Answer
The exact value is $\frac{5}{12}$.
Work Step by Step
Let us assume $\theta $ represents the angle.
$\theta ={{\sin }^{-1}}\left( \frac{5}{13} \right)$
Then,
$\sin \theta =\frac{5}{13}$
Since, $\sin \theta $ is positive, $\theta $ is in the first quadrant.
Now, using the Pythagorean identity, we get
$\begin{align}
& {{x}^{2}}+{{y}^{2}}={{r}^{2}} \\
& {{x}^{2}}+{{5}^{2}}={{13}^{2}} \\
& {{x}^{2}}=169-25 \\
& x=12
\end{align}$
Thus,
$\begin{align}
& \tan \left( {{\sin }^{-1}}\left( \frac{5}{13} \right) \right)=\tan \theta \\
& =\frac{x}{y} \\
& =\frac{5}{12}
\end{align}$
