Answer
The exact value of the provided expression is $-\frac{\sqrt{10}}{10}$.
Work Step by Step
Let us assume $\theta ={{\tan }^{-1}}\left( -\frac{1}{3} \right)$. Then,
$\tan \theta =-\frac{1}{3}$
As we know that $\tan \theta $ is negative, $\theta $ lies in the fourth quadrant.
Now, by using the Pythagorean theorem:
$\begin{align}
& {{r}^{2}}={{x}^{2}}+{{y}^{2}} \\
& {{r}^{2}}={{3}^{2}}+{{\left( -1 \right)}^{2}} \\
& r=\sqrt{9+1} \\
& r=\sqrt{10}
\end{align}$
Then, the value of the given expression is
$\begin{align}
& \sin \left[ {{\tan }^{-1}}\left( -\frac{1}{3} \right) \right]=\sin \theta \\
& =\frac{y}{r} \\
& =\frac{-1}{\sqrt{10}} \\
& =-\frac{1}{\sqrt{10}}
\end{align}$
And rationalize the denominator as follows:
$\begin{align}
& -\frac{1}{\sqrt{10}}=-\frac{1}{\sqrt{10}}\times \frac{\sqrt{10}}{\sqrt{10}} \\
& =-\frac{\sqrt{10}}{10}
\end{align}$
