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: 17


$2 \sin \theta \cos 3 \theta$

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 $(b)$, we have: $\sin 4 \theta -\sin 2 \theta =2 \sin \dfrac{4 \theta - 2 \theta}{2} \ \cos \dfrac{4 \theta + 2 \theta }{2} \\=2 \sin \theta \cos 3 \theta$
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