783 8.1 The Law of Sines 37. B = 74.3°, a = 859m, b = 783m 38. C = 82.2°, a = 10.9km, c = 7.62km 39. A = 142.13°, b = 5.432 ft, a = 7.297 ft 40. B = 113.72°, a = 189.6yd, b = 243.8yd Concept Check Answer each question. 51. Apply the law of sines to the following: a = 25, c = 225, A = 30°. What is the value of sin C? What is the measure of C? Based on its angle measures, what kind of triangle is triangle ABC? 52. What condition must exist to determine that there is no triangle satisfying the given values of a, b, and B, once the value of sin A is found by applying the law of sines? 53. Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20′, a = 14.6 ft, b = 20.4 ft. 54. Apply the law of sines to the following: A = 104°, a = 26.8, b = 31.3. What happens when we try to find the measure of angle B using a calculator? Solve each triangle ABC that exists. See Examples 4–6. 41. A = 42.5°, a = 15.6 ft, b = 8.14 ft 42. C = 52.3°, a = 32.5yd, c = 59.8yd 43. B = 72.2°, b = 78.3m, c = 145m 44. C = 68.5°, c = 258 cm, b = 386 cm 45. A = 38° 40′, a = 9.72m, b = 11.8m 46. C = 29° 50′, a = 8.61m, c = 5.21m 47. A = 96.80°, b = 3.589 ft, a = 5.818 ft 48. C = 88.70°, b = 56.87m, c = 112.4m 49. B = 39.68°, a = 29.81m, b = 23.76m 50. A = 51.20°, c = 7986 cm, a = 7208 cm Solve each problem. See Examples 2 and 3. 55. Distance across a River To find the distance AB across a river, a surveyor laid off a distance BC = 354m on one side of the river. It is found that B = 112° 10′ and C = 15° 20′. Find AB. See the figure. 56. Distance across a Canyon To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582yd. She then finds that T = 32° 50′ and R = 102° 20′. Find RS. See the figure. 57. Distance a Ship Travels A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse? 58. Distance between Radio Direction Finders Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A. A B C 354 m 1128 109 158 209 328 509 1028 209 T R S 582 yd
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