SECTION 7.6 Double-angle and Half-angle Formulas 527 θ θ θ ( ) ( ) = − + tan 1 cos 2 1 cos 2 2 (8) Formulas (6) through (8) do not have to be memorized since their derivations are straightforward. Formulas (6) and (7) are important in calculus. The next example illustrates a problem that arises in calculus requiring the use of formula (7). Establishing an Identity Write an equivalent expression for θ cos4 that does not involve any powers of sine or cosine greater than 1. Solution EXAMPLE 3 The idea here is to use formula (7) twice. θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) = = + = + + = + + = + + ⋅ + ⋅ = + + + = + + cos cos 1 cos 2 2 1 4 1 2cos 2 cos 2 1 4 1 2 cos 2 1 4 cos 2 1 4 1 2 cos 2 1 4 1 cos 2 2 2 1 4 1 2 cos 2 1 8 1 cos 4 3 8 1 2 cos 2 1 8 cos 4 4 2 2 2 2 2 Formula (7) Formula (7) Now Work PROBLEM 43 Solving a Trigonometric Equation Using Identities Solve the equation: θ θ θ π = − ≤ < sin cos 1 2 , 0 2 EXAMPLE 4 Solution The left side of the equation, except for a factor of 2, is in the form of the Doubleangle Formula, θ θ θ ( ) = 2 sin cos sin 2 . Multiply both sides by 2. θ θ θ θ θ ( ) = − = − = − sin cos 1 2 2 sin cos 1 sin 2 1 The argument is θ2 . Write the general formula that gives all the solutions of this equation, and then list those that are in the interval π [ ) 0, 2 . Because π π ( ) + = − k sin 3 2 2 1, for any integer k , this means that θ π π θ π π = + = + k k 2 3 2 2 3 4 Multiply both sides by 2. Double-angle Formula k an integer θ π π π θ π π π θ π π π θ π π π ( ) = + − = − = + ⋅ = = + ⋅ = = + ⋅ = 3 4 1 4 , 3 4 0 3 4 , 3 4 1 7 4 , 3 4 2 11 4 ↑ ↑ ↑ ↑ =− = = = k k k k 1 0 1 2 (continued)
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