A-41 Answers to Selected Exercises 31. b = 9.53 in., A = 64.6°, C = 40.6° 33. a = 15.7 m, B = 21.6°, C = 45.6° 35. A = 30°, B = 56°, C = 94° 37. The value of cos u will be greater than 1. A calculator will give an error message (or a nonreal complex number) when the inverse cosine function is used. 39. 257 m 41. 163.5° 43. 281 km 45. 438.14 ft 47. 10.8 mi 49. 40° 51. 26° and 36° 53. second base: 66.8 ft; first and third bases: 63.7 ft 55. 39.2 km 57. 47.5 ft 59. 5500 m 61. 16.26° 63. 2423 sq units 65. 78 m2 67. 12,600 cm2 69. 3650 ft2 71. Area and perimeter are both 36. 73. 437,000 mi2 75. (a) 87.8° and 92.2° are possible angle measures. (b) 92.2° (c) With the law of cosines we are required to find the inverse cosine of a negative number. Therefore, we know that angle C is greater than 90°. 77. x y B (–1, 3) 0 A (2, 5) C (4, 0) a b c a = 234, b = 229, c = 213 78. 9.5 sq units 79. 9.5 sq units 80. 9.5 sq units Chapter 8 Quiz [8.1] 1. 131° [8.2] 2. 201 m 3. 48.0° [8.1] 4. 15.75 sq units [8.2] 5. 189 km2 [8.1] 6. 41.6°, 138.4° 7. a = 648, b = 456, C = 28° 8. 3.6 mi [8.2] 9. 25.24983 mi 10. 3921 m 8.3 Exercises 1. m and p; n and r 3. m and p equal 2t, or t equals 1 2 m and 1 2 p. Also m= 1p and n = 1r. 5. –b 7. 2c 9. b a a + b 11. a –c a – c 13. b a + (b + c) a b + c c 15. d c c + d 17. Yes, it appears that vector addition is associative (and this is true, in general). 19. 278 12 20 21. 308 20 30 23. 94.2 lb 25. 24.4 lb 27. 530 newtons 29. 88.2 lb 31. 2640 lb at an angle of 167.2° with the 1480-lb force 33. 93.9° 35. second force: 190 lb; resultant: 283 lb 37. 18° 39. 2.4 tons 41. 17.5° 43. 226 lb 45. 13.5 mi; 50.4° 47. 39.2 km 49. current: 3.5 mph; motorboat: 19.7 mph 51. bearing: 237°; ground speed: 470 mph 53. ground speed: 161 mph; airspeed: 156 mph 55. bearing: 74°; ground speed: 202 mph 57. bearing: 358°; airspeed: 170 mph 59. ground speed: 230 km per hr; bearing: 167° 8.4 Exercises 1. 2 3. 22 2 5. 84, 99 7. ac + bd 9. 274; 54.5° 11. 17; 331.9° 13. 8; 120° 15. 47, 17 17. 38.8, 28.0 19. -123, 155 21. 8 5 23 2 , 5 2 9 23. 8-3.0642, 2.57129 25. 84.0958, -2.86799 27. (a) 8-4, 169 (b) 8-12, 09 (c) 88, -89 29. (a) 88, 09 (b) 80, 169 (c) 8-4, -89 31. (a) 80, 129 (b) 8-16, -49 (c) 88, -49 33. 8-6, 29 35. 88, -209 37. 8-30, -39 39. 88, -79 41. (a) 4i (b) 7i + 3j (c) -5i + j 43. (a) 8-2, 49 (b) 87, 49 (c) 86, -69 45. -5i + 8j 47. 2i, or 2i + 0j 49. 7 51. 0 53. 20 55. 135° 57. 90° 59. 150° 61. 17.59° 63. 36.87° 65. 180° 67. -6 69. -24 71. orthogonal 73. not orthogonal 75. not orthogonal 77. orthogonal 79. (a) R = 25 ≈2.2, A = 21.25 ≈1.1; 2.2 in. of rain fell. The area of the opening of the rain gauge is 1.1 in.2. (b) V = 1.5; The volume of rain was 1.5 in.3. In Exercises 81–85, answers may vary due to rounding. 81. magnitude: 9.5208; direction angle: 119.0647° 82. 8-4.1042, 11.27639 83. 8-0.5209, -2.95449 84. 8-4.6252, 8.32199 85. magnitude: 9.5208; direction angle: 119.0647° 86. They are the same. Preference of method is an individual choice. Summary Exercises on Applications of Trigonometry and Vectors 1. 29 ft; 38 ft 2. 38.3 cm 3. 5856 m 4. 15.8 ft per sec; 71.6° 5. 42 lb 6. 7200 ft 7. (a) 10 mph (b) 3v = 18i + 24j; This represents a 30-mph wind in the direction of v. 8. 380 mph; 64° 9. It could not exist. 10. Other angles can be 36° 10′, 115° 40′, third side 40.5, or other angles can be 143° 50′, 8° 00′, third side 6.25. (Lengths are in yards.) 8.5 Exercises 1. (a) 2 (b) 21cos 0° + i sin 0°2 3. (a) 2i (b) 21cos 90° + i sin 90°2 5. (a) 2 + 2i (b) 2221cos 45° + i sin 45°2 7. multiply; add 9. 10; 180°; 180°; -10; 0 11. 0°; 1; 0
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