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 - Page 693: 12


$\displaystyle \cos 2\theta=\frac{119}{169}$ $\sin 2\displaystyle \theta=-\frac{120}{169}$

Work Step by Step

First, using the Pythagorean Identity $\sin^{2}\theta+\cos^{2}\theta=1, $with $\sin\theta > 0,$ $\displaystyle \sin\theta=+\sqrt{1-(-\frac{12}{13})^{2}}=\sqrt{1-\frac{144}{169}}$ $=\sqrt{\dfrac{25}{169}}=\dfrac{5}{13}$ We now use the Double-Angle Identities: $\displaystyle \cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=\frac{144}{169}-\frac{25}{169}=\frac{119}{169}$ $\sin 2\displaystyle \theta=2\sin\theta\cos\theta=2\cdot\frac{-12}{13}\cdot\frac{5}{13}=-\frac{120}{169}$
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