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Stitz-Zeager_College_Algebra_e-book

# Provided x 0 or 0 x 2 2 additionally arccosecant

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Unformatted text preview: rigonometric Identities 663 Finally, we exchange sin2 (θ) for 1 − cos2 (θ) courtesy of the Pythagorean Identity, and get cos(3θ) = = = = 2 cos3 (θ) − cos(θ) − 2 sin2 (θ) cos(θ) 2 cos3 (θ) − cos(θ) − 2 1 − cos2 (θ) cos(θ) 2 cos3 (θ) − cos(θ) − 2 cos(θ) + 2 cos3 (θ) 4 cos3 (θ) − 3 cos(θ) and we are done. In the last problem in Example 10.4.3, we saw how we could rewrite cos(3) as sums of powers of cos(θ). In Calculus, we have occasion to do the reverse; that is, reduce the power of cosine and sine. Solving the identity cos(2θ) = 2 cos2 (θ) − 1 for cos2 (θ) and the identity cos(2θ) = 1 − 2 sin2 (θ) for sin2 (θ) results in the aptly-named ‘Power Reduction’ formulas below. Theorem 10.18. Power Reduction Formulas: For all angles θ, • cos2 (θ) = 1 + cos(2θ) 2 • sin2 (θ) = 1 − cos(2θ) 2 Example 10.4.4. Rewrite sin2 (θ) cos2 (θ) as a sum and diﬀerence of cosines to the ﬁrst power. Solution. We begin with a straightforward application of...
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