Answer
The exact value of the provided expression is $-\frac{\sqrt{3}}{2}$.
Work Step by Step
We know that $\theta =\frac{22\pi }{3}$ lies in quadrant III, so the reference angle for $\theta =\frac{22\pi }{3}$ is:
$\begin{align}
& \theta '=\frac{22\pi }{3}-\left( 6\pi +\pi \right) \\
& =\frac{22\pi -21\pi }{3} \\
& =\frac{\pi }{3}
\end{align}$
Now, the function value for the reference angle is:
$\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$
Thus, we see that the angle $\theta =\frac{22\pi }{3}$ lies in quadrant III, so the sine function is negative:
$\begin{align}
& \sin \frac{22\pi }{3}=-\sin \frac{\pi }{3} \\
& =-\frac{\sqrt{3}}{2}
\end{align}$
