Use the fundamental identities to simplify the expression.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta

Question

Use the fundamental identities to simplify the expression.
-$\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$ Find $\sin ( 2 \theta )$
A) $\frac { - 12 \sqrt { 10 } } { 49 }$
B) $\frac { 31 } { 49 }$
C) $- \frac { 31 } { 49 }$
D) $\frac { 12 \sqrt { 10 } } { 49 }$

Quizplus · Accepted Answer

To simplify $\sin(2\theta)$, we use the double angle formula: $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. 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 $\tan(\theta) < 0$, implying $\sin(\theta)$ and $\cos(\theta)$ have opposite signs in the given quadrant). Substituting these values into the double angle formula gives $\sin(2\theta) = 2 \cdot \frac{2\sqrt{10}}{7} \cdot -\frac{3}{7} = \frac{-12\sqrt{10}}{49}$.

Use the Fundamental Identities to Simplify the Expression sin⁡θ=2107,tan⁡θ<0\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quadsinθ=7210​​,tanθ<0

Use the Fundamental Identities to Simplify the Expression $\sin \theta = \frac { 2 \sqrt { 10 } } { 7 } , \tan \theta < 0 \quad$