6. If sin A = angles and less than 90°, find sin (A+B+C). , sin B13, and sin C = 5, where A, B, C are positive 4 cos3 x 3 cos x, find sin 18°. 7. Assuming the equation cos 3x [SOLUTION: 54° + 36° = 90°. .. cos 54° = sin 36°; i. e. cos 3. 18° = sin 2.18°. Hence, 4 cos3 18° - 3 cos 18° - 2 sin 18° cos 18°. .. 4 cos2 18° - 3 = 2 sin 18°. On putting 1-sin2 18° for cos2 18°, and solving for sin 18°, there is obtained the result, sin 18° : = √5-1 8. Assuming the result in Ex. 7, find the other trigonometric functions of 18° and the functions of 72°. √5+1 9. Assuming the result in Ex. 7, show that cos 36° sin 54°. Hence, deduce the other trigonometric functions of 36° and 54°. Also, deduce the trigonometric functions of 9° and 81°. (The results in Exs. 8, 9, can be verified by means of the tables.) 10. Prove the formulas : sin (36° + A) – sin (36° — A) — sin (72° + A) + sin (72° — A) = sin A, cos (36° + A) + cos(36° — A) — cos (72° + A) — cos (72° — A) = cos A, and explain their use. (See Art. 97.) 11. (a) Show that sin2 30° = sin 18° sin 54°. (b) Solve x3 cot 108° = 128 sin 72° cos 18°, without tables. (c) Find the trigonometric functions of 48°. 12. Two parallel chords of a circle lying on the same side of the centre of a circle subtend angles of 72° and 144° at the centre. Show that the distance between the chords is equal to half the radius of the circle, (a) using tables, (b) not using tables. 13. (a) Solve: (i.) cos 0 = 0; (ii.) sin x cos x = 1; = = 14. (a) If in triangle ABC, A = 3 B, show that sin B (b) Given cos A = .28, find tan A that presents itself in the result. 15. Prove the following: Explain the reason of the ambiguity (c) If sin A = 2 ab' a2 + b2' find tan (iii.) sin A+ sin 3 A + sin 5 A + sin 7 A : 16 sin A cos2 A cos2 2 A. (iv.) cos 6 A (vi.) 4 cos3 A sin 3 A + 4 sin3 A cos 3 A = 3 sin 4 A. (vii.) cos 20° cos 40° cos 80° = }. (viii.) sin3 A + sin3 (120° + A) + sin3 (240° + A) = − & sin 3 A. (ix.) 1 + cos 2 (A – B) cos 2 B = cos2 A + cos2 (A – 2 B). (x.) 2 cosec 4 A+ 2 cot 4 A = cot A - tan A. 16. Show that cos (36° + A) cos (36° — A) + cos (54° + A) cos (54° — A) = cos 2 A ; sin 3 A = 4 sin A sin (60° + A) sin (60° — A). 18. Prove that the following equations are true for certain values of the angles : (i.) 3 sin-1 x = sin-1 (3x-4x3). (ii.) 3 cos-1x = cos-1 (4x3- 3 x). (iii.) tan-1x + tan−1y + tan-1z = tan-1 x + y + z − xYz 1xyyz yxyπ 1 + x + y − xy 4 (v.) Given tan α = · 1, tan ß = }, tan y = }, find tan (α + ß + y). (vii.) tan-11= tan-1 (viii.) sin-1 13 5 65 2 4 √5 11 √146 12 (xii.) tan-tan-112. 19. The hypotenuse and shortest side of a right-angled triangle are 5 ft. and 3 ft., respectively. Find the length of the perpendicular from the right angle upon the hypotenuse, and show that it is inclined at sin-1 to the straight line drawn from the right angle to the middle point of the hypotenuse. 25 20. If a triangle ABC is to be solved from given parts A, a, b, show that the solution is sometimes ambiguous; and that in such a case the difference b sin A of the two values of C is 2 cos-1 α 21. The tangent of an angle is 2.4. Find the cosecant of the angle, the cosecant of half the angle, and the cosecant of the supplement of double the angle. 22. The angle of elevation of a tower at a distance of 20 yd. from its foot is three times as great as the angle of elevation 100 yd. from the same point; show that the height of the tower is 300: √7 ft. 23. DE is a tower on a horizontal plane. ABCD is a straight line in the plane. The tower subtends an angle @ at A, 20 at B, and 30 at C. If AB 50 ft., and BC= 20 ft., find the height of the tower and the distance CD. 24. A ship sailing at a uniform rate was observed to bear N. 30° 57' 30" E. After 20 minutes she bore N. 35° 32′ 15′′ E., and after 10 minutes more, N. 37° 52′ 30′′ E. Find the direction in which she was sailing. [Ans. S. 44° 38' E.] 25. A spectator observes the explosion of a meteor, due south of him, at an elevation of 28° 45'. To another spectator, 11 mi. S.S.W. of the former, it appears at the same instant to have an altitude of 42° 15' 30". Show that there are two possible heights above the earth's surface at which it may have exploded, and find these heights. [Ans. 4.33 mi. or 13.21 mi.] ANSWERS TO THE EXAMPLES. N.B. Not all of the answers to the exercises are given. In various ways, the student should test, or check, every result that he obtains in working the problems. CHAPTER I. Art 2. 1. logs 27 = 3, log4 256 = 4, log11 121 = 2, 2. 288, 54 = = 625, ***, na P. 3. 0, 1, 2, 3, 4, 5, 6, 7, 8. 64, 256, 1024. 7. 0, 1; 1, 2; 2, 3 ; 3, 4; 3, 4; 0, −1; −1, Art. 6. 6(a). 1.4007. 6(d). .09856. 7(b). 7.2767. 10. 9.214. 12. .6443. 13(a). 3.236. 13(c). 1.5563. logm p = b. 6. 1, 4, 16, −2; −2, −3. 8(a). 7.937. Art. 11. (In these answers, h, p, b represent hypotenuse, perpendicular, Р b p .466, -=2.14. 4. 56° 19', 33° 41′ (nearly), h = 54.08, ¿=1.8, h .75, -1.33, b b Ρ 5. 41° 25', 48° 35' (nearly), p=39.7, =.88, 1.13. 6. 50°, p=59.6, h=77.8, Ρ b h = .643, =1.56, =.766, h 1.31,2=1 b = .839. h h Ρ Ρ Art. 15. 1. 2.28025. 2. 2.3333. 3. 5.846. 9. 2.75. 10. - .708. Art. 18. 15. 90°, 36° 52′ 12′′. 16. 45°. 17. 45°, 71° 34'. 18. 53° 7'48". 19. 30°, 48° 35' 25". 20. 36° 52' 12", 16° 15' 36". CHAPTER III. Art. 27. 5. A=65° 14', b=7.834. 6. B=50° 12' 24". 9. A=30° 12' 12". 11. b 215.6. 12. a 312.23. CHAPTER IV. Art. 28. 1. Art. 29. 10. 86.6, 50. 24.948, 12.71. 2. 58.78. 4. 398.19 ft. 5. 228.4, 258 ft. 6. 63.88 ft. 7. 276.95 ft. 12. 219.45 ft. 1. 26.172, 52.345 mi., second ship bears E. 19° 42'.1 N. from 14.197 mi. Art. 30. first. 2. LB = 2, 3. 2392.18 sq. ft. 5. 22.5 sq. ft. Art. 33. 1. 14.54 ft., 16.13 ft., 48.45 sq. ft., 20.415 ft., 133.94 sq. ft., 318.4 sq. ft. Art. 34 b. 2. 10.954 mi. 3. 96 ft. Art. 31. 6. 435.7 sq. ft. Art. 32. 2. Base = 187.9 ft.; height = 350.63 ft.; area = 3. Base 358.21 ft.; height = 161.26 ft.; area = 28,881 sq. ft. 105.2 sq. ft. 32,943 sq. ft. 2. 16.516 ft., 4. 14.454 mi. 5. 67.08 mi., 19. 60°, Art. 44. 18. 45°, 135°, — 225°, — 315°; 45°, 135°, 405°, 495°. 240°, 120°, - 300°; 60°, 240°, 420°, 600°. 20. 135°, 225°, - 135°, - 225°; 135°, 225°, 495°, 585°. 21. 150°, 330°, — 30°, — 210°; 150°, 330°, 510°, 690°. CHAPTER VI. Art. 46. 8. cos(x + y) = .7874, sin (x + y) = .6164. (Verify by tables.) Art. 47. tables.) (Verify by 3. sin (xy) = -.1582, cos (x − y) = .9874. = 1-2 sin2 x = 2 cos2x - 1. √1 - cos 12x. 12. sin x = Art. 50. 7. cos 6 x = cos2 3x - sin23 x - 2 sin2 3 x = 2 cos2 3 x 1, sin 6 x 2 sin 3 x cos 3x. 9. sin x=2 sin x cos x, cos x=cos2 x-sin2 3 x 10. cos 6x= √1+ cos 12x, sin 6x √1 – cos 3 x, cos § x = √1 + cos § x. 1 = |