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: 11

Answer

$\displaystyle \cos 2\theta=\frac{17}{25}$ $\sin 2\displaystyle \theta=-\frac{4\sqrt{21}}{25}$

Work Step by Step

First, using the Pythagorean Identity$ \sin^{2}\theta+\cos^{2}\theta=1,$ with $\cos\theta < 0,$ $\displaystyle \cos\theta=-\sqrt{1-(\frac{2}{5})^{2}}=-\sqrt{1-\frac{4}{25}}=-\sqrt{\frac{21}{25}}=-\frac{\sqrt{21}}{5}$ We now use the Double-Angle Identities: $\displaystyle \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=\frac{21}{25}-\frac{4}{25}=\frac{17}{25}$ $\sin 2\displaystyle \theta=2\sin\theta\cos\theta=2\cdot\frac{-\sqrt{21}}{5}\cdot\frac{2}{5}=-\frac{4\sqrt{21}}{25}$
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