnieger. 
 
 Squaring the quantity r (cos a -\-i sin a), we have 
 [r (cos a + i sin a)] 2 = r 2 (cos 2 sin 2 a-\-2i sin a cos a) 
 
 = r 2 (cos 2 a + i sin 2 a) by Art. 71. (1) 
 
 Multiplying each member of (1) by r (cos a + i sin a) we 
 
 have 
 
 [r (cos a + 1 sin a)] 3 = r 3 [(cos 2 a cos sin 2 a sin a) 
 + /(sin 2 a cos a -f- sin a cos 2 a)] 
 = j- 3 (cos 3 a + i sin 3 a) by Art. 68. (2) 
 
 From (2) we have, similarly, 
 
 [r (cos a-\-i sin a)] 4 = r 4 (cos 4 a -f- i sin 4 a). (3) 
 Equations (1), (2), and (3) have the form 
 
 [r (cos + i sin )] tt = r n (cos na + i sin w). (4) 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 115 
 
 Multiplying both members of (4) by r (cos a + i sin a), we 
 have 
 [r (cos a+i sin a)] n+1 = r n+1 [cos (w + 1) <*+i sin (n +!)]. (5) 
 
 Hence, assuming the law expressed in (4) to be true, 
 equation (5) shows that it is true when n is increased by 
 unity. But the law is true for n = 4, by equation (3) ; 
 hence it is true for n = 5. Being true for n 5, it must 
 also be true for n = 6, etc. Hence equation (4) is true for 
 all positive integral values of n. 
 
 It can be shown that equation (4) is still true when n is a 
 negative integer, or a fraction. 
 
 From equation (4) it is seen that the ?ith power of a com- 
 plex number is a complex number having an argument n 
 times the argument of the given number and a modulus 
 equal to the nth power of the given modulus. 
 
 Letting r=l, equations (1), (2), (3), and (4) become 
 
 (cos a + i sin a) 2 = cos 2 a + * sin 2 a (5) 
 
 (cos a + i sin a) 3 = cos 3 a + i sin 3 a (6) 
 
 (cos a + i sin a) 4 = cos 4 a + i sin 4 a (7) 
 
 ( cos a + / sin a) n = cos /ia + / sin /ia. (8) 
 
 The last equation is known as De Moivre's theorem. 
 
 PROB. Show De Moivre's theorem is true when n = 3. 
 
 106. Geometric Interpretation, 
 
 Since each of the complex num- 
 bers cos a + i sin a 
 
 cos 2 a + i sin 2 a 
 
 cos 3 a + i sin 3 a 
 
 cos 4 a + i sin 4 a 
 has a modulus equal to unity, 
 the lines representing these num- 
 bers terminate in points lying 
 on the circumference of a circle whose radius is unity. 
 The arguments of any two consecutive integral powers of 
 cos a + i sin a differ by a, hence the lines representing any 
 two consecutive powers differ in direction by a. 
 
116 TRIGONOMETRY 
 
 107. Applications of De Moivre's theorem. De Moivre's 
 theorem may be used to find the various roots of unity, to 
 extract any root of a complex number, to obtain the sine 
 and cosine of any multiple of an angle, and to expand the 
 sine and cosine of an angle in a series of powers of the 
 angle. 
 
 108. To find the cube roots of unity. 
 
 If the cube roots of unity are real numbers, or complex 
 numbers, we may assume, by Art. 104, that 
 
 VI = r (cos a + i sin a). (1) 
 
 Then, cubing, 1 = r 3 (cos 3 a 4- i sin 3 a). (2) 
 
 Also l = r 3 cos(3a-2n7r) + zV 3 sin(3a-2ri7r), (3) 
 
 n being an integer. 
 
 Equating the real and imaginary parts of equation (3), we 
 have 
 
 and ?* 3 sin (3 a 2 nir) = 0. 
 
 These equations of condition are satisfied when 
 3 a 2 mr = and r = 1, 
 
 whence a =2-^. 
 
 3 
 
 When n = 0, 1, 2, 3, 4, etc. 
 
 - 0, =, if, ^, S- etc., respectively. (4) 
 
 o o o o 
 
 Every angle in this series is coterminal with either 0, -^, 
 
 o 
 
 4 7T 
 
 or -; hence all the values of sin a and cos a are obtained 
 o 
 
 by using only the first three values of a. Substituting 
 these values of a in equation (1), and remembering that r=l, 
 we have 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 117 
 
 ^1=1 forn = 0, 
 
 \/I = -i + iV3 forn = l, 
 </l = - - 1 i V3 f or n = 2. 
 
 The three cube roots of unity 
 are represented geometrically by 
 P 1? P 2 , and P 3 . 
 
 We can now write the three 
 cube roots of any real number 
 a, for letting a l5 2 , and a 3 be the cube roots of a, we have 
 
 Oi =</a- 1, 02 =%(- i + H V3), a 3 =^a (- i - t V3), 
 where -\/a is the arithmetical cube root. 
 
 PROBLEM. Show that ( 1 -f i -J- V3) 3 = 1, thus justifying 
 the assumption made in Eq. 1. 
 
 109. To find the fifth roots of unity. 
 
 Let -v/1 = r (cos a + i sin a). (1 ) 
 
 Then, raising each member to the fifth power, 
 
 1 = r 5 (cos 5 a + i sin 5 a). (2) 
 
 Also 1 = r 5 cos (5 a 2 mi) + it 6 sin (5 a 2 nw), (3) 
 
 n being an integer. 
 
 Equating the real and imaginary parts of equation (3), we 
 have r 5 cos (5 a 2 mi) = 1, 
 
 and r 6 sin (5 a 2 mi) = 0. 
 
 These equations of condition are satisfied when 
 5 a 2 mr = 0, and r = 1. 
 
 Therefore = . 
 
 o 
 
 When w = 0, 1, 2, 3, 4, etc. 
 
 -jr, -jr, -jr> etc., respectively. (4) 
 
118 TRIGONOMETRY 
 
 Substituting the values from (4) in equation (1), and 
 remembering that r = 1, we have 
 
 j-> 2 -v/1 = 1 for n = 0, 
 
 P, ^ 
 
 il + isini? f or n = 2, 
 o o 
 
 5 5 
 
 = cos + isin forn = 4. 
 5 5 
 
 110. To extract the square root of a + ib. 
 Let Va + ib = r (cos a + i sin a). (1) 
 
 Squaring, a + ib = r 2 (cos 2 a + i sin 2 a). (2) 
 
 Also a + ib = ^[003(2 a 2nir)+i sin (2 a 2 rnr)], (3) 
 
 where n is any integer. 
 
 Equating the real and imaginary parts, 
 
 ^008(2 a - 2tt7r)= a, (4) 
 
 r 2 sin(2a-2ri7r) = 6. (5) 
 
 Squaring and adding, we have 
 
 r 4 =a 2 + & 2 , (6) 
 
 and r = Va 2 + & 2 . (8) 
 
 Equation (8) gives the value of r in terms of the known 
 
 numbers a and 6. 
 
 From equations (4) and (7) 
 
 whence 2 a 2 nir cos' 
 
 the quadrant of 2 a 2 n-ir being determined by (4) and (5). 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 119 
 
 Then a = cos- 1 a forn = 0, (9) 
 
 Va 2 + b 2 
 
 and = TT + cos" 1 a for n = 1. (10) 
 
 The values of a for n = 2, 3, 4, etc., are coterminal with 
 the values of a in equations (9) or (10). Hence there are 
 only two values for sin a and cos a ; namely those for which 
 n = and n = l. 
 
 Substituting these values of a in equation (1), we have, 
 finally, 
 
 Va + ib = \/a 2 -f 6 2 [~cos / cos- 1 CT 
 
 and Va-M& = -\/a 2 + 6 2 |cos f + 4 cos' 1 a 
 L V Va 2 + 
 
 + i sin f TT 4 i cos- 1 - - - \\ . (12) 
 V Va 2 + 6V J 
 
 These values of Va + 16, while more complicated than 
 Va H- ib itself, are nevertheless in the standard form of 
 complex numbers, 
 
 r (cos a + i sin a). 
 
 111. To extract the Mh root ofa + ib. 
 
 Let -fya + ib = r (cos a + i sin a). (1) 
 
 Then a + 6 = r* (cos fca + i sin fca) (2) 
 
 = r*[cos (Tea - 2 rnr) + i sin(fca - 2 WTT)], (3) 
 
 where n is an integer. 
 
 Equating the real and imaginary parts of equation (3), we 
 
 have r* cos (ka 2 HIT) = a, (4) 
 
 and i* sin (& 2 WTT) = 6. (5) 
 
120 TRIGONOMETRY 
 
 -Squaring and adding equations (4) and (5), we have 
 
 r 2 * = a 2 + b 2 . (6) 
 
 Then r* = Va 2 + 6 2 , (7) 
 
 and r= ^a 2 + 6 2 . (8) 
 
 From equations (4) and (7), we have 
 
 cos (ka- 
 
 or fca 2 nir = cos' 1 
 
 the quadrant of ka 2 mr being determined by the signs of 
 (4) and (5). 
 
 Therefore = ?^Z[ + ! cos" 1 a 
 k k 
 
 whence a = 
 
 =^ + icos- 1 
 k k 
 
 4?r 1 
 
 c^-r+'-cos- 1 
 
 k k Va 2 + b* 
 
 Substituting the values for r and a as found in equations 
 (8), (10), (11), (12), etc., in equation (1), we have the several 
 kih roots of a + ib. 
 
 When A; is an integer there are k roots. Consecutive 
 
 values of a, as given by (9), differ by , hence all the 
 
 fc 
 
 values of a after the fcth are coterminal with one of the 
 first k values. 
 
 112. To express sin na and cos na in terms of sin a and 
 cosa. 
 
 We have 
 
 cos na + i sin na = (cos a + i sin ) n . 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 121 
 
 Expanding the second member by the binomial theorem 
 and equating the real and imaginary parts, the problem is 
 solved. 
 
 Thus, for sin 4 a and cos 4 a we have 
 cos 4 a 4- i sin 4 a = (cos a -f i sin a) 4 = cos 4 a, 
 -h 4 i cos 3 a sin a 6 cos 2 a sin 2 a 4 i cos a sin 3 a + sin 4 a. 
 Therefore sin 4 a = 4 cos 3 a sin a 4 cos a sin 3 a. 
 and cos 4 a = cos 4 a 6 cos 2 a sin 2 a -f- sin 4 a. 
 
 113. Comparison of the values of sin a, a, and tan a, a being 
 an acute angle. Let a be any acute angle expressed in ra- 
 dians. With the vertex as 
 a center and any radius OB, 
 describe the arc EC. Draw AG 
 and BD perpendicular to OB, 
 and join B with O. 
 
 The area of the triangle OBC 
 is less than the area of the 
 sector OBC, and the sector OBC is less than the triangle 
 OBD. 
 
 But since 
 
 AC= OC sin a, BD = OB tan a, and arc BC = OB a, Art. 17, 
 the area of the triangle OBC is equal to | OB OC sin a, 
 the area of the sector OBC is equal to ^ OB OB - a, 
 and the area of the triangle OBD is equal to |- OB - OB tan . 
 Hence 
 
 or sin a < a < tan a. 
 
 114. Value of for small values of a. In Art. 40 we 
 
 a 
 
 saw that sin a approaches as a approaches 0. The value 
 
 of therefore approaches . as a approaches 0. 
 a (J 
 
122 TRIGONOMETRY 
 
 But from the previous article 
 
 sin a. < a < tan a. 
 Dividing by sin a, we have 
 
 a 1 
 
 sin a cos a 
 
 sin a 
 
 or 1 > > cos a. 
 
 a 
 
 sin a , . . sin a 
 
 Since lies between 1 and cos a, must approach 
 
 a a. 
 
 1 as a approaches 0, since cos a approaches 1. 
 
 Then for very small angles sin a may be replaced by , 
 
 expressed in radians. The error thus introduced is so small 
 
 that it may be neglected in many problems. Thus, to five 
 
 decimal places, 
 
 sin 1 = 0.01745 1 = 0.01745 radians 
 
 sin 2 = 0.03490 2 = 0.03491 radians 
 
 sin 3 = 0.05234 3 = 0.05236 radians 
 
 sin 4 = 0.06976 4 = 0.06981 radians 
 
 115. To develop sin a and cos a in terms of a. 
 
 By De Moivre's theorem, 
 
 cos nO + i sin nO = (cos -f i sin 0) n . 
 
 On expanding the second member by the binomial theorem, 
 we have 
 cos nO + * sin n& = cos n -{-in cos"" 1 sin 6. 
 
 ^j2 -cos n ~ 2 sin 2 ?! j^ ' cos n ~ 3 sin 3 
 
 | n(n-l)(n~2)(n-8) cos ,_ 4gsin4g | ... (1) 
 
 11 
 Equating the imaginary parts, we have ' 
 
 sin n$ = n cos"" 1 sin r^ cos w ~ 3 sin 3 
 
 tt(n-l)(n-2)(n-3)(tt-4) 
 + _A & A A / cos n 
 
 l 
 Let nO = a. Then equation (2) may be written 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 123 
 
 sin a = ? cos"- 1 sin - ^ ^ ^ cos w ~ 3 sin 3 
 
 |3 
 
 
 ---- 
 
 " " 
 
 Let a remain constant while n increases indefinitely. 
 Then necessarily decreases indefinitely, since n0 = a, a 
 
 constant. By Art. 114, when approaches 0, ^ ap- 
 
 proaches 1, and cos 6 approaches 1. Making these substi- 
 tutions in equation (4), we have 
 
 a 3 a 5 a 7 
 
 Equating the real parts of equation (1), we have 
 cos n6 = cos" - n ^~ 1 ) cos"- 2 6 sin 2 6 
 
 e 
 
 n(n- l)(n-2)(tt-3) **-.* /K N 
 
 4. _1 - d\_^ /i -- ? C os n ~ 4 6 sm 4 ^ -. (5) 
 
 By the same process as above, equation (5) becomes 
 
 The series for sin a, and cos a are convergent for all finite 
 values of a* They enable us to compute the sine and 
 cosine of any angle. It is then possible to construct a table 
 of natural functions, from which the logarithmic functions 
 may be obtained. In using these series a must, of course, 
 be expressed in radian measure. 
 
 * See any College Algebra on the convergency of series. 
 
124 TRIGONOMETRY 
 
 116. EXAMPLES 
 
 1. Find the four fourth roots of unity by De Moivre's 
 theorem. 
 
 2. Find the six sixth roots of unity by De Moivre's 
 theorem. 
 
 3. Find the square root of 5 3 i. 
 
 SOLUTION. Let V5 3 i = r (cos a + I'sin a). (1) 
 
 Then 5 - 3 1 = r 2 (cos 2 a + i sin 2 ) 
 
 = r 2 [cos (2 a - 2 WTT) +isin(2a 2nir)]. 
 Equating the real and the imaginary parts, 
 
 r 2 cos (2 a - 2 WTT) = 5, (2) 
 
 and r 2 sin (2 a - 2 WTT) = - 3. (3) 
 
 Squaring and adding (2) and (3), we have 
 
 r* = 34, . . r 2 = V&, and r = ^34. (4) 
 
 Then cos (2 a 2 nir) = 
 
 V34 
 
 and 2 a - 2 nir = 329 2', (5) 
 
 the quadrant being determined by (2) and (3). 
 
 When n = 0, a = 164 31' ; (6) 
 
 = !, = 844 31'. (7) 
 
 Substituting from (4) and (6) in (1), we have 
 \/5^3l = - 2.3271 + .6446 L 
 Substituting from (4) and (7) in (1), we have 
 V5^3l = 2.3271 -.6446i. 
 
 4. Find the square root of 3 -f 4 i. 
 
 5. Find the square root of 3 4 i. 
 
 6. Find the square root of 1 + 2 1. 
 
 7. Find the square root of i. 
 
 8. Find the square root of i. 
 
 9. Find the cube root of 2 3 i. 
 
 SOLUTION. Let \/2 - 3 i - r (cos a + i sin a). (1) 
 
 Then 2 - 3 i =r s (cos 3 a + i sin 3 a) 
 
 (3a-2n7r)]. 
 
DE MOIVRE'S THEOREM WITH APPLICATIONS 125 
 
 Equating the real and imaginary parts, 
 
 r 3 cos (3 a - 2 WTT) = 2 ; (2) 
 
 and r 8 sin (3 a - 2 nir) = - 3. (3) 
 
 Squaring and adding (2) and (3), we have 
 
 i* = 13, .-. r 8 = Vl3~ and r = \/13T (4) 
 
 2 
 Then cos (3 a - 2 WTT) =- , 
 
 and 3 a - 2 nir = 303 41' ; (5) 
 
 the quadrant being determined by (2) and (3). 
 
 When n = 0, a = 101 14' ; (6) 
 
 n = l, a = 221 14'; (7) 
 
 n = 2, a = 341 14'. (8) 
 
 Substituting from (4) and (6) in (1), we have 
 v / 2^3l = - .2987 + 1.5041 i. 
 Substituting from (4) and (7) in (1), we have 
 
 v 2-3i = - 1.1530 - 1.0107 *. 
 Substituting from (4) and (8) in (1), we have 
 
 2/2 - 3 i = 1.4518 - .4933 i. 
 We have thus found the three cube roots of 2 3 i. 
 
 10. Find the cube root of 1 + i. 
 
 11. Find the cube root of 1 +i. 
 
 12. Find the cube root of 2 -f 3 i. 
 
 13. Find the values of sin 3x and cos 3 a; in terms of 
 sin a? and cos a;. 
 
 14. Find the values of sin 5 a; and cos 5 a; in terms of 
 sin x and cos x. 
 
 15. Prove by De Moivre's theorem that 
 
 sin a = 2 sin ~ cos ~> also cos a = cos 2 ~ sin 2 - 
 2, L & & 
 
 I a . a\2 
 SUGGESTION, cos a + i sin a = I cos - + i sm - 1 . f 
 
 16. Show that cos a = cos 3 ^ 3 cos f sin 2 1, 
 
 o o o 
 
 sin a = 3 cos 2 ^ sin ^ sin 3 
 o o o 
 
SPHERICAL TRIGONOMETRY 
 
 CHAPTER X 
 FUNDAMENTAL FORMULAS 
 
 117. The spherical triangle.* Spherical trigonometry 
 treats of the relations between the various parts of a 
 spherical triangle and of the methods of solving the spheri- 
 cal triangle. 
 
 The sides of a spherical triangle 
 are always arcs of great circles. 
 
 Having given a spherical triangle 
 ABC, situated upon a sphere S, a 
 triedral angle 0- ABC may be formed 
 by passing planes through 0, the 
 center of the sphere, and through 
 the sides of the triangle. 
 
 It is known from geometry that the arc AB and the 
 angle AOB contain the same number of degrees, and that 
 the angle CAB and the diedral angle C-AO-B contain the 
 same number of degrees. 
 
 The sides and angles of a spherical triangle may have 
 any values between and 360. 
 
 A triangle having one or more of its parts greater than 
 180 is called a general spherical triangle. 
 
 A triangle having each of its parts less than 180 is called 
 a spherical triangle. 
 
 We shall consider only those triangles whose parts are 
 each less than 180. 
 
 * For a course on the right spherical triangle read Arts. 117 and 126 
 and from Art. 128 to end of Chapter XI. 
 
 127 
 
128 
 
 TRIGONOMETRY 
 
 118. Law of sines. To find the relation between two sides 
 of a spherical triangle and the angles opposite. 
 
 /3 > 90, 6 > 90. 
 
 Given a spherical triangle and its accompanying triedral 
 angle ; through the vertex P pass planes perpendicular to 
 OR and OQ, intersecting in PF a line perpendicular 
 to the plane ORQ. 
 
 Then 
 
 
 sin ft 
 
 w 
 
 
 Also 
 
 sin a 
 
 -- 
 
 *-=& 
 
 therefore 
 Hence 
 Likewise 
 
 sin a 
 
 RP 
 
 
 sin b 
 sin a 
 
 QP 
 
 sin a 
 
 sin/3 
 
 sin 6 
 
 sn y sn c 
 Uniting these equations, we have 
 
 sing _ sin b _ sine 
 sin a sin p sin Y 
 
FUNDAMENTAL FORMULAS 
 
 129 
 
 This demonstration applies to similar figures drawn for 
 all possible cases,* hence the theorem is always true. 
 
 To find the relation between the three 
 
 A 
 
 J 
 v 
 
 119. Law of cosines. 
 
 sides and an angle. 
 
 Given a spherical tri- 
 angle and its accompany- 
 ing triedral angle ; pass 
 a plane through the ver- 
 tex A perpendicular to 
 OA, intersecting the 
 planes of the triedral 
 angle in the lines AB, 
 AC, and BC. 
 
 b < 90, c < 90 ; a < 180, a < 180. 
 
 Then AB = r tan c, OB = r sec c, AC = r tan b, OC = r sec b. 
 From the triangle OBC, by Art. 90, 
 
 B& = (r sec b) 2 + (r sec c) 2 2(r sec &) (r sec c) cos a. (1) 
 Likewise from the triangle ABC, 
 
 BC 2 = (r tan b) 2 + (r tan c) 2 - 2 (r tan 6) (r tan c) cos a. (2) 
 Subtracting (2) from (1), we have 
 
 = r 2 (sec 2 6 - tan 2 6) + r 2 (sec 2 c - tan 2 c) 
 
 2 r 2 sec b sec c cos a + 2 r 2 tan 6 tan c cos <*, 
 which reduces to 
 
 == 1 sec 6 sec c cos a + tan b tan c cos , 
 or cos a = cos b cos c + sin b sin c cos a. (3) 
 
 Also cos b = cos c cos a -f- sin c sin a cos p, (4) 
 
 and cos c = cos a cos 6 + sin a sin 6 cos y. (5) 
 
 * The following seven cases can arise : 
 
 (1) 3 sides < 90, 3 angles < 90 (4) 1 side < 90, 2 angles < 90 
 
 (2) 3 sides < 90, 2 angles < 90 (5) 1 side < 90, 1 angle < 90 
 (8) 2 sides < 90, 2 angles < 90 (6) 1 side < 90, angle < 90 
 
 (7) side < 90, angle <90. 
 
 It is to be understood that all parts not mentioned are greater 
 than 90. 
 
130 
 
 TRIGONOMETRY 
 
 120- To extend the law of cosines. 
 In the derivation of the formula 
 
 cos a = cos b cos c -f- sin b sin c cos a, 
 
 b and c were less than 90, while a and a were less than 180. 
 To show that the formula is true in general, it is necessary 
 to consider two additional cases : 
 
 1st. Both b and c greater than 90. 
 2d. Either b or c greater than 90. 
 
 Since ft and y do not enter the formula, they may have 
 any value consistent with the above conditions. 
 
 First. Given the triangle ABC in 
 which b > 90 and c > 90. 
 
 Extend the sides 6 and c of the tri- 
 angle ABC, forming the lune whose 
 angle is a. Then in the triangle A'BC, 
 the sides A'B and A'C are each less 
 than 90, hence by Art. 119 
 cos a = cos (180 - b) cos (180 - c) 
 
 + sin (180 - 6) sin (180 - c) cos , 
 or cos a = cos b cos c + sin b sin c cos a. 
 Hence the law of cosines holds when 
 both b and c are greater than 90. 
 Second. Given the triangle ABC } 
 in which b < 90 and c > 90. 
 
 Extend the sides a and c of the 
 triangle ABC, forming the lune 
 whose angle is ft. Then in the tri- 
 angle AB'C, the sides AB' and AG 
 are each less than 90, and the 
 angle B'AC is equal to 180 - a. 
 
 Then, by Art. 119, 
 cos (180 - a) = cos b cos (180 - c) 
 
 + sin b sin (180 - c) cos (180 - a), 
 or cos a = cos b cos c -f sin b sin c cos a. 
 
FUNDAMENTAL FORMULAS 131 
 
 Hence the law of cosines holds when either b or c is greater 
 than 90. The law of cosines is therefore true in general. 
 
 121. To find the relation between one side and the three 
 angles. 
 
 Let a, b, and c be the sides of any 
 spherical triangle, and a', b' } c' the sides 
 of its polar triangle. 
 
 Applying the law of cosines to the 
 polar triangle, we have 
 cos a' = cos b' cos c' + sin b' sin c' cos a'. 
 
 But 
 
 a' = 180 - a, b'= 180 - /?, etc. 
 
 Therefore 
 cos (180 - a) = cos (180 - /?) cos (180 - y) 
 
 + sin (180 - ) sin (180 - y) cos (180 - a), 
 or cos a = cos /3 cos y + sin /? sin y cos a. (1) 
 
 Also cos (3 = cos y cos a + sin y sin a cos 6, (2) 
 
 and cos y = cos a cos ft + sin a sin /? cos c. (3) 
 
 122. The sine- cosine law. To find the relation between 
 three sides and two angles. 
 
 We have 
 
 cos a = cos 6 cos c + sin b sin c cos a, (1) 
 
 and cos b = cos c cos a -f sin c sin a cos /?. (2) 
 
 Eliminating cos a by substitution, 
 
 cos b = cos b cos 2 c + sin b sin c cos c cos a + sin c sin a cos /?. 
 Transposing and factoring, 
 
 cos b (1 cos 2 c) = sin b sin c cos c cos a -f- sin a sin c cos /?. 
 Eeplacing (1 cos 2 c) by sin 2 c, and dividing by sin c, we 
 have 
 
 cos b sin c = sin b cos c cos a + sin a cos ft. 
 
 Kearranging terms, 
 
 sin a cos /3 = cos 6 sin c sin 6 cos c cos . (3) 
 
132 TRIGONOMETRY 
 
 Also sin 6 cos y = cos c sin a sin. c cos a cos ft, (4) 
 
 and sin c cos a = cos a sin b sin a cos 6 cos y. (5) 
 
 Interchanging ft and y and consequently b and c, we have 
 from (3) 
 
 sin a cos y = cos c sin b sin c cos 6 cos a. (6) 
 
 Similarly from (4) 
 
 sin 6 cos a = cos a sin c sin a cos c cos /?, (7) 
 
 and from (5) 
 
 sin c cos ft = cos 6 sin a sin & cos a cos y. (8) 
 
 123. To find the relation between two sides and the three 
 angles. 
 
 Applying the sine-cosine law to the polar of the given 
 triangle, we have 
 
 sin a' cos ft' = cos b f sin c' sin b ! cos c r cos ex! . 
 But a'= 180 - a, 0' = 180- 6, etc. 
 Then 
 sin (180 -a) cos (180-&)=cos (180 -0) sin (180 -y) 
 
 -sin (180- /?) cos (180- y) cos (180- a). 
 
 Therefore sin a cos 6 = cos ft sin y -f- sin ft cos y cos a. (1) 
 
 Also sin /? cos c = cos y sin a + sin y cos a cos &, (2) 
 
 sin y cos a = cos a sin /3 -f sin a cos /? cos c, (3) 
 
 sin a cos c = cos y sin ft + sin y cos /? cos a, (4) 
 
 sin ft cos a = cos a sin y 4- sin a cos y cos b, (5) 
 
 sin y cos b = cos /3 sin a + sin /? cos a cos c. (6) 
 
 124. To find the relation between two sides and two angles t 
 one of the angles being included between the given sides. 
 
 From Art. 122 
 
 sin a cos ft = cosb sin c sin 6 cos c cos a. 
 Dividing this equation by 
 
 sin a sin ft = sin b sin a, Art. 118 
 
FUNDAMENTAL FORMULAS 133 
 
 member by member, we have 
 
 , Q _ cot b sin c cos c cos a 
 
 sin a 
 
 Therefore sin a cot (3 = cot b sin c cos c cos a. , (1) 
 
 Similarly sin ft cot y = cot c sin a cos a cos ft (2) 
 
 and sin y cot a = cot a sin b cos 6 cos y. (3) 
 
 Interchanging a and ft and consequently a and 6, we 
 
 have from (1) 
 
 sin /? cot a = cot a sin c cos c cos ft (4) 
 
 Similarly from (2) 
 
 sin y cot ft = cot b sin a cos a cos y, (5) 
 
 and from (3) 
 
 sin a cot y = cot c sin 6 cos b cos a. (6) 
 
 125. Formulas independent of the radius of the sphere. It 
 
 will be noticed that r, the radius of the sphere, does not 
 enter any of the formulas thus far developed ; hence they 
 are independent of the radius of the sphere, and may be 
 applied, without modification, to triangles on any sphere. 
 The fact is serviceable in problems of Astronomy and Ge- 
 odesy where the formulas are applied to triangles situated 
 upon the celestial and terrestrial spheres. 
 
CHAPTER XI 
 SPHERICAL RIGHT TRIANGLE 
 
 126. The spherical right triangle is a spherical triangle 
 one of whose angles is a right angle. The other parts may 
 have any values between and 180. 
 
 In the work that follows the angle y will be taken as the 
 right angle. 
 
 127. Formulas for the solution of right triangles. The 
 formulas for the solution of any spherical right triangle are 
 obtained from the general formulas of Chapter X by letting 
 y = 90. We thus have 
 
 from eq. (1) Art. 118 sin a = sin c sin a 
 
 from-eq. (1) Art. 118 sin b = sin c sin /3 
 
 from eq. (5) Art. 119 cos c = cos a cos b 
 
 from eq. (1) Art. 121 cos a = sin ft cos a 
 
 from eq. (2) Art. 121 cos ft = sin a cos b 
 
 from eq. (3) Art. 121 cos c = cot a cot ft 
 
 from eq. (2) Art. 124 cos ft = tan a cot c 
 
 from eq. (3) Art. 124 sin b = cot a tan a 
 
 from eq. (5) Art. 124 sin a =cot ft tan b 
 
 from eq. (6) Art. 124 cos a = tan b cot c. 
 
 128. Direct geometric derivation of formulas. Let a and b 
 be the sides of a given spherical right triangle, a and ft the 
 angles opposite, and c its hypotenuse. 
 
 Let 0-ABC be its accompanying triedral angle. 
 
 Through the vertex B pass a plane perpendicular to OA, 
 intersecting the planes of the triedral angle in AB } BC, 
 and CA. 
 
 134 
 
SPHERICAL RIGHT TRIANGLE 
 
 135 
 
 Then Z BAO=a, Z BOC=a, Z AOC=b, and Z.AOB=c. 
 Also Z jB(L4, Z BOO, ZCAO,^BAO are each a right angle. 
 From the triangle ABC we have 
 OB 
 
 ; sina ? (1) 
 
 OB 
 AC 
 
 AC OA tan 6 
 
 cos a. = = -pr- = , 
 
 AB AB tan c 
 
 OA 
 CB 
 
 CB OC tana 
 tana = __ = ^ = ___. 
 
 OC 
 
 By interchanging a and /? and consequently a and 6, or 
 by passing a plane through D JL to OB and proceeding as 
 above, we have 
 
 (2) 
 
 (3) 
 
 o 
 cos/? = 
 
 sin c 
 
 tana 
 - -- , 
 tan c 
 
 sin a 
 Also from the figure. 
 
 OA 
 
 OA OC cos b 
 = OB = 0^ = seca 
 OC 
 
 Dividing (1) by (5) and reducing by (7) we have 
 
 , 
 
 COS 
 
 and by interchange of letters as before 
 
 (5) 
 (6) 
 
 (7) 
 (8) 
 
 sin (3 
 
 cos a 
 cos a 
 
136 
 
 TRIGONOMETRY 
 
 Substituting the values of cos a and cos b from (8) and (9) 
 in (7), we have, after reduction, 
 
 cos c = cot a cot ft. (10) 
 
 In the demonstration of formulas (1) to (10) the parts 
 a, ft, a, b, and c were assumed less than 90. To show that 
 these formulas are true in general, it is necessary to con- 
 sider two additional cases : viz. (1) when one side and the 
 hypotenuse are each greater than 90, and (2) when the two 
 sides are each greater than 90. 
 
 1. Wlien a > 90 and c > 90. 
 
 Let ABC be the given 
 spherical right triangle. 
 Draw the lune BB'. Then 
 in the right triangle AB'C 
 a each part is less than 90. 
 
 and formulas (1) to (10) are applicable. 
 
 sm(180-a) = sin ( 180 - a ) 
 ' sin (180 - c) 
 
 sin a 
 
 From (1) 
 
 or 
 
 sin a = 
 
 sine 
 
 which shows that (1) holds when a and c are each greater 
 than 90. 
 
 From (2) cos (180 - a) = tan b 
 
 or 
 
 cos a = 
 
 tan (180 -c) 
 tan b 
 
 tanc' 
 which shows that (2) also holds in this case. 
 
 Similarly it may be shown that formulas (3) to (10) hold 
 when a and c are each greater than 90. 
 
 2. When a > 90 and b > 90. 
 
 Let ABC be the given spherical 
 right triangle. Draw the lune (7(7. 
 Then in the right triangle ABC' c 
 each part is less than 90 and for- 
 mulas (1) to (10) are applicable. 
 
SPHERICAL RIGHT TRIANGLE 
 
 137 
 
 From (1) sin (180 - a) = Bin (180 -a) 
 
 sine 
 
 or 
 
 sin a = 
 
 sin a 
 sin G 
 
 which shows that (1) holds when a and b are each greater 
 than 90. 
 
 From (2) cos (180 - ) = 
 tan b 
 
 tan c 
 
 or 
 
 cos a = 
 
 tan c' 
 
 which shows that (2) also holds in this case. 
 
 Similarly it may be shown that formulas (3) to (10) hold 
 when a and 6 are each greater than 90. 
 
 129. Sufficiency of formulas. It will be noticed that the 
 ten formulas of Arts. 127 and 128 contain all possible com- 
 binations of the five parts of a spherical right triangle, taken 
 three at a time ; hence they are sufficient to solve any spheri- 
 cal right triangle directly from two given parts. 
 
 130. Comparison of formulas of plane and spherical right 
 triangles. By rearranging the formulas of the previous 
 article, the analogy between the formulas of the plane and 
 the spherical right triangles is made apparent. 
 
 IN PLANE RIGHT TRIANGLES* 
 
 a 
 sin a = 
 
 cos a = - 
 c 
 
 tan=- 
 o 
 
 sin /?=* 
 
 C 
 
 cos/3 = - 
 c 
 
 tan/?=- 
 a 
 
 sin a = cos ft sin ft = cos a 
 c 2 = a 2 + b 2 
 
 IN SPHERICAL RIGHT TRIANGLES 
 
 cos a = 
 
 sin a 
 sin c 
 tan 6 
 tanc 
 tana 
 sin b 
 
 sin ft = 
 cos ft = 
 
 sin b 
 sin c 
 tana 
 tanc 
 tan b 
 sin a 
 
 sn = 
 
 1 = cot a cot ft. 
 
 *The above comparison is taken from Chauvenet's 
 Spherical Trigonometry." 
 
 cos b cos a 
 
 cos c = cos a cos b 
 cos c = cot a cot /?. 
 
 Plane and 
 
138 
 
 TRIGONOMETRY 
 
 131. Napier's Rules. The ten formulas used in the solution 
 of spherical right triangles can all be expressed by means 
 of two rules, known as Napier's rules 
 of circular parts. 
 
 Napier's circular parts are the 
 sides a and 5, the complements of the 
 angles opposite or 90 a, 90 /?, 
 and the complement of the hypot- 
 enuse or 90 c. 
 
 They are usually written 
 
 c o a 
 
 a, b, GO a, co/3, coc. 
 
 It will be noticed that the right angle 
 is not one of the circular parts. 
 
 Let the five circular parts be placed 
 in the sectors of a circle in the order 
 in which they occur in the triangle. 
 Whenever any three parts are considered, 
 it is always possible to select one of 
 them in such a manner that the other two parts will either 
 be adjacent to this part, or opposite this part. The part 
 having the other two parts adjacent to it or opposite it is 
 called the middle part. 
 
 Thus let co a, 6, and a be the parts under consideration. 
 Then b is the middle part and co a and a are adjacent parts. 
 
 If co c, co /?, and b are the parts under consideration, b is 
 the middle part and co c and co y8 are opposite parts. 
 
 Napier's rules may now be stated as follows : 
 
 The sine of the middle part is equal to the product of the 
 cosines of the opposite parts. 
 
 The sine of the middle part is equal to the product of the 
 tangents of the adjacent parts.* 
 
 * To associate cosine with opposite and tangent with adjacent, it 
 may be noticed that the words cosine and opposite have the same 
 vowels ; likewise the words tangent and adjacent. 
 
SPHERICAL RIGHT TRIANGLE 139 
 
 132. Theorem. In a spherical right triangle, a side and 
 the angle opposite terminate in the same quadrant. 
 
 From the equation 
 
 cos a = cos a sin ft 
 
 it is seen that cos a and cos a must always have the same 
 sign, since sin j3 is always positive. Hence a and a termi- 
 nate in the same quadrant. 
 
 133. Theorem. Of the three parts a, b, c, if any two ter- 
 minate in the same quadrant, the third terminates in the first 
 quadrant ; if any two terminate in different quadrants, the 
 third terminates in the second quadrant. 
 
 This follows directly from the equation 
 
 cos c = cos a cos b 
 
 by noticing that if any two of the quantities cos a, cos 6, 
 and cos c have like signs, the third is positive ; if any two 
 have unlike signs, the third is negative. 
 
 134. Two parts determine a triangle. In order to solve a 
 spherical right triangle two parts, in addition to the right 
 angle, must be given. Each of the required parts should be 
 obtained directly from the given parts. 
 
 Thus, given 
 
 6 = 50, c = 110 
 
 we have cos a = - , 
 
 cos 50 ' 
 
 cos a = tan 50 cot 110, 
 
 using the formulas for spherical right triangles, or Napier's 
 Rules. If, in the solution of a problem, the sine of any 
 required part is found to be negative, no triangle is possible, 
 since no part of a spherical triangle can be greater than 180. 
 Likewise if the logarithmic sine or cosine of any required 
 part is found to be greater than zero, the triangle is impossi- 
 ble, since no sine or cosine is numerically greater than unity. 
 
140 TRIGONOMETRY 
 
 135. The quadrant of any required part. Since the parts 
 of a spherical triangle may have any value between and 
 180, it is always necessary to determine whether the required 
 parts are greater or less than 90. This can be done by the 
 theorems of Arts. 132 and 133. 
 
 Thus, given 
 
 6 = 50, c=110, 
 
 we have a > 90, a > 90, and ft < 90. 
 
 The quadrant in which any required part terminates may 
 also be determined from the formula used in calculating that 
 part, by observing the signs of the functions involved. But 
 when the unknown part is determined from its sine, the part 
 terminates in both quadrants, giving two solutions, unless 
 limited by the theorems of Arts. 132 and 133. 
 
 Thus, given 
 
 6 = 50, c = 110, 
 
 by writing the signs of each function above the function, we 
 have 
 
 cos 110 
 
 cos a = - , 
 
 cos 50 
 cos a = tan 50 cot 110, 
 
 sin 110 
 
 Then a > 90 and a > 90, since their cosines are negative, 
 and (3 < 90 by Art. 132. 
 
 136. Check formula. The formula containing the three 
 computed parts may always be used as a check formula. 
 
 Thus, having given 
 
 6 = 50, c = 110, 
 the check formula is 
 
 cos a = cos a sin . 
 
SPHERICAL RIGHT TRIANGLE 
 
 141 
 
 137. Solution of a right triangle. 
 
 Given b = 77 35' 16" and a = 112 19' 42" 
 
 tan a = ^=-^- 
 cota 
 
 6 
 a 
 
 log sin b 
 log cot a 
 log tan a 
 
 a 
 
 77 35' 16" 
 112 19' 42'' 
 9.98973 - 10 
 9.61353 - 10 
 
 0.37620 
 67 11' 30" 
 112 48' 30" 
 
 cos a 
 
 + 
 tan b 
 
 log cos a 
 log tan b 
 log cot c 
 
 c 
 
 9.57969 - 10 
 0.65740 
 
 8.92229 - 10 
 85 13' 13'' 
 94 46' 47" 
 
 cos /3 = cos & sin a 
 
 log cos 6 
 
 log sin a 
 
 log cos j3 
 
 
 
 9.33233 - 10 
 9.96615 - 10 
 
 9.29848 - 10 
 78 31' 53" 
 
 Check 
 
 + 
 cos /3 = tan a cot c 
 
 log tan a 
 log cot c 
 log cos /3 
 
 0.37620 
 8.92229 - 10 
 
 9.29849 - 10 
 
 138. Two solutions or the ambiguous case. Whenever a 
 side and the angle opposite are given, there are two solutions. 
 
 Thus if a and a are given, the only formulas by which 6, 
 /3, and c can be determined are 
 
 sin 6 = tan a cot a, 
 
 sin/?.; 
 
 cos a 
 cos a 3 
 
 sin a 
 
 Since the unknown parts are obtained from their sines, 
 each may have two values, giving two solutions, the 
 theorems of Arts. 132 and 133 not restricting the values of 
 the parts to one solution. 
 
 Having found the two values for each part, the theorems 
 
142 
 
 TRIGONOMETRY 
 
 of Arts. 132 and 133 determine the values that belong to 
 each solution. 
 
 Thus, given a = 155 27' 45" to find b 
 a = 100 21' 50" c 
 
 ft 
 
 SOLUTION 
 
 sin b = tan a cot a 
 
 a 
 a 
 
 log tan a 
 log cot a 
 log sin b 
 b 
 
 155 27' 45" 
 100 21' 50" 
 9.65945 - 10 
 9.26217 - 10 
 
 8.92162 - 10 
 4 47' 20" and 
 175 12' 40" 
 
 stn 
 
 cos a 
 
 P _ 
 cos a 
 
 log cos a 
 log cos a 
 log sin /3 
 18 
 
 9.25503 - 10 
 9.95889 - 10 
 
 9.29614 - 10 
 11 24' 22" and 
 168 35' 38" 
 
 sin c = 
 
 sing 
 
 -4- 
 
 sina 
 
 log sin a 
 log sin a 
 log sin c 
 c 
 
 9.61835 
 9.99286 
 
 - 10 
 -10 
 
 9.62549 
 24 58' 
 155 1' 
 
 - 10 
 18" and 
 
 42" 
 
 Check 
 
 sin b = sin c sin /3 
 
 log sin c 
 log sin /3 
 log sin 6 
 
 9.62549 - 10 
 9.29614 - 10 
 
 8.92163 - 10 
 
 FIRST SOLUTION 
 
 &!= 4 47' 20" 
 ft= 11 24' 22" 
 C = 155 1'42" 
 
 SECOND SOLUTION 
 
 6 2 = 17512'40" 
 A = 168 35' 38" 
 c 2 = 24 58' 18" 
 
SPHERICAL RIGHT TRIANGLE 
 
 143 
 
 139. 
 
 EXAMPLES 
 
 Solve the following spherical right triangles, right- 
 angled at y. 
 
 9. a = 132 25' 
 a = 107 30' 
 
 1. 6= 10 32' 
 a= 12 3' 
 
 2. a = 25 18' 
 b= 32 41' 
 
 3. c = 12037' 
 ft = 9 49' 
 
 4. c= 46 40' 
 a= 20 50' 
 
 5. a = 115 6' 
 b = 123 14' 
 
 6. a = 112 43' 30" 
 c= 85 10' 10" 
 
 7. a = 15 18' 20" 
 c= 21 30' 40" 
 
 8. 6 = 168 13' 45" 
 c = 150 9' 20" 
 
 140. Quadrantal triangles. A quadrantal triangle is a 
 triangle one of whose sides is 90. Its polar triangle is 
 then a right triangle. The solution of a quadrantal tri- 
 angle is effected through the solution of its polar triangle. 
 
 141. Isosceles triangles. The solution of an isosceles 
 triangle is effected by solving the two equal right triangles 
 formed by dropping a perpendicular from the vertex to the 
 base. 
 
 10. c= 80 3' 20" 
 = 135 16' 30" 
 
 11. 6=171 3' 15" 
 c= 12 20' 30" 
 
 12. a= 35 54' 20" 
 a= 47 6' 10" 
 
 13. b= 15 2' 30" 
 J3= 20 11' 40" 
 
 14. = 20 26' 20" 
 
 = 84 41' 40" 
 
 15. a= 25 41' 30" 
 a= 34 25' 40" 
 
144 TRIGONOMETRY 
 
 
 
 142. EXAMPLES 
 
 Solve the following triangles, 
 
 1. a = 117 54' 30" 3. =153 16' 
 
 b= 95 42' 20" a= 19 3' 
 
 c= 90 c= 90 
 
 2. a = 69 45' 4. a = 15933'40" 
 (3= 94 40' 6= 95 18' 20" 
 c= 90 c= 90 
 
 5. The base of an isosceles triangle is 51 8'. The equal 
 angles are each 41 57'. Find the equal sides and the 
 angle at the vertex. 
 
 6. The base angles of an isosceles triangle are each 
 100 12' 30", the vertical angle is 50 19' 40". Find the 
 equal sides and the base. 
 
CHAPTER XII 
 OBLIQUE SPHERICAL TRIANGLE 
 
 143. In the present chapter the general formulas of 
 spherical trigonometry, already developed (Chap. X) are 
 transformed into standard formulas adapted to logarithmic 
 computation; and the problem of the solution of the spheri- 
 cal triangle is discussed. 
 
 GENEEAL SOLUTION 
 
 144. To find the angles when the three sides are given. 
 From Art. 119, we have 
 
 cos a = cos b cos c + sin b sin c cos a. 
 
 Therefore cos = cos a .- cos b cos c . (1) 
 
 sin b sin c 
 
 But 2 sin 2 i a = 1 cos a 
 
 _ sin b sin c cos a -f- cos b cos c 
 sin b sin c 
 
 cos a cos (6 c) 
 
 sin b sin c 
 Applying formula (4) of Art. 73, we have 
 
 sin b sin c 
 Letting 2s = a + b-\-c, we have 
 
 8in i a= 
 
 sin '6 sin c 
 145 
 
146 TRIGONOMETRY 
 
 Similarly uniting (1) with 
 
 2 cos 2 1 a = 1 4- cos a, 
 
 we have 2 cos 2 4 = sin 6 sin c + cos -cos 6 cos c 
 
 sin b sin c 
 
 cos (b 4- c) cos CT 
 sin 6 sin c 
 
 2 sin.a + b c sin- a 
 
 sin 6 sin c 
 
 Therefore cos^ a=J sin * h (*-). (3) 
 
 * sin b sin c 
 
 Uniting (2) and (3), 
 
 ten 1 q = J 8in ~ *) Sin (' ~ C ) = tan ** (4) 
 
 ^ 
 
 sin 5 sin (s a) sin (s a) 
 
 Similarly tan * p = J sin ( J ~ c > f < ~ ">= . ^ n \. (5) 
 
 \ sin . sin fx h\ sin <^.c A^ ^ ' 
 
 and tan H = J 8in (^ ~ ") 8in ( 
 ^ 
 
 sin 5 sin (s 6) sin (s b) 
 tan 
 
 sin 5 sin (5 (?) sin (5 c) 
 
 where tan r = /sin (s - a) sin (s - fr) sin (3 - c) . 
 v sins 
 
 145. To find the sides when the three angles are given. 
 Following the method of the last article, the equation 
 cos a= cos ft cos y 4- sin ft sin y cos a 
 
 gives 
 
 . - sin 6 sin Y 
 
 and cos J a = J(*-P) eo. (J- 
 ^ sin p sin -y 
 
 and tan J a = J ~' ct s ^ Q cos ( ^~ a ) = tan /? cos (5 -a) 
 ^- - 
 
 ctQ 
 cos(S-P) 
 
OBLIQUE SPHERICAL TRIANGLE 147 
 
 where 2# = a 
 
 tan R 
 
 146. Delambre's or Gauss's formulas express relations be- 
 tween the six parts. 
 By Art. 68 
 
 sin i (a + (3) = sin i a cos i'/3 + cos ^ a sin i. /?. 
 
 Substituting for sin \ a, cos 1 a, sin 1 y8, and cos | /8 their 
 values in terms of the sides of the triangle, and simplifying, 
 we have 
 
 sin i (a 1 ff) = sin ( s ~ 6 ) + sin (* ~ a ^ A sln S Sln ( .' 
 sin c ^ sin a si 
 
 2 sin c cos c 
 
 sin b 
 
 *r- 
 
 Then sin \ (a + p) = cos i (a -6) . CQs ^ ^ (1) 
 
 cos^c 
 
 Similarly sin (a - p) = sin ^ g ~^ . cos i Y , (2) 
 
 sin^c 
 
 and cos J- (a + p) = . 8in J Y , (3) 
 
 COS C 
 
 and co8i(a-p) = .si n | Y . (4) 
 
 147. Napier's analogies express relations between five parts 
 of a triangle. They are easily obtained from Gauss's formulas. 
 Dividing (1) by (2), Art. 146, 
 
 8 in|(a_p) tan|(a-6) 
 Dividing (3) by (4), 
 
 . tan c 
 
 cos|(a-p) 
 
148 TRIGONOMETRY 
 
 Dividing (4) by (2), 
 
 cotjH 
 
 Dividing (3) by (1), 
 
 148. Formulas collected. The following formulas are suf- 
 ficient to solve a spherical triangle when any three parts 
 are given : 
 
 c 1nr 
 
 sin s sin (s a) sin (s a) 
 ) = tan 
 
 sin !(<* + /?) _ tan^c 
 sin i (a ft) tan 1 (a 6) 
 
 cos - + ) tan c 
 
 cos i (a ./8) tan 1 (a + 6) 
 
 sin 1 (a + 6) _ cot ^- y 
 sin i (a 6) tani( a ^) 
 
 1-6) tanf 
 
 sin a _ sin b 
 sin a sin /? 
 
 VII 
 
 Formula I is used to determine an angle when the three 
 sides are given. 
 
 Formula II is used to determine a side when the three 
 angles are given. 
 
 Formulas III and IV are used when two angles and the 
 
 * These formulas are typical. Other formulas of the same type are 
 obtained by a cyclic change of letters. 
 
OBLIQUE SPHERICAL TRIANGLE 149 
 
 included side are given. Formula III determines ^ (a b) 
 and formula IV determines -j- (a -j- b), from which a and b 
 are obtained. Either formula may also be used to deter- 
 mine the side c when the other two sides and their opposite 
 angles are given. 
 
 Formulas V and VI are used when two sides and the in- 
 cluded angle are given. Formula V determines ^ (a ft) 
 and formula VI determines J (a + (3), from which a and ft 
 are obtained. Either formula may also be used to deter- 
 mine the angle y when the other two angles and their op- 
 posite sides are given. 
 
 Formula VII is used when an angle and the side opposite 
 are among the given parts. 
 
 149. Whenever the formulas I to VI are employed in the 
 solution of a spherical triangle as indicated above, the 
 quadrant in which any part terminates may always be deter- 
 mined by noticing the signs of the functions involved. 
 
 But when the law of sines is employed, two values are 
 found for the required part. This leads to two solutions 
 unless limited to one solution by the following principles. 
 
 150. Theorem. Half the sum of any two angles is in the 
 same quadrant as half the sum of the sides opposite. 
 
 This follows from a consideration of the signs of the 
 functions involved in 
 
 cos \(a + ft) _ tan^- c 
 cos %(a ft) tan |(a -f b) ' 
 
 Since each part is less than 180, tan 1 c and cos ^( ft) 
 are always positive. Hence cos|( + /?) an( i tan i (a + 6) 
 must always have the same sign. Hence i(a + ft) and 
 |-(a + b) terminate in the same .quadrant. 
 
 151. Theorem. A side which differs more from 90 than an- 
 other side, terminates in the same quadrant as its opposite angle. 
 
150 
 
 TRIGONOMETRY 
 
 We have from Art. 119 
 
 cos a cos b cos c 
 
 cos a 
 
 sin b sin c 
 
 If a differs more from 90 than 6, cos a is numerically 
 greater than cos b. Cos a is also greater than cos b cos c, 
 since cos c is not greater than unity. Hence the numerator 
 of this fraction has the same sign as cos a. The denominator 
 being always positive, cos a and cos a have the same sign. 
 Hence a and a terminate in the same quadrant. 
 
 The negative of this theorem is not true. 
 
 Thus given, a = 165, b = 120, ft = 135, 
 a terminates in the second quadrant, since a differs more 
 from 90 than 6. 
 
 152. Theorem. An angle which differs more from 90 than 
 another angle, terminates in the same quadrant as its opposite 
 side. 
 
 This follows from 
 
 sff cosy 
 
 sin ft sin y 
 by considerations similar to those of the previous article. 
 
 Thus given, a = 80, y = 140, and a = 120, 
 c terminates in the second quadrant, since y differs more 
 from 90 than a. 
 
 153. Illustrative examples. 
 1. Given the three sides, a = 105 27' 20", 6 
 c = 96 53' 10", to find a, ft, and y. 
 
 83 14' 40", 
 
 a 
 
 105 27' 20" 
 
 b 
 
 83 14' 40" 
 
 c 
 
 96 53' 10" 
 
 2s 
 
 284 94' 70" 
 
 s 
 
 142 47' 35" 
 
 s a 
 
 37 20' 15" 
 
 s-b 
 
 59 32 '55" 
 
 s c 
 
 45 54' 25" 
 
 Check-: s 
 
 142 47' 35" 
 
 tan r = J sin (-) sin (s-&)sin Q-c) 
 * sins 
 
 tan \ a = 
 
 tanr 
 
 tan \ 8 = 
 
 sin (s a) 
 
 tanr 
 sin (s b) 
 
 tan r 
 sin (s c) 
 
OBLIQUE SPHERICAL TRIANGLE 
 
 151 
 
 log sin (s a) 
 
 9.78284-10 
 
 log sin (s 6) 
 
 9.93553-10 
 
 log sin (s c) 
 
 9.85623-10 
 
 colog sin s 
 
 0.21846 
 
 2 log tan r 
 
 9.79306-10 
 
 log tan r 
 
 9.89653-10 
 
 log tan A a 
 
 0.11369 
 
 log tan A /3 
 
 9.96100-10 
 
 log tan A 7 
 
 0.04030 
 
 i 
 
 52 24' 55" 
 
 i 
 
 42 25' 51" 
 
 IT 
 
 47 39' 17" 
 
 0E 
 
 104 49' 50" 
 
 
 
 84 51' 42" 
 
 T 
 
 95 18' 34" 
 
 Check 
 
 ^ U2/ siiiK-&) 
 
 a + b 
 a-b 
 *(' + ) 
 *(*-&) 
 
 K-0) 
 
 188 42' 
 22 12' 40' 
 94 21' 0' 
 11 6' 20' 
 9 59' 4' 
 
 log sin (a + &) 
 
 log tan (-) 
 colog sin A (a 6) 
 log cot 7 
 log tan 1 7 
 
 9.99875 - 10 
 9.24563 - 10 
 0.71531 
 
 9.95969 - 10 
 0.04031 
 
 2. Given two sides and the included angle, a = 29 18', 
 b = 37 30', y = 51 52', to find a, (3, and c. 
 
 *>=!!! J ( ,g + ??*K6-) 
 
 b 
 
 37 30' 
 
 
 a 
 
 29 18' 
 
 
 y 
 
 51 52' 
 
 
 b-a 
 
 8 12' 
 
 
 b + a 
 
 66 48' 
 
 
 4 (ft - ) 
 
 4 6' 
 
 
 i (& + a ) 
 
 33 24' 
 
 
 17 
 
 25 56' 
 
 
 log sin |(6 a) 
 
 8.85429 
 
 -10 
 
 log cot A 7 
 
 0.31310 
 
 
 colog sin i(6 + a) 
 
 0.25926 
 
 
 log tan ( cc) 
 
 9.42665 
 
 - 10 
 
 1 03 - a) 
 
 14 57' 14" 
 
 log cos A (& _ a) 
 log cot \ 7 
 colog cos \ (6 + a) 
 log tan A (|8 + ) 
 103 + 0) 
 
 a. 
 
 9.99889 - 10 
 0.31310 
 0.07839 
 
 0.39038 
 67 51' 8" 
 14 57' 14" 
 
 82 48' 22" 
 52 53' 54" 
 
 log tan \ (b - a) 
 log sin A (|8 + a) 
 colog sin A (^3 ) 
 log tan A c 
 
 c 
 
 8.85540 - 10 
 9.96671 - 10 
 0.58831 
 
 9.41042 - 10 
 14 25' 43" 
 28 51' 26" 
 
152 
 
 TRIGONOMETRY 
 
 
 
 Check 
 
 
 
 
 
 sin a _ sin 6 _ sin c 
 sin a sin j8 sin 7 
 
 
 
 log sin a 
 log sin a 
 
 9.68965 
 9.90177 
 
 -10 
 -10 
 
 log sin 6 
 log sin /3 
 
 9.78445-10 
 0.99656 - 10 
 
 log sin c 
 log sin 7 
 
 9.68361 
 9.89574 
 
 -10 
 -10 
 
 9.78788 
 
 -10 
 
 9.78789 - 10 
 
 9.78787 
 
 -10 
 
 3. Given two sides, and an angle opposite one of them, 
 a = 63 24' 50", b = 17 36' 40", a = 44 48' 20", to find ft y, 
 and c. 
 
 sin a 
 
 taDic = *'"}( g + ^tanK-6) 
 
 a 
 
 63 24' 50" 
 
 log sin J (a -f 
 
 &) 
 
 9.71445 - 10 
 
 a 
 
 44 48' 20" 
 
 log tan K - /3) 
 
 9.57083 - 10 
 
 b 
 
 17 36' 40" 
 
 colog sin \ (a 
 
 ft) 
 
 0.62876 
 
 a + b 
 
 62 25' 0" 
 
 log cot \ 7 
 
 9.91404 - 10 
 
 a-b 
 
 27 11' 40" 
 
 i 
 
 7 
 
 50 38' 0" 
 
 \ (a + 6) 
 
 31 12' 30" 
 
 
 7 
 
 101 16' 0' 
 
 ( a _ 5) 
 
 13 35' 50" 
 
 
 
 log sin a 
 
 9.95147 - 10 
 
 
 
 log sin b 
 
 9.48081 - 10 
 
 log sin (a + ) 
 
 9.83375 - 10 
 
 colog sin a 
 
 0.15200 
 
 log tan (a 
 
 &) 
 
 9.38359 - 10 
 
 log sin /3 
 
 9.58428 - 10 
 
 colog sin \ (a /3) 
 
 0.45735 
 
 /3 
 
 22 34' 44" . 
 
 log tan , 
 
 c 
 
 9.67469 - 10 
 
 a /3 
 
 40 50' 6" 
 
 
 
 j C 
 
 25 18' 20" 
 
 + /3 
 
 85 59' 34" 
 
 
 c 
 
 50 36' 4(X 
 
 !(*-/) 
 
 20 25' 3" 
 
 
 
 i (a + /3) 
 
 42 59' 47" 
 
 
 
 
 Cftecfc 
 
 * 
 
 
 sin ft _ sin c 
 
 
 
 sin /3 sin 7 
 
 
 log sin 6 9.48081 - 
 
 10 log sin c 
 
 9.88810 - 10 
 
 log sin 9.58428- 
 
 10 log sin 7 
 
 9.99155 - 10 
 
 9.89653 - 
 
 10 
 
 9.89655 - 10 
 
OBLIQUE SPHERICAL TRIANGLE 
 
 153 
 
 154. Two solutions. There are two solutions, if any, when- 
 ever two sides and an angle opposite one of them, or two an- 
 gles and a side opposite one of them, are given, unless limited 
 to one solution by the principles of Arts. 150, 151, and 152. 
 
 Thus, having given = 45 15' 12", b = 56 49' 46", a = 
 68 52' 48", to find a, y, and c. 
 
 sin 6 
 
 a 
 
 b 
 
 68 52' 48" 
 56 49' 46" 
 45 15' 12" 
 
 a 
 
 a + P 
 a-p 
 
 ci + b 
 a-b 
 
 fo-&) 
 
 52 19' 33" 
 97 34' 45" 
 7 4' 21" 
 48 47' 22" 
 3 32' 10" 
 125 42' 34" 
 12 3' 2" 
 62 51' 17" 
 6 1'31" 
 
 127 40' 27" 
 172 55' 39" 
 82 25' 15" 
 86 27' 50" 
 41 12' 38" 
 
 log sin p 
 log sin a 
 colog sin b 
 . log sin a 
 a 
 
 9.85140 
 9.96980 
 0.07725 
 
 9.89845 
 52 19' 33", or 
 127 40' 27" 
 
 log sin ( + ) 
 log tan $ (a - 6) 
 colog sin & (a /9) 
 log tan c 
 
 I 
 
 c 
 
 9.87639 
 9.02346 
 1.20984 
 
 9.99917 
 9.02346 
 0.18124 
 
 0.10969 
 52 9' 35" 
 104 19' 10'' 
 
 9.20387 
 9 5' 7" 
 18 10' 14" 
 
 log sin (a + 6) 
 logtanO-) 
 colog sin } (a ft) 
 log cot 1 7 
 i7 
 7 
 
 9.94932 
 8.79099 
 0.97895 
 
 9.94932 
 9.94238 
 0.97895 
 
 9.71926 
 6221'1" 
 124 42' 2" 
 
 0.87065 
 7 40' 16" 
 15 20' 32" 
 
154 
 
 TRIGONOMETRY 
 
 Check 
 
 cosJ(a-/8) _ ; 
 
 logcos|(a + j3) 
 log tan %(a + 6) 
 cologcos(ot /3) 
 log tan \ c 
 
 9.81877 
 0.29012 
 0.00083 
 
 8.79013 
 0.29012 
 0.12361 
 
 0.10972 
 &) tan^r 
 
 9.20386 
 a 4- B) 
 
 cos J(a - b) 
 
 FIRST SOLUTION 
 
 = 52 19' 33" 
 
 c = 104 19' 10" 
 
 = 12442' 2" 
 
 SECOND SOLUTION 
 
 a = 127 40' 27" 
 c = 18 10' 14" 
 y = 15 20' 32" 
 
 155. Area of spherical triangle. Representing the area of 
 a sphere, S, by 720 spherical degrees, it is demonstrated in 
 geometry that the area of a spherical triangle, A, in terms 
 of spherical degrees, is equal to its spherical excess ; or 
 >f=(a + p+y 180) spherical degrees 
 
 -' 
 
 log cos ( a + 6) 
 log tan ( + ) 
 cologcos (o &) 
 log cot 7 
 
 9.65920 
 0.05762 
 0.00241 
 
 9.65920 
 0.20904 
 0.00241 
 
 9.71923 
 
 0.87065 
 
 Let A' and S' represent the area of the triangle and the 
 sphere respectively, in terms of the unit in which r is ex- 
 pressed. Since the ratio of the area of the spherical tri- 
 angle to the area of the sphere is independent of the units 
 used, we have A' __A 
 
 Therefore 
 
 But the area of the sphere expressed in terms of r is 
 4 ?rr 2 , therefore the area of the triangle is given by 
 
 A' = 
 
 180 
 
 156. 
 
 EXAMPLES 
 
 Solve the following spherical triangles and check the 
 results : 
 
OBLIQUE SPHERICAL TRIANGLE 
 
 155 
 
 1. 
 
 6= 10 O'lO" 
 c = 114 40' 40" 
 a= 92 28' 20" 
 
 11. = 61 8' 
 
 J3= 59 12' 
 y= 78 25' 
 
 2. 
 
 b= 85 4' 19" 
 c = 139 58' 25" 
 = 12 20' 31" 
 
 12. a= 76 43' 15" 
 b= 83 35' 27" 
 c= 98 26' 38" 
 
 3. 
 
 c= 82 3' 4" 
 a= 70 14' 12" 
 = 84 20' 9" 
 
 13. = 11035' 
 ^8 = 135 42' 
 
 y = 146 8' 
 
 4. a=, 95 3' 30" 
 b = 128 38' 50" 
 y = 170 52' 20" 
 
 5. o= 29 18' 
 6= 37 30' 
 y = 51 52' 
 
 6. a= 11 21' 10" 
 y = 19 0'20" 
 6= 66 19' 30" 
 
 7. y = 179 22' 11" 
 a = 148 17' 17" 
 6= 25 39' 34" 
 
 8. a = 108 5' 18" 
 6 = 170 30' 46" 
 c= 85 50' 22" 
 
 9. a = 105 27' 20" 
 6= 83 14' 40" 
 c= 96 53' 10" 
 
 10. a= 34 19' 30" 
 
 6= 28 37' 10" 
 c= 22 44' 40" 
 
 14. y= 11 34' 10" 
 
 6= 82 56' 30" 
 c= 27 9' 40" 
 
 15. a= 32 4' 10" 
 /? = 12856'20" 
 a= 39 50' 30" 
 
 16. /?= 80 40' 2" 
 c= 75 54' 0" 
 6 = 100 21' 28" 
 
 17. a= 21 I'lO" 
 y= 17 22' 50" 
 a= 14 13 '30" 
 
 18. )8= 77 44' 55" 
 y= 92 17 '24" 
 a= 26 29' 39" 
 
 19. a = 114 23' 9" 
 0= 88 41' 11" 
 y= 79 0' 4" 
 
 20. 0= 99 4' 12" 
 
 y = 106 0' 9" 
 a = 161 2' 10" 
 
156 TRIGONOMETRY 
 
 21. c = 100 10' 40" 23. b= 42 15' 20" 
 a= 65 20' 30" c = 127 3' 30" 
 y= 94 30' 10" (3= 31 44' 20" 
 
 22. a = 103 19' 50" 24. y = 127 4' 10" 
 /?= 92 37' 30" a= 88 12' 0" 
 y = 128 54' 20" c = 141 20' 30" 
 
 SOLUTION WHEN ONLY ONE PART IS REQUIRED 
 
 157. In many problems of astronomy and geodesy, it is re- 
 quired to find only one or two of the unknown parts of a spher- 
 ical triangle, the other unknown parts being of no importance 
 in the problem. It then becomes desirable to have a method 
 whereby the required parts can be computed without being 
 under the necessity of first computing any part not desired. 
 
 It has already been shown that any angle can be found 
 directly from three given sides (Art. 144), and that any 
 side can be found directly from three given angles (Art. 145). 
 
 It is evident that any part can be found from any three 
 given parts by the use of the general formulas containing 
 the required part and the three given parts. By the intro- 
 duction of auxiliary quantities, these formulas will now be 
 adapted to logarithmic computation. 
 
 158. Given two sides and the included angle, to find any 
 one of the remaining parts. 
 
 Let a, b, y be the given parts. 
 
 First. To find c. 
 
 The relation between a, 6, y, and c is (Art. 119), 
 
 cos c = cos a cos 6 -f sin a sin b cos y. (1) 
 
 To adapt this formula to logarithmic computation, let 
 
 m sin M= sin b cos y, (2) 
 
 and m cos M = cos b. (3) 
 
OBLIQUE SPHERICAL TRIANGLE 
 
 157 
 
 Then eliminating b by uniting (1), (2), and (3), we have 
 
 cos c = m (cos a cos M+ sin a sin M ), 
 or cos c = m cos (a M ). 
 
 From (2) and (3) 
 
 tan Jf = tan 6 cos y, 
 and from (3) and (4) 
 
 cos b cos (a M") 
 cos c = = t 
 
 cos 
 
 (4) 
 (5) 
 (6) 
 
 Equations (5) and (6) enable us to find c. 
 ILLUSTRATION. Given o= 75 38' 20", & = 54 54' 38", and 
 = 30 17' 43". 
 
 tan M = tan & cos 7 
 
 a 
 
 75 38' 20" 
 
 6 
 
 54 54' 38" 
 
 7 
 
 30 17' 43" 
 
 log tan 6 
 log cos 7 
 log tan Jf 
 M 
 
 0.15333 
 9.93623 
 0.08956 
 
 50 51' 58" 
 
 a- M 
 
 24 46' 22" 
 
 cos c = cos 5 cos (a -Jf) 
 cos Jf 
 
 log cos b 
 log cos (a M) 
 colog cos M 
 log cos c 
 c 
 
 9.75956 
 9.95808 
 0.19987 
 
 9.91751 
 34 12' 27" 
 
 Second. To find ft. 
 
 The relation between a, b, y, and ft is given by Art. 124, 
 (equation 5), from which we have 
 
 , Q _ cot b sin a cos a cos y x^ 
 
 siny 
 
 Multiplying numerator and denominator by sin b, 
 , ~_ cos b sin a sin b cos a cos y 
 
 sin b siiiy 
 Again using equations (2) and (3) we have 
 
 m (cos M sin a sin Jf cos a) 
 sin 6 sin y 
 
 (8) 
 
 , 
 cot = 
 
 or 
 
 sin b sm y 
 
 (9) 
 (10) 
 
158 TRIGONOMETRY 
 
 As before tan M = tan 6 cos y. (11) 
 
 From (2) and (10) 
 
 . (12) 
 
 . 
 sin M 
 
 Equations (11) and (12) enable us to find ft. 
 Third. To find y. 
 
 By interchanging a and 6 and consequently a and ft in 
 (11) and (12) we have, calling the auxiliary angle N 9 
 
 tan JV= tan a cos y (13) 
 
 and cota = tysin(6-^) (14) 
 
 sm^" 
 
 to determine a. 
 
 159. Two parts required. It will be noticed that the same 
 auxiliary quantity M is used to find both c and ft. We thus 
 have a convenient method, much used in astronomy, for 
 finding a side and an angle when two sides and the included 
 angle are given. 
 
 For finding c and ft we have, collecting our formulas, 
 
 tan M = tan 6 cos y 
 
 cos M 
 
 cot/?= cotysin(q-Jf) t 
 sin M 
 
 Dividing equation (10) of Art. 158 by equation (4), we have 
 
 cot/?_tan (a Jf) 
 cos c sin b sin y ' 
 which serves as a check upon c and ft. 
 Similarly, for finding c and a, we have 
 
OBLIQUE SPHERICAL TRIANGLE 
 
 tan N= tan a cos y 
 
 cos a cos (b N} 
 cos c = 5s L 
 
 cos N 
 
 159 
 
 sin N 
 
 160. 
 
 PROBLEMS 
 
 An arc of l r on the earth's surface is equal to one English 
 geographical mile. 
 
 1. Find the distance between Boston, latitude 42 21' N., 
 longitude 71 41' W., and San Francisco, latitude 37 48' N. 
 and longitude 122 28' W. P 
 
 SOLUTION. Let APD be the me- 
 ridian of Greenwich from which longi- 
 tude is measured, ABCD the equator, 
 and P the north pole. 
 
 Let the positions of San Francisco 
 and Boston be represented by E and F 
 respectively. The desired distance is 
 EF. 
 
 Then 
 
 But 
 
 Letting 
 
 angle DPE = 122 28', 
 
 angle DPF = 71 41', 
 
 arc EB= 37 48', 
 
 arc CF= 42 21', 
 
 arc PB = arc PC = 90. 
 
 PE=PB-EB = 52 12', 
 
 PF = PC - FC = 47 39', 
 Z FPE = /. DPE - Z DPF = 50 47'. 
 Z FPE = 7, PE = a, PF= &, 
 
 we may find EF or c by the formulas 
 
 tan M = tan 6 cos 7, cos c = cos&cos ( 
 
 cos M 
 
 -M) 
 
160 
 
 TRIGONOMETRY 
 
 _ cos b cos (a M ) 
 
 cos M 
 
 a 
 
 b 
 y 
 
 52 12' 
 47 39' 
 50 47' 
 
 log cos b 
 log cos (a M ) 
 colog cos M 
 log cos c 
 c 
 
 9.82844 - 10 
 9.97951 - 10 
 0.08526 
 
 log tan b 
 log cos y 
 log tan M 
 M 
 a- M 
 
 0.04023 
 9.80089 - 10 
 
 9.89321 - 10 
 38 33' 20", or 
 2313 miles 
 
 9.84112 - 10 
 34 44' 23" 
 17 27' 37'' 
 
 2. Find the distance between New York (lat. 40 43' K, 
 long. 74 0' W.) and San Francisco (lat. 37 48' K, long. 
 122 28' W.). 
 
 3. Find the distance between Calcutta (lat. 22 33 f N., 
 long. 88 19' E.) and Greenwich (lat. 51 29' N.). 
 
 4. Find the distance between Baltimore (lat. 39 17' N., 
 long. 76 37' W.) and Calcutta. 
 
 161. Given two angles and the included side, to find any 
 one of the rema ining parts. 
 Let a, ft c be the given parts. 
 
 First. To find y. 
 
 The relation between a, ft c, and y is, Art. 121, eq. (3), 
 
 cos y = cos a cos ft + sin a sin ft cos c. - (1) 
 Let ra sin M = cos a, (2) 
 
 and m cos M = sin a cos c. (3) 
 
 Uniting (1), (2), and (3) 
 
 cos y = m ( sin M cos /? -f cos M sin /?), 
 or cos y = m sin (ft M). (4) 
 
 From (2) and (3) 
 
 cot M = tan a cos c, (5) 
 
OBLIQUE SPHERICAL TRIANGLE 161 
 
 and from (2) and (4) 
 
 cos a sin (8 
 
 sinM 
 Equations (5) and (6) enable us to find y. 
 
 (6) 
 
 Second. To find a. 
 
 The relation between a } (3, c, and a is given by Art. 124, 
 equation (4), from which 
 
 sin 8 cot a + cos c cos 8 ^ 
 
 cot a = . ( 1 1 
 
 sine 
 
 , _ sin ft cos a + sin a cos c cos ft t ,^ 
 
 sin a sin c 
 
 Uniting equations (2), (3), and (8), 
 
 cot a = ^ " "" ^ ' , (9) 
 
 sin a sin c 
 
 sin a sin c 
 From (2), (3), and (10) 
 
 cot M = tan a cos c (11) 
 
 cot c cos (8 Jf) /., \ 
 
 and cota = ^ L , (12) 
 
 cos Jf 
 
 from which a is found. 
 
 To find b. 
 
 From (11) and (12) by interchanging a and /?, and conse- 
 quently a and 6, we have, calling the auxiliary quantity JV, 
 
 cot N= tan cos c (13) 
 
 and cot6 = - (14) 
 
 cos N 
 
 to determine &. 
 
 162. Given two sides and an angle opposite one of them, to 
 find any one of the remaining parts. 
 Let a, bj a be the given parts. 
 
162 TRIGONOMETRY 
 
 First. To find c. 
 
 The relation between a, b } a and c is 
 
 cos a = cos b cos c + sin 6 sin c cos a. (1) 
 
 Let m sin Jf = sin b cos a, (2) 
 
 and m cos Jf = cos 6. (3) 
 
 Then cos a = m cos (c M ). (4) 
 
 From equations (2), (3), and (4), 
 
 tan M = tan b cos a (5) 
 
 and cos (c - M) = Cos a cos ^ (6) 
 
 cos b 
 
 Equation (5) determines M and equation (6) determines 
 c M. Adding these values, we have c. 
 
 In general there are 2 solutions for c. We may limit M 
 to positive values less than 180. By equation (6) c M 
 may have two values, numerically equal but opposite in 
 sign, giving two values for c unless the sum M -\- (c M ) 
 is greater than 180 or negative, in which case there is but 
 one solution. 
 
 Second. To find y. 
 
 The relation between a, 6, a, and y is 
 
 sin y cot a = cot a sin b cos b cos y. 
 Multiplying by sin a and rearranging, we have 
 
 sin y cos a + cos y sin cos b = sin a cot a sin 6. (7) 
 Let n cos JV= cos a, (8) 
 
 and n sin ^= sin a cos 6. (9) 
 
 From (7), (8), and (9) we have 
 
 n sin (y -f- JV) = sin a cot a sin 6. (10) 
 
 Then from (8), (9), and (10), 
 
 tan N== tan a cos 6, (11) 
 
 and sin (y -f JV) = sin N cot a tan 6. (12) 
 
 Equation (12) determines y + N and equation (11) deter- 
 mines H. Subtracting the second value from the first gives y. 
 
OBLIQUE SPHERICAL TRIANGLE 163 
 
 In general there are two solutions, since y + N may have 
 two values. 
 
 Third. To find ft. 
 
 The angle ft is found from 
 
 . sin 6 sin a ,* o\ 
 
 sm ft = : , (lo) 
 
 sin a 
 
 which in general gives two values. 
 
 163. Given two angles and a side opposite one of them, to 
 find any one of the remaining parts. 
 Let a, ft, a be the given parts. 
 
 First. To find y. 
 
 The relation between a, ft, a, and y is 
 
 cos a = cos ft cos y + sin ft sin y cos a. (1) 
 
 Let m sin M = cos ft, (2) 
 
 and m cos M = sin ft cos a. (3) 
 
 Then cos a = msin(y M). (4) 
 
 From (2), (3), and (4) 
 
 cot M = tan ft cos a, (5) 
 
 -, . / , f ^ cos a sin J/" //>. 
 
 and sm(y M)= (6) 
 
 cosp 
 
 Equations (5) and (6) determine M and y M, from 
 which y is found. 
 
 Two solutions may be possible, as in Art. 162. 
 
 Second. To find c. 
 
 The relation between a, ft, a, and c is 
 
 sin ft cot a = cot a sin c cos c cos /?. (7) 
 
 Multiplying by sin a and transposing, 
 
 cos a sin c sin a cos c cos /3 = sin /? sin a cot a. (8) 
 
 Let n cos JV= cos a, (9) 
 
 and w sin JV= sin a cos ft. (10) 
 
164 TRIGONOMETRY 
 
 Then uniting (8), (9), and (10) 
 
 n sin (c JV) = sin ft sin a cot a. (11) 
 
 From (9), (10), and (11) 
 
 tan N= tan a cos /?, (12) 
 
 and sin (c N) = tan ft cot a sin N. (13) 
 
 Equations (12) and (13) determine N and c JV, from 
 which c is found. 
 
 There may be two solutions. 
 
 Third. To find b. 
 
 We have sin b = sin ? sin * (14) 
 
 Bin 
 
 There are two values of b unless restricted to one solution 
 by the principles of Arts. 150, 151, and 152. 
 
 164. The general triangle. The parts of the general spheri- 
 cal triangle are not restricted to values less than 180. It 
 can be shown that all the formulas developed for the oblique 
 spherical triangle are true for the general spherical triangle 
 if the double sign is introduced in the formulas of Arts. 
 144, 145, and 146. 
 
ANSWERS 
 
 Art. 19 ; Page 10 
 
 5. 114 35' 28", 286 28' 40", etc. 
 
 6. 60, 135, - 300, 57 17' 44", 36 28' 31", etc. 
 
 7. 7ift. 
 
 8. 2f radians, 137 .30' 34". 12. 247.16 E. P. M. 
 
 25.882. 
 
 9. 94 3' 24". 13. 18.33 mi. per sec. 
 
 10. ifTrft. 14. 5236. 
 
 11. 2.7216 radians. 15. 9.6 TT 
 
 240 it ft. per min. 
 
 Art. 27 ; Page 20 
 1. 3d and 4th. 2. 1st and 4th. 3. 1st and 3d 
 
 7. 2d. 8. 3d. 20. -?^. 21. -^. 
 
 o o 
 
 28. sin r = -g% V85, cos ^ = ^ V85, cot a 1 = ^ ) 
 sec ! = ^ V85, esc 06! = ^- V85, 
 sin 2 = -^- V85, cos 2 = -g 7 - A/ 85, cot 02 = -J-, 
 sec 2 = -J- V85, esc c^ = 1 V85. 
 
 35. if 
 
 Art. 36 ; Page 29 
 
 1. 6 = 16.5 3. c-=.869 5. a = 27 
 c = 17.5 a = .225 ft = 63 
 ft = 70. a = 15. b = 72.6 
 
 2. a = 1.44 4. c = 65 6. a = 18 
 b = 2.05 a = 23 ft = 71| 
 
 = 55. = 67 & = . 00867 
 
 165 
 
166 
 
 TRIGONOMETRY 
 
 7. a = 346 8. a = 27 9. a = .029 10. a 
 
 c = 507 ft = 63 b = .089 a = 8.11 
 
 = 47 a = 3.50 0=72 6 = 9.48 
 
 11. 6 = 5161 
 c = 5489 
 =70 5' 
 
 14. c=. 00006294 
 a = 72 26' 
 = 17 34' 
 
 12. a = .1384 
 6 = .2878 
 ft = 64 19' 
 
 13. a = 1.446 
 c = 1.719 
 a = 57 17' 
 
 15. b = 810.80 16. 17. etc. 
 
 a = 47 31' 32" check your 
 
 = 42 28' 28" results. 
 
 31. Base 1331.1, vertical angle 149* 19' 10". 
 
 32. Base angles 39 23' 56", base 1477.0. 
 
 33. Equal sides 1622.9, base angles 37 59' 37". 
 
 34. Equal sides 219.75, base angles 68 27' 19". 
 
 35. 37.504. 
 
 Art. 38; Page 31 
 
 1. 
 
 290.83 
 
 ft. 
 
 7. 
 
 2nrtai 
 
 i 
 
 12. 
 
 54.775 
 
 mi. 
 
 
 2. 
 
 405.24 
 
 ft. 
 
 
 
 n 
 
 13. 
 
 153.72 
 
 Ibs. 
 
 
 3. 
 
 263.92 
 
 ft. 
 
 8. 
 
 130.99 
 
 ft. 
 
 
 38 31' 
 
 46". 
 
 4. 
 
 289.93 
 21.442 
 
 ft. 
 ft. 
 
 9. 
 10. 
 
 226.11 
 2572.5 
 
 ft. 
 ft. 
 
 14. 
 
 739.38 
 hi. 
 
 mi. 
 
 per 
 
 5. 
 
 2 nr sii 
 
 180 
 
 11. 
 
 21.360 
 22.638 
 
 in. 
 in. 
 
 15. 
 
 16 ft. 
 
 l& 
 
 o Z 
 
 in. 
 
 n 
 
 6. 
 
 132.52 
 
 ft. 
 
 
 69 26' 
 
 36". 
 
 16. 
 
 35 16'. 
 
 
 
 Art. 50; Page 46 
 
 2. - cos 10, - sin 80. 5. cos 20, sin 70. 
 
 3. - cot 20, - tan 70. 6. - tan 80, - cot 10. 
 
 4. -.cot 20, - tan 70. 7. - sin 60, - cos 30. 
 8. cos B. 9. -tan B. 10. tan0. 11. -cos 
 
ANSWERS 
 
 16T 
 
 11. 
 
 12. 
 
 15. 
 18. 
 20. 
 21. 
 35. 
 37. 
 
 33. 
 39. 
 
 41. 
 
 Art. 56 ; Page 51 
 
 _, tan = - V5, cot0 = -V5, 
 sec = | V5, esc = f . 
 
 sin = ^VUfl, cos = T |^ V149, tan = J^, 
 sec = I V149, esc = T V VI49. 
 
 z = 30. 16. = 45. 17. z=0, 60. 
 
 = 45, 60. 19. sina = ^-i. + V5. 
 
 Identity. 22. = 120. 24. Identity. 
 
 x = 60, 120. 23. = 30. 25. y = 30, 150. 
 
 a = 30, 150. 36. a = 30, 60, 120, 150. 
 
 x = 45, 135. 38. x = 45. 
 
 Art. 60; Page 57 
 
 tan 2 a-l. 34. sinzTVl-sin 2 *. 35. 1 + COS X 
 tan 2 # + l 1 cos 2 a; 
 
 ' vr r o 2 ' 
 
 _i_->/1 __ /S 1 1 
 
 -, sec0 
 
 a 
 
 Vl - a 2 
 
 COS0 
 
 cot0 
 
 CSC0 = 
 
 , sec0 
 
 COS0' 
 
 43. 
 
 50. 
 55. 
 58. 
 59. 
 60. 
 
 ? 47. 0, 90. 48. 135. 49. 0, 30, 60, 150. 
 
 2 
 
 30, 150. 51. 45. 54. 36 52'. (See tables.) 
 46 24', 90. 57. 65 54', 114 6'. 
 
 0, _ 30, - 150. 
 
 tanu=|V5, iV2. 61. 195, 345. 
 
 j, |V2. 62. 54, 234. 
 
168 
 
 TRIGONOMETRY 
 
 Art. 70 ; Page 66 
 
 2. ^V2(V3 - 1). 3. iV2( V3 - 1). 4. 
 3+V3 
 
 1). 
 
 5. 
 
 1. 
 2. 
 
 3. 
 
 32. 
 
 34. 
 35. 
 
 1. 
 3. 
 5. 
 
 7. 
 10. 
 
 13. 
 
 14. 
 
 20. 
 
 3-V3 
 
 7. cos ft. 8. cot a. 9. cos a. 
 
 Art. 75 ; Page 71 
 
 * V2 - V2, fV2 + V2, V3 - 2V2, V3 + 2 V2. 
 |V15, - |, | V15, 
 
 20 
 
 /O O /?^O O/W H fr 
 
 .raj;;- 400 65 !; 2 
 
 30, 90, 150. 42. 0, 30, 90, 150. 
 
 0, 120. 43. 0, 90, 120. 
 
 Art. 85 ; Page 82 
 
 60, 120, 420, etc. 2. 150, 210, - 150, etc. 
 
 45, 225, - 135, etc. 4. A. 
 
 6. 
 
 u 
 
 Vl 4- % 2 
 2 ?^ Vl - 
 
 Vw 2 + 1 
 
 9. vr^: 
 u. i 
 
 u 
 
 15. 
 
 2 Vw 2 - 1 
 
 v*v^/l u 2 . 21. Vl u 2 Vl v 2 uv. 
 
ANSWERS 
 
 169 
 
 - V 2 
 
 
 ' V 2u 
 
 2Q SoT TTTTOV T 
 
 
 1 1 
 
 30 z a - 
 
 cot 
 31. 1 
 
 1 34. .: 
 
 V 
 
 cot" 1 a a 
 
 rt 2 
 
 VI + a 2 
 32 VT=rf-aV 
 
 .4 a . 
 
 \Xl-62_j_5Vl-a 2 
 
 35. -i (1 + Va 2 1 
 a& 
 
 
 V6 2 -l). 
 
 Art. 101 ; Page 98 
 
 1. 6 = 24.5 
 c = 31.8 
 
 y=78. 
 
 2. a 
 
 & 
 c 
 
 = 50 
 = 54.9 
 
 = 58.6 
 
 3. a = 61.4 
 c = 47.7 
 
 y = 48. 
 
 4. /? 
 
 y 
 
 c 
 
 = 48 20' 
 = 62 40' 
 = 76.1 
 
 5. = 68 22' 
 y = 56 58' 
 a = 107 
 
 6. 
 
 c 
 
 = 40 48' 
 = 104 52' 
 = 141. 
 
 7. a = 69 22' 
 = 38 38' 
 
 c = 42.7 
 
 8. ft 
 
 y 
 
 c 
 
 = 38 37' 
 = 31 23' 
 = 173. 
 
 9. a = 44 42' 
 = 60 20' 
 
 y = 74 54' 
 
 10. a 
 
 y 
 
 = 72 21' 
 
 = 49 38' 
 = 58 V 
 
 11. /3 = 34 44' or 
 y = 125 16' or 
 c = 35.8 or 
 
 146 16' 12. a 
 13 44' ft 
 10.4 6 
 
 = 51 44' or 128 16' 
 = 84 26' or 7 54' 
 = .431 or .059 
 
170 TRIGONOMETRY 
 
 Art. 101 ; Page 98 
 
 13. y = 39 49' 50" 14. a = 13.081 
 
 a = 41.581 c = 13.620 
 
 c = 41.432 /? = 972'15" 
 
 15. a = 90 20' 34" 16. a = 126 59' 18" 
 
 ft = 65 17' 34" ft = 27 29' 38" 
 
 y= 24 21' 50" y = 2531'4" 
 
 17. a = 1.9555 18. y = 77 22' 16" 
 
 a = 43 36' 35" ft = 51 57' 20" 
 
 ft = 15 37' 7" a = 83.732 
 
 19. a = 87 33' 58" 20. a = 78 40' 32" 
 
 y = 18 6' 24" ft = 41 20' 47" 
 
 b = 1.4033 y = 59 58' 41'^ 
 
 21. a = 32 24' 0" 22. Impossible. 
 
 ft = 55 0' 28" 
 y = 92 35' 32" 
 
 23. a = 142 28' 9" or 21 31' 11" 
 ft = 29 31' 31" or 150 28' 29" 
 a = .080746 or .04862 
 
 24. y = 76 50' 20" or 103 9' 40" 
 a =69 39' 35" or 43 20' 15" 
 a = 17.405 or 12.739 
 
 25. c = 502.28 26. a = 96 9' 32" 
 b = 300.25 . ft = 41 11' 10" 
 
 y = 23 7' 3" y = 42 39' 18" 
 
 27. .4 = 150. 30. ^ = 108.61 
 
 33. A = 368.91. 35. 172.8 ft. 
 
 36. 106.1ft. 37. 3.710 mi. 
 
 38. 97.14 
 
 124.59 39. 60.1ft. 
 178.64 
 
 40. 57.93 mi. per Lr. -41. 3888.0ft. 
 
 42. 6328.7 ft. 43. 239600 mi. 
 
ANSWERS 
 
 171 
 
 Art. 102 ; Page 102 
 
 1. 
 
 4. 4258 / 18 // . 5. 35 48' 35". 6. 13.754 in. 
 7. . 8. 
 
 2. 60',72M35'. 3. , , , 
 
 7T 
 
 9* 
 
 7T 7T 7T 
 
 6' 3' 2 
 
 55 7T 73 7T 
 
 144 144 
 
 9. 
 
 4 7T 
 
 9 ' 3' 9 
 
 12. 
 50. 
 51. 
 54. 
 56. 
 57. 
 59. 
 60. 
 61. 
 62. 
 63. 
 64. 
 66. 
 67. 
 68. 
 
 69. 
 73. 
 
 11 tE 10?r 16 22 TT 28 TT 34 TT 40 * 
 
 ' 77 ' 77 ' 77 ' 77 ' 77 ' 77 ' 77 ' 
 
 9fin. 49. 0, 180. 
 
 30, 150, 210, 330, 45, 135, 225, 315. 
 
 30, 150. 52. tan-\(2 V3). 53. 0, 180, cos' 1 f 
 
 45, 135, 225, 215, sin' 1 Vf 55. 60, 300, 90, 270. 
 
 60, 120, 240, 300, 45, 135, 225, 315. 
 
 tan- 1 !. 58. 90, 270, sin- 1 (-|i). 
 
 0, 45, 90, 135, 180, 225 a , 270, 315. 
 
 0, 45, 60, 90, 120, 180, 225, 240, 270, 300, 315. 
 
 7i, 37i, 67$., 97i, 127^, 1571, etc. 
 
 30, 150, 210, 330. 
 
 90, 270, 70, 110, 190, 230, 310, 350. 
 
 90, 180. 65. 210, 330. 
 
 45, 215, 671, 1571, 247f, 337^. 
 
 60, 120, 240, 300, 18, 54, 90, 126, 162, etc. 
 
 60, 90, 120, 240, 270, 300. 
 
 70. 0, V|. 71. 
 
172 
 
 TRIGONOMETRY 
 
 99. 532. 100. 50.0. 101. 1478.5. 
 
 102. 93,470,000 mi. 103. 51.9 ft. 104. 27.925. 
 
 105. 112.36. 106. - 107. 129.90 ft. 
 
 106. f 
 
 108. 8.2596. 109. 70 31' 43". 
 
 111. 196. 112. 89.431. 
 
 115. 90 ft., 40 ft. 116. 13.66. 
 
 118. 820.54. 119. 535.4. 
 
 121. 567.3. 122. 132.6. 
 
 124. 641. 125. 962.605. 
 
 128. a. 129. 4009. 
 
 110. 5296 ft., 251 ft. 
 114. V2. 
 117. 103.97. 
 120. 25.43. 
 123. 36.7 ft. 
 127. 16.33 N. 75 36' E. 
 130. 1674.3. 
 
 Art. 116 ; Page 124 
 
 1. 1, i, 1, i. 
 2.1, i 
 
 4. (2 + 0- 
 
 6. (1.272 + .786 1). 
 
 7. 
 
 8. _ 
 10. 1.0842 + .29051 i, 
 
 - .79370 + .79370 1, 
 
 - .29051 - 1.0842 1. 
 
 5. (l-2t). 
 
 Art. 139 ; Page 143 
 
 1. /3 = 789'22" 
 c = 10 45' 55" 
 a = 2 14' 5" 
 
 3. 6 = 8 26' 14" 
 a = 120 59' 19" 
 a = 95 2' 10" 
 
 2 ? a= 41 11' 53" 
 
 = 56 19' 56" 
 c = 40 27' 11" 
 
 4. ^ = 75 21' 53" 
 6 = 44 43' 49" 
 a = 14 59' 33" 
 
ANSWERS 173 
 
 5. a = 111 23' 47'.' 6. ft =101 38' 28" 
 = 120 40' 56" a = 112 13'. 48" 
 
 c = 76 33' 24" b = 102 35' 26" 
 
 7. = 46r28" 8. a = 68 42' 11" 
 b = 15 18' 0" = 15548'0" 
 
 a = 46 2' 40" a = 27 37' 26" 
 
 9. p = 153 31' 29" or 26 28' 31" 
 c = 50 43' 22" or 129 16' 38" 
 b = 159 48' 44" or 20 11' 16" 
 
 Art. 142; Page 144 
 
 1. a = 117 45' 28" 
 
 2. y = 88 23' 11" 
 
 = 9627'1" 
 
 a = 69 48' 42" 
 
 y = 930'61" 
 
 b = 94 22' 46" 
 
 3. a = 8 49' 46" 
 
 4. a =160 13' 48" 
 
 y = 283'4" 
 
 = 105 21' 16" 
 
 b = 10.6 56' 53" 
 
 y = 104 25' 45" 
 
 5. Sides, 32 45' 6" 
 
 6. Sides, 112 32' 20" 
 
 angles, 105 49' 32" 
 
 base, 46 15' 12" 
 
 Art. 
 
 156 ; Page 154 
 
 1. = 110'47" 
 
 2. = 14 53' 47" 
 
 y=928'27" 
 
 y = 170 26' 51" 
 
 a = 114 42' 50" 
 
 a = 55 56' 0" 
 
 3. = 71r23" 
 
 6. a = 24 35' 10" 
 
 y = 84 22' 25" 
 
 c = 43 29' 48" 
 
 6 = 821'30" 
 
 = 154 19' 20" 
 
 8. a = 133 28' 34" 
 
 9. a = 104 49' 50" 
 
 = 169 38' 12" 
 
 = 84 51' 42" 
 
 y = 132 6' 14" 
 
 y = 95 18' 24" 
 
 10. 
 
 a = 84 57' 8" 
 
 
 = 57 47' 44" 
 
 
 v = 434'36" 
 
174 TRIGONOMETRY 
 
 14. a = 155 14' 24" or 21 52' 40" 
 = 25 50' 58" or 154 9' 2" 
 a = 107 34' 50" or 58 0' 44" 
 15. Two solutions. 24. No solution. 
 
 Art. 160 ; Page 159 
 2. 2229| miles. 3. 4291J miles. 
 

 
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