Answer
The exact value of the trigonometric function $\sin 2\theta $ is $\frac{24}{25}$.
Work Step by Step
The figure is the right-angle triangle; in this triangle, the base is $4$, the perpendicular is $3$, and the hypotenuse is $5$.
Calculate the value of $\sin 2\theta $.
Recall the double angle formula.
$\begin{align}
& \sin 2\theta =2\sin \theta \cos \theta \\
& =2\left( \frac{\text{perpendicular}}{\text{hypotenuse}} \right)\left( \frac{\text{base}}{\text{hypotenuse}} \right)
\end{align}$
Substitute $4$ for the base, $3$ for the perpendicular and $5$ for the hypotenuse.
$\begin{align}
& \sin 2\theta =2\left( \frac{\text{3}}{\text{5}} \right)\left( \frac{\text{4}}{\text{5}} \right) \\
& =\frac{24}{25}
\end{align}$
Therefore, the exact value of the trigonometric function $\sin 2\theta $ is $\frac{24}{25}$.
