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: 16

Answer

$\displaystyle \cos 2\theta=-\frac{19}{25}$ $\displaystyle \sin 2\theta=\frac{2\sqrt{66}}{25}$

Work Step by Step

Plan: work out $\sin\theta$, then apply the double-angle identities. Pythagorean Identity ($\sin\theta$ is positive): $\sin\theta=+\sqrt{1-\cos^{2}\theta}=\sqrt{1-\dfrac{3}{25}}$ $=\displaystyle \sqrt{\frac{22}{25}}=\frac{\sqrt{22}}{5}$ Double-Angle Identities: $\displaystyle \cos 2\theta=2\cos^{2}\theta-1=2(\frac{\sqrt{3}}{5})^{2}-1$ $=2\displaystyle \cdot\frac{3}{25}-1=\frac{6}{25}-1=-\frac{19}{25}$ $\sin 2\theta=2\sin\theta\cos\theta$ $=2(\displaystyle \frac{\sqrt{22}}{5})(\frac{\sqrt{3}}{5}$)$=\displaystyle \frac{2\sqrt{66}}{25}$
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