AN62 Answers: Chapter 8 CHAPTER 8 Applications of Trigonometric Functions 8.1 Assess Your Understanding (page 553) 4. F 5. b 6. angle of elevation 7. T 8. F 9. sin 5 13 ; cos 12 13 ; tan 5 12 ; cot 12 5 ; sec 13 12 ; csc 13 5 θ θ θ θ θ θ = = = = = = 11. sin 2 13 13 ; cos 3 13 13 ; tan 2 3 ; cot 3 2 ; sec 13 3 ; csc 13 2 θ θ θ θ θ θ = = = = = = 13. sin 3 2 ; cos 1 2 ; tan 3; cot 3 3 ; sec 2; csc 2 3 3 θ θ θ θ θ θ = = = = = = 15. sin 6 3 ; cos 3 3 ; tan 2; cot 2 2 ; sec 3; csc 6 2 θ θ θ θ θ θ = = = = = = 17. sin 5 5 ; cos 2 5 5 ; tan 1 2 ; cot 2; sec 5 2 ; csc 5 θ θ θ θ θ θ = = = = = = 19. 0 21. 1 23. 0 25. 0 27. 1 29. a c A 13.74, 14.62, 70 ≈ ≈ = ° 31. b c A 5.03, 7.83, 50 ≈ ≈ = ° 33. a c B 0.71, 4.06, 80 ≈ ≈ = ° 35. b c B 10.72, 11.83, 65 ≈ ≈ = ° 37. b a A 3.08, 8.46, 70 ≈ ≈ = ° 39. c A B 5.83, 59.0 , 31.0 ≈ ≈ ° ≈ ° 41. b A B 4.58, 23.6 , 66.4 ≈ ≈ ° ≈ ° 43. 23.6° and 66.4° 45. 4.59 in.; 6.55 in. 47. (a) 5.52 in. or 11.83 in. 49. 70.02 ft 51. 985.91 ft 53. 137.37 m 55. 80.5° 57. (a) 111.96 ft sec or 76.3 mi h (b) 82.42 ft sec or 56.2 mi h (c) Under 18.8° 59. (a) 2.4898 10 miles 13 × (b) 0.000214° 61. 554.52 ft 63. S76.6 E° 65. The embankment is 30.5 m high. 67. 3.83 mi 69. 1978.09 ft 71. 60.27 ft 73. The buildings are 7984 ft apart. 75. 69.0° 77. 38.9° 79. The white ball should hit the top cushion 3.34 ft from the upper left corner. 81. 76.94 in. 86. Yes 87. 6 2 4 or 2 4 3 1 ( ) − − 88. 0.236, 0.243, 0.248 89. 2 , 7 6 , 11 6 π π π { } 90. 21 4 5.25 = 91. 3 5 + 92. 1 93. 65 94. x y 4 5 2 2 ( ) + + = 95. , ( ) −∞∞ 8.2 Assess Your Understanding (page 566) 5. a 6. A a B b C c sin sin sin = = 7. d 8. F 9. F 10. ambiguous case 11. a b A 3.23, 3.55, 40 ≈ ≈ = ° 13. a c B 3.25, 4.23, 45 ≈ ≈ = ° 15. C c a 95 , 9.86, 6.36 = ° ≈ ≈ 17. A a c 40 , 2, 3.06 = ° = ≈ 19. C c a 95 , 9.86, 6.36 = ° ≈ ≈ 21. A a c 69 , 6.23, 4.88 = ° ≈ ≈ 23. B a b 40 , 5.64, 3.86 = ° ≈ ≈ 25. C a b 100 , 1.31, 1.31 = ° ≈ ≈ 27. One triangle; B C c 30.7 , 99.3 , 3.86 ≈ ° ≈ ° ≈ 29. One triangle; C A a 23.8 , 41.2 , 6.55 ≈ ° ≈ ° ≈ 31. One right triangle; B C c 90 , 60 , 12.12 = ° = ° ≈ 33. Two triangles; C A a 30.9 , 129.1 , 9.07 1 1 1 ≈ ° ≈ ° ≈ or C A a 149.1 , 10.9 , 2.20 2 2 2 ≈ ° ≈ ° ≈ 35. No triangle 37. Two triangles; A B b 29 , 139 , 9.47 1 1 1 ≈ ° ≈ ° ≈ or A B b 151 , 17 , 4.22 2 2 2 ≈ ° ≈ ° ≈ 39. 1490.48 ft 41. 335.16 ft 43. 153.42 ft;136.59 ft 45. The tree is 39.39 ft high. 47. Naomi receives 100.6 more frequent flyer miles. 49. (a) Station Able is about 143.33 mi from the ship: Station Baker is about 135.58 mi from the ship. (b) Approximately 41 min 51. 84.7 ; 183.72 ft ° 53. 2.64 mi 55. 38.5 in. 57. 449.36 ft 59. 187,600,000 km or 101,440,000 km 61. The diameter is 252 ft. 63. a b c a c b c A C B C A B C A B A B C C A B C C C A B C sin sin sin sin sin sin sin 2sin 2 cos 2 2sin 2 cos 2 sin 2 cos 2 2 sin 2 cos 2 sin 2 cos 2 π ( ) ( ) ( ) ( ) ( ) − = − = − = − = − + = − − = − 65. a b a b a b c a b c A B C A B C A B C A B C A B A B sin 1 2 cos 2 cos 1 2 sin 2 tan 1 2 cot 2 tan 1 2 tan 2 2 tan 1 2 tan 1 2 π( ) ( ) ( ) ( ) ( ) ( ) ( ) − + = − + = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ 71. 3, 4 3 , 3 { } − − 72. 3 5 6.71 ≈ 73. 15 7 − 74. y 5 x 5P (0, 0) (23P, 4) (24P, 0) (22P, 0) (4P, 0) (2P, 0) (2P, 24) (3P, 24) (P, 4) 75. x log 100 0.2 a = 76. 199.668 77. y 1 3 = 78. y x 3 2 2 = − 79. Neither 80. 6 13 , ( )∞ 8.3 Assess Your Understanding (page 573) 3. Cosines 4. a 5. b 6. F 7. F 8. T 9. b A C 2.95, 28.7 , 106.3 ≈ ≈ ° ≈ ° 11. c A B 3.75, 32.1 , 52.9 ≈ ≈ ° ≈ ° 13. A B C 48.5 , 38.6 , 92.9 ≈ ° ≈ ° ≈ ° 15. A B C 127.2 , 32.1 , 20.7 ≈ ° ≈ ° ≈ ° 17. c A B 2.57, 48.6 , 91.4 ≈ ≈ ° ≈ ° 19. a B C 3.98, 29 , 76 ≈ ≈ ° ≈ ° 21. b A C 6.46, 48.4 , 26.6 ≈ ≈ ° ≈ ° 23. c A B 2.98, 39.1 , 70.9 ≈ = ° = ° 25. A B C 43.6 , 90 , 46.4 ≈ ° = ° ≈ ° 27. A B C 60 , 60 , 60 = ° = ° = ° 29. A B C 29.8 , 65.8 , 84.3 ≈ ° ≈ ° ≈ ° 31. A B C 127.1 , 43.8 , 9.2 ≈ ° ≈ ° ≈ ° 33. A a c 85 , 14.56, 14.12 = ° = = 35. A B C 40.8 , 60.6 , 78.6 = ° = ° = ° 37. A b c 80 , 8.74, 13.80 = ° = = 39. C a b 90 , 4.93, 6.30 = ° = = 41. B C c B C c Two triangles: 35.4 , 134.6 , 12.29; 144.6 , 25.4 , 7.40 1 1 1 2 2 2 = ° = ° = = ° = ° = 43. B C a 24.5 , 95.5 , 10.44 = ° = ° = 45. 165 yd 47. (a) 26.4° (b) 30.8 hr 49. (a) 63.7 ft (b) 66.8 ft (c) 92.8° 51. (a) 492.58 ft (b) 269.26 ft 53. (a) 59.2 mm (b) male 55. (a) 9.9 , 8.3 α β ≈ ° ≈ ° (b) 21.73 yd (c) 0.36 yd or 13 in. ≈ 57. 342.33 ft 59. The footings should be 7.65 ft apart.
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