Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 6 - Analytic Trigonometry - Section 6.7 Product-to-Sum and Sum-to-Product Formulas - 6.7 Assess Your Understanding - Page 524: 26


$\cos \theta$

Work Step by Step

Recall the Sum to Product Identities: $a) \sin x +\sin y =2 \sin \dfrac{x+y}{2} \cos \dfrac{x-y}{2} \\ b) \sin x - \sin y =2 \sin \dfrac{x -y}{2} \cos \dfrac{x +y}{2} \\ c) \cos x +\cos y =2 \cos \dfrac{x+y}{2} \cos \dfrac{x-y}{2} \\ d) \cos x - \cos y = -2 \sin \dfrac{x+y}{2} \sin \dfrac{x-y}{2}$ By identity $(c)$, we have: $\dfrac{\cos 3 \theta + \cos \theta }{2 \cos 2 \theta}=\dfrac{2 \cos \dfrac{ 3 \theta + \theta}{2} \ \cos \dfrac{3 \theta - \theta }{2} }{2 \cos 2 \theta}\\=\dfrac{2 \cos 2 \theta \ \cos \theta}{2 \cos 2 \theta} \\=\cos \theta$
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