Answer
The exact value of the trigonometric function $\tan 2\theta $ is $\frac{24}{7}$.
Work Step by Step
The figure shows the right-angle triangle. In this triangle, the base is $4$, the perpendicular is $3$, and the hypotenuse is $5$.
Calculate the value of $\tan 2\theta $.
Recall the double angle formula.
$\begin{align}
& \tan 2\theta =\frac{2\tan \theta }{1-{{\tan }^{2}}\theta } \\
& =\frac{2\left( \frac{\text{perpendicular}}{\text{base}} \right)}{1-{{\left( \frac{\text{perpendicular}}{\text{base}} \right)}^{2}}}
\end{align}$
Substitute $4$ for the base and $3$ for the perpendicular.
$\begin{align}
& \tan 2\theta =\frac{2\left( \frac{\text{3}}{\text{4}} \right)}{1-{{\left( \frac{\text{3}}{\text{4}} \right)}^{2}}} \\
& =\frac{2\left( \frac{\text{3}}{\text{4}} \right)}{1-\frac{9}{16}} \\
& =\frac{\frac{\text{3}}{\text{2}}}{\frac{7}{16}} \\
& =\frac{24}{7}
\end{align}$
Therefore, the exact value of the trigonometric function $\tan 2\theta $ is $\frac{24}{7}$.
