4-7 The Law of Sines and the Law of Cosines
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1 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 27. ABC, if A = 42, b = 12, and c = 19 Use the Law of Cosines to find the missing side measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, B 39, C 99 and a esolutions Manual - Powered by Cognero Page 1
2 29. PQR, if P = 73, q = 7, and r = 15 Use the Law of Cosines to find the missing side measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, Q 27, R 80 and p esolutions Manual - Powered by Cognero Page 2
3 31. RST, if r = 35, s = 22, and t = 25 Use the Law of Cosines to find the missing angle measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, R 96, S 39 and T 45. esolutions Manual - Powered by Cognero Page 3
4 33. BCD, if B = 16, c = 27, and d = 3 Use the Law of Cosines to find the missing side measure. Use the Law of Sines to find a missing angle measure. Find the measure of the remaining angle. Therefore, C 162, D 2 and b esolutions Manual - Powered by Cognero Page 4
5 35. AIRPLANES During her shift, a pilot flies from the Columbus to Atlanta, a distance of 448 miles, and then on to the Phoenix, a distance of 1583 miles. From Phoenix, she returns home to Columbus, a distance of 1667 miles. Determine the angles of the triangle created by her flight path. Draw a diagram to represent the situation. Use the Law of Cosines to find an angle measure. Use the Law of Sines to find a second angle measure. Find A. Therefore, the angles of the triangle created by the flight path are about 15.6, 71.5, and esolutions Manual - Powered by Cognero Page 5
6 Use Heron s Formula to find the area of each triangle. Round to the nearest tenth. 37. x = 9 cm, y = 11 cm, z = 16 cm First, find the value of s. Next, use Heron's Formula find the area of. Therefore, the area of is about 47.6 cm x = 58 ft, y = 40 ft, z = 63 ft First, find the value of s. Next, use Heron's Formula find the area of. Therefore, the area of is about ft 2. esolutions Manual - Powered by Cognero Page 6
7 41. x = 8 yd, y = 15 yd, z = 8 yd First, find the value of s. Next, use Heron's Formula find the area of. Therefore, the area of is about 20.9 ft 2. esolutions Manual - Powered by Cognero Page 7
8 43. LANDSCAPING The Steele family want to expand their backyard by purchasing a vacant lot adjacent to their property. To get a rough measurement of the area of the lot, Mr. Steele counted the steps needed to walk around the border and diagonal of the lot. a. Estimate the entire area in steps. b. Mr. Steele measured his step to be 1.8 feet. Determine the area of the lot in square feet. a. Find the area of the Steele s property. First, find s. Use Heron's Formula find the area of the triangle. Next, find the area of the vacant lot. Use Heron's Formula find the area of the triangle. Therefore, the total area is or about square steps. b. Use dimensional analysis to convert the area from square steps to square feet. Therefore, the area is about 14,617 square feet. esolutions Manual - Powered by Cognero Page 8
9 Find the area of each triangle to the nearest tenth. 45. ABC, if A = 98, b = 13 mm, and c = 8 mm Therefore, the area of ABC is about 51.5 mm RST, if R = 35, s = 42 ft, and t = 26 ft Therefore, the area of ABC is about mm FGH, if F = 41, g = 22 in., and h = 36 in. Therefore, the area of ABC is about mm LIGHTHOUSES The bearing from the South Bay lighthouse to the Steep Rock lighthouse 25 miles away is N 28 E. A small boat in distress spotted off the coast by each lighthouse has a bearing of N 50 W from South Bay and S 80 W from Steep Rock. How far is each tower from the boat? Draw a diagram to represent the situation. esolutions Manual - Powered by Cognero Page 9
10 Recall from Geometry that when two parallel lines are cut by a transversal, then alternate interior angles are congruent. Therefore, C in ABC is = 52. B = = 78 A = 180 ( ) = 50 Use the Law of Sines to find the distance from the lighthouse at South Bay to the boat. Use the Law of Sines again to find the distance from the lighthouse at Steep Rock to the boat. Therefore, South Bay is about miles from the boat and Steep Rock is about miles from the boat. esolutions Manual - Powered by Cognero Page 10
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