Solve the problem.
-If $\sin \theta = - \frac { 4 } { 5 }$, and $\theta$ terminates in quadrant IV, then find $\sin 2 \theta$.
A) $- \frac { 24 } { 25 }$

B) $\frac { 24 } { 25 }$

C) $- \frac { 7 } { 25 }$

D) $\frac { 7 } { 25 }$

Question

Solve the problem.
-If $\sin \theta = - \frac { 4 } { 5 }$, and $\theta$ terminates in quadrant IV, then find $\sin 2 \theta$.
A) $- \frac { 24 } { 25 }$

B) $\frac { 24 } { 25 }$

C) $- \frac { 7 } { 25 }$

D) $\frac { 7 } { 25 }$

Quizplus · Accepted Answer

The formula for $\sin 2\theta$ is $2\sin\theta\cos\theta$. Given $\sin\theta = -\frac{4}{5}$ and $\theta$ is in the fourth quadrant, where sine is negative and cosine is positive. To find $\cos\theta$, we use the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$. Substituting $\sin\theta = -\frac{4}{5}$ gives $\left(-\frac{4}{5}\right)^2 + \cos^2\theta = 1$, leading to $\cos\theta = \frac{3}{5}$. Thus, $\sin 2\theta = 2\left(-\frac{4}{5}\right)\left(\frac{3}{5}\right) = -\frac{24}{25}$.

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