Answer
The exact value of the trigonometric function $\tan 2\alpha $ is $\frac{336}{527}$
Work Step by Step
Calculate the value of $\tan 2\alpha $.
Recall the double angle formula.
$\begin{align}
& \tan 2\alpha =\frac{2\tan \alpha }{1-{{\tan }^{2}}\alpha } \\
& =\frac{2\left( \frac{\text{perpendicular}}{\text{base}} \right)}{1-{{\left( \frac{\text{perpendicular}}{\text{base}} \right)}^{2}}}
\end{align}$
Substitute $24$ for the base and $7$ for the perpendicular.
$\begin{align}
& \tan 2\alpha =\frac{2\left( \frac{\text{7}}{\text{24}} \right)}{1-{{\left( \frac{\text{7}}{\text{24}} \right)}^{2}}} \\
& =\frac{\left( \frac{\text{7}}{\text{12}} \right)}{1-\frac{49}{576}} \\
& =\frac{\left( \frac{\text{7}}{\text{12}} \right)}{\frac{527}{576}} \\
& =\frac{336}{527}
\end{align}$
Therefore, the exact value of the trigonometric function $\tan 2\alpha $ is $\frac{336}{527}$.
