Precalculus (6th Edition)

Published by Pearson
ISBN 10: 013421742X
ISBN 13: 978-0-13421-742-0

Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises: 15

Answer

$\displaystyle \cos 2\theta=\frac{39}{49}$ $\displaystyle \sin 2\theta=-\frac{4\sqrt{55}}{49}$

Work Step by Step

Plan: work out $\cos\theta$, then apply the double-angle identities. Pythagorean Identity ($\cos \theta$ is positive): $\cos\theta=+\sqrt{1-\sin^{2}\theta}=\sqrt{1-\dfrac{5}{49}}$ $=\displaystyle \sqrt{\frac{44}{49}}=\frac{2\sqrt{11}}{7}$ Double-Angle Identities: $\displaystyle \cos 2\theta=1-2\sin^{2}\theta=1-2(-\frac{\sqrt{5}}{7})^{2}$ $=1-2\displaystyle \cdot\frac{5}{49}=1-\frac{10}{49}=\frac{39}{49}$ $\displaystyle \sin 2\theta=2\sin\theta\cos\theta=2(-\frac{\sqrt{5}}{7})(\frac{2\sqrt{11}}{7})$ $=-\displaystyle \frac{4\sqrt{55}}{49}$
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