Question

use identities to find the indicated value for each angle measure.
find \\( \cos ( 2 \theta ) \\) if \\( \sin \theta = \frac { 15 } { 17 } \\), \\( \cos \theta > 0 \\)
a. \\( - \frac { 163 } { 289 } \\)
b. \\( - \frac { 161 } { 289 } \\)
c. \\( \frac { 240 } { 289 } \\)
d. \\( \frac { 161 } { 289 } \\)

Explanation:

Step1: Find $\cos\theta$ using the Pythagorean identity

We know that $\sin^{2}\theta+\cos^{2}\theta = 1$. Given $\sin\theta=\frac{15}{17}$, then $\cos^{2}\theta=1-\sin^{2}\theta$.
Substitute $\sin\theta=\frac{15}{17}$ into the formula:
$\cos^{2}\theta=1 - (\frac{15}{17})^{2}=1-\frac{225}{289}=\frac{289 - 225}{289}=\frac{64}{289}$.
Since $\cos\theta>0$, then $\cos\theta=\sqrt{\frac{64}{289}}=\frac{8}{17}$.

Step2: Use the double - angle formula for cosine

The double - angle formula for cosine is $\cos(2\theta)=1 - 2\sin^{2}\theta$ (we could also use $\cos(2\theta)=2\cos^{2}\theta - 1$).
Substitute $\sin\theta=\frac{15}{17}$ into the formula $\cos(2\theta)=1-2\sin^{2}\theta$:
$\cos(2\theta)=1-2\times(\frac{15}{17})^{2}=1 - 2\times\frac{225}{289}=1-\frac{450}{289}=\frac{289-450}{289}=-\frac{161}{289}$.

Answer:

B. $-\frac{161}{289}$