Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta

Question

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$

Quizplus · Accepted Answer

The identity for $\sin(2\theta)$ is $2\sin(\theta)\cos(\theta)$. Given $\sin(\theta) = \frac{2\sqrt{10}}{7}$, we find $\cos(\theta)$ using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$. Solving for $\cos(\theta)$, we get $\cos(\theta) = -\frac{3}{7}$ (negative because $\tan(\theta) < 0$ implies $\sin(\theta)$ and $\cos(\theta)$ have opposite signs). Thus, $\sin(2\theta) = 2 \cdot \frac{2\sqrt{10}}{7} \cdot -\frac{3}{7} = -\frac{12\sqrt{10}}{49}$.

Use Identities to Find the Indicated Value for Each Angle sin⁡θ=2107,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quadsinθ=7210​​,tanθ<0

Use Identities to Find the Indicated Value for Each Angle $\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$