Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta

Question

Use identities to find the indicated value for each angle measure.
-$\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$
A) $\frac { - 4 \sqrt { 6 } } { 25 }$
B) $\frac { 23 } { 25 }$
C) $- \frac { 23 } { 25 }$
D) $\frac { 4 \sqrt { 6 } } { 25 }$

Quizplus · Accepted Answer

Using the double-angle formula for sine, we have:$$\sin ( 2 \theta ) = 2 \sin \theta \cos \theta$$Since $\tan \theta < 0$, we know that $\theta$ is in the second quadrant, where $\cos \theta < 0$. Therefore, we have:$$\cos \theta = - \sqrt { 1 - \sin^2 \theta } = - \sqrt { 1 - \frac { 24 } { 25 } } = - \frac { \sqrt { 1 } } { 5 } = - \frac { 1 } { 5 }$$Substituting this into the double-angle formula, we get:$$\sin ( 2 \theta ) = 2 \sin \theta \cos \theta = 2 \left( \frac { 2 \sqrt { 6 } } { 5 } \right) \left( - \frac { 1 } { 5 } \right) = \frac { - 4 \sqrt { 6 } } { 25 }$$Therefore, the correct answer is A.

Use Identities to Find the Indicated Value for Each Angle sin⁡θ=265,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quadsinθ=526​​,tanθ<0

Use Identities to Find the Indicated Value for Each Angle $\sin \theta = \frac { 2 \sqrt { 6 } } { 5 } , \tan \theta < 0 \quad$