572 CHAPTER 8 Applications of Trigonometric Functions Figure 38 N S W E Key West Naples 60 A 150 u x 208 Using the Law of Cosines to Solve an SSS Triangle Solve the triangle: a b c 4, 3, 6 = = = Solution EXAMPLE 2 See Figure 37. To find the angles A B , , and C, use the Law of Cosines. For A: A b c a bc A cos 2 9 36 16 2 3 6 29 36 cos 29 36 36.3 2 2 2 1 = + − = + − ⋅ ⋅ = = ≈ ° − For B: B a c b ac B cos 2 16 36 9 2 4 6 43 48 cos 43 48 26.4 2 2 2 1 = + − = + − ⋅ ⋅ = = ≈ ° − Now use A and B to find C: C A B 180 180 36.3 26.4 117.3 = °−−≈ °− °− °= ° Figure 37 B A C 6 4 3 Now Work PROBLEMS 15 AND 25 Correcting a Navigational Error A motorized sailboat leaves Naples, Florida, bound for Key West, 150 miles away. Maintaining a constant speed of 15 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds, after 4 hours, that the sailboat is off course by 20 .° (a) How far is the sailboat from Key West at this time? (b) Through what angle should the sailboat turn to correct its course? (c) How much time has been added to the trip because of this? (Assume that the speed remains at 15 miles per hour.) EXAMPLE 3 Solution See Figure 38.With a speed of 15 miles per hour, the sailboat has gone 60 miles after 4 hours. The distance x of the sailboat from Key West is to be found, along with the angle θ that the sailboat should turn through to correct its course. (a) To find x, use the Law of Cosines, because two sides and the included angle are known. x x 150 60 2 150 60 cos20 9185.53 95.8 2 2 2 = + − ⋅ ⋅ ⋅ ° ≈ ≈ The sailboat is about 96 miles from Key West. (b) With all three sides of the triangle now known, use the Law of Cosines again to find the angle A opposite the side of length 150 miles. A A A A 150 96 60 2 96 60 cos 9684 11,520cos cos 0.8406 147.2 2 2 2 = + − ⋅ ⋅ ⋅ = − ≈ − ≈ ° So, A 180 180 147.2 32.8 θ = °− ≈ °− ° = ° The sailboat should turn through an angle of about 33° to correct its course. 3 Solve Applied Problems
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