## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 5 - Trigonometric Identities - Section 5.5 Double-Angle Identities - 5.5 Exercises - Page 236: 8

#### Answer

$$\cos2\theta=\frac{119}{169}$$ $$\sin2\theta=-\frac{120}{169}$$

#### Work Step by Step

$$\cos\theta=-\frac{12}{13}\hspace{2cm}\sin\theta\gt0$$ $$\sin2\theta=?\hspace{2cm}\cos2\theta=?$$ To find $\sin2\theta$ and $\cos2\theta$, it would be wise to find the unknown $\sin\theta$. Using Pythagorean Identities: $$\sin^2\theta=1-\cos^2\theta=1-\Big(-\frac{12}{13}\Big)^2=1-\frac{144}{169}=\frac{25}{169}$$ $$\sin\theta=\frac{5}{13}\hspace{1.5cm}(\sin\theta\gt0)$$ Now we apply the Double-Angle Identities for $\cos2\theta$ and $\sin2\theta$: $$\cos2\theta=\cos^2\theta-\sin^2\theta=\Big(-\frac{12}{13}\Big)^2-\Big(\frac{5}{13}\Big)^2=\frac{144}{169}-\frac{25}{169}$$ $$\cos2\theta=\frac{119}{169}$$ $$\sin2\theta=2\sin\theta\cos\theta=2\times\frac{5}{13}\times\Big(-\frac{12}{13}\Big)$$ $$\sin2\theta=-\frac{120}{169}$$

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