576 CHAPTER 8 Applications of Trigonometric Functions where θ is the angle of rotation of rod OA. A L x O B u r 61. Challenge Problem Geometry Show that the length d of a chord of a circle of radius r is given by the formula d r2 sin 2 θ = where , 0 , θ θ π < < is the central angle formed by the radii to the ends of the chord. See the figure. Use this result to derive the fact that sin , θ θ < where θ is measured in radians. 62. Challenge Problem For any triangle, show that C s s c ab cos 2 ( ) = − where s a b c 1 2 . ( ) = + + 63. Challenge Problem For any triangle, show that C s a s b ab sin 2 ( )( ) = − − where s a b c 1 2 . ( ) = + + 64. Challenge Problem Use the Law of Cosines to prove the identity A a B b C c a b c abc cos cos cos 2 2 2 2 + + = + + 57. Wrigley Field, Home of the Chicago Cubs The distance from home plate to the fence in dead center in Wrigley Field is 400 feet (see the figure). How far is it from the fence in dead center to third base? 90 ft 90 ft 400 ft 58. Little League Baseball The distance from home plate to the fence in dead center at the Oak Lawn Little League field is 280 feet. How far is it from the fence in dead center to third base? [Hint: The distance between the bases in Little League is 60 feet.] 59. Building a Swing Set Clint is building a wooden swing set for his children. Each supporting end of the swing set is to be an A-frame constructed with two 10-foot-long 4 by 4’s joined at a 45° angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame to concrete footings. How far apart should the footings for each A-frame be? 60. Rods and Pistons See the figure (top, right). Rod OA rotates about the fixed point O so that point A travels on a circle of radius r. Connected to point A is another rod AB of length L r2 , > and point B is connected to a piston. Show that the distance x between point O and point B is given by x r r L r cos cos 2 2 2 2 θ θ = + + − d r r u O Explaining Concepts 65. What do you do first if you are asked to solve a triangle and are given two sides and the included angle? 66. What do you do first if you are asked to solve a triangle and are given three sides? 67. Make up an applied problem that requires using the Law of Cosines. 68. Write down your strategy for solving an oblique triangle. 69. State the Law of Cosines in words. Retain Your Knowledge Problems 70–79 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 70. Graph: R x x x 2 1 3 ( ) = + − 71. Solve 4 3 . x x 1 = + Express the solution in exact form. 72. If tan 2 6 5 θ = − and cos 5 7 , θ = − find the exact value of each of the four remaining trigonometric functions. 73. Find an equation for the graph. x y 3 23 p –– 4 3p––– 8 5p––– 8 2 p –– 8 p –– 8 p –– 4 p –– 2 2
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