Use the figure to find the exact value of the trigonometric function.
-$\cos \theta = \frac { 4 } { 5 } , \theta$ lies in quadrant IV Find $\sin 2 \theta$.
A) $- \frac { 24 } { 25 }$
B) $- \frac { 7 } { 25 }$
C) $\frac { 7 } { 25 }$
D) $\frac { 24 } { 25 }$

Question

Quizplus · Accepted Answer

The formula for $\sin 2\theta$ is $2\sin\theta\cos\theta$. Since $\cos\theta = \frac{4}{5}$ and $\theta$ is in the fourth quadrant (where sine is negative), we find $\sin\theta$ using the Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$, which gives $\sin\theta = -\frac{3}{5}$. Thus, $\sin 2\theta = 2(-\frac{3}{5})(\frac{4}{5}) = -\frac{24}{25}$.

Use the Figure to Find the Exact Value of the Trigonometric