Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.5 Double-Angle Identities - 5.5 Exercises - Page 230: 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 $\sin2\theta$: $$\cos2\theta=\cos^2\theta-\sin^2\theta=\Big(-\frac{12}{13}\Big)^2-\Big(\frac{5}{13}\Big)^2=\frac{144}{169}-\frac{25}{169}$$ $$\cos2\theta=\frac{119}{169}$$ $$\sin2\theta=2\sin\theta\cos\theta=2\times\frac{5}{13}\times\Big(-\frac{12}{13}\Big)$$ $$\sin2\theta=-\frac{120}{169}$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.