Answer
The exact value of the provided expression is $\frac{4}{5}$.
Work Step by Step
Let us assume $\theta ={{\cos }^{-1}}\left( \frac{3}{5} \right)$. Then,
$\cos \theta =\frac{3}{5}$
As $\cos \theta $ is positive, $\theta $ lies in the first quadrant.
Now, by using the Pythagorean theorem:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& {{5}^{2}}={{3}^{2}}+{{y}^{2}} \\
& y=\sqrt{25-9} \\
& y=4
\end{align}$
Then, the value of the given expression is
$\begin{align}
& \sin \left[ {{\cos }^{-1}}\frac{3}{5} \right]=\sin \theta \\
& =\frac{y}{r} \\
& =\frac{4}{5}
\end{align}$
