Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = \frac { 4 } { 5 }$ and $\theta$ terminates in quadrant I Find $\sin \theta$.
A) $\sin \theta = - \frac { \sqrt { 10 } } { 10 }$
B) $\sin \theta = 0$
C) $\sin \theta = \frac { \sqrt { 10 } } { 10 }$
D) $\sin \theta = \sqrt { 10 }$

Question

Quizplus · Accepted Answer

Given $\cos 2\theta = \frac{4}{5}$, we can use the double angle identity for cosine, $\cos 2\theta = 2\cos^2\theta - 1$, to find $\cos\theta$. Solving for $\cos^2\theta$ gives $\cos^2\theta = \frac{1 + \frac{4}{5}}{2} = \frac{9}{10}$. Since $\theta$ is in the first quadrant, where all trigonometric functions are positive, $\cos\theta = \sqrt{\frac{9}{10}}$. Using the Pythagorean identity, $\sin^2\theta + \cos^2\theta = 1$, we find $\sin^2\theta = 1 - \frac{9}{10} = \frac{1}{10}$, so $\sin\theta = \sqrt{\frac{1}{10}} = \frac{\sqrt{10}}{10}$.

Use Identities to Find the Indicated Value for Each Angle cos⁡2θ=45\cos 2 \theta = \frac { 4 } { 5 }cos2θ=54​

Use Identities to Find the Indicated Value for Each Angle $\cos 2 \theta = \frac { 4 } { 5 }$