528 CHAPTER 7 Analytic Trigonometry The solutions in the interval π [ ) 0, 2 are θ π θ π = = 3 4 7 4 The solution set is π π { } 3 4 , 7 4 . FUN FACT When movie-makers film action scenes in which a car jumps off a ramp, laws of physics involving projectile motion are often used to make scenes look as realistic as possible. j Projectile Motion An object is propelled upward at an angle θ to the horizontal with an initial velocity of v0 feet per second. See Figure 39. If air resistance is ignored, the range R —the horizontal distance that the object travels—is given by the function θ θ θ ( ) = R v 1 16 sin cos 0 2 (a) Show that θ θ ( ) ( ) = R v 1 32 sin 2 . 0 2 (b) Find the angle θ for which R is a maximum. Solution EXAMPLE 5 (a) Rewrite the expression for the range using the Double-angle Formula θ θ θ ( ) = sin 2 2 sin cos . Then θ θ θ θ θ θ ( ) ( ) = = = R v v v 1 16 sin cos 1 16 2 sin cos 2 1 32 sin 2 0 2 0 2 0 2 (b) In this form, the largest value for the range R can be found. For a fixed initial speed v ,0 the angle θ of inclination to the horizontal determines the value of R . The largest value of a sine function is 1, which occurs when the argument θ2 is ° 90 . For maximum R , it follows that θ θ = ° = ° 2 90 45 An inclination to the horizontal of ° 45 results in the maximum range. Figure 39 R u Now Work PROBLEM 73 3 Use Half-angle Formulas to Find Exact Values Another important use of formulas (6) through (8) is to prove the Half-angle Formulas . In formulas (6) through (8), let θ α = 2 . Then • α α α α α α α = − = + = − + sin 2 1 cos 2 • cos 2 1 cos 2 • tan 2 1 cos 1 cos 2 2 2 (9) THEOREM Half-angle Formulas i α α = ± − sin 2 1 cos 2 (10) i α α = ± + cos 2 1 cos 2 (11) i α α α = ± − + tan 2 1 cos 1 cos (12) where the + or − sign is determined by the quadrant of the angle α 2 . Solving for the trigonometric functions on the left sides of equations (9) gives the Half-angle Formulas.
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