Use identities to find the indicated value for each angle measure.
-$\cos 2 \theta = \frac { 2 } { 3 }$ and $\theta$ terminates in quadrant $\mathrm { I } \quad$ Find $\sin \theta$

Question

Quizplus · Accepted Answer

Using the identity $\cos^2 \theta = 1 - \sin^2 \theta$, and given $\cos 2\theta = \frac{2}{3}$, we can use the double angle formula for cosine, $\cos 2\theta = 2\cos^2 \theta - 1$, to find $\cos^2 \theta$. Solving $\frac{2}{3} = 2\cos^2 \theta - 1$ gives $\cos^2 \theta = \frac{5}{6}$. Thus, $\sin^2 \theta = 1 - \cos^2 \theta = \frac{1}{6}$, and $\sin \theta = \pm\sqrt{\frac{1}{6}}$. Since $\theta$ is in quadrant I, where sine is positive, $\sin \theta = \frac{\sqrt{6}}{6}$.

Use Identities to Find the Indicated Value for Each Angle cos⁡2θ=23\cos 2 \theta = \frac { 2 } { 3 }cos2θ=32​

Use Identities to Find the Indicated Value for Each Angle $\cos 2 \theta = \frac { 2 } { 3 }$