Answer
The exact value of the expression is $\frac{1-\sqrt{3}}{2}$.
Work Step by Step
Consider the following equation.
$x=\sin \frac{\pi }{3}\cos \pi -\cos \frac{\pi }{3}\sin \frac{3\pi }{2}$
The values of the trigonometric functions are:
$\begin{align}
& \sin \frac{\pi }{3}=\frac{\sqrt{3}}{2} \\
& \cos \pi =1 \\
& \cos \frac{\pi }{3}=\frac{1}{2} \\
& \sin \frac{3\pi }{2}=-1
\end{align}$
Now, substitute $\frac{\sqrt{3}}{2}$ for $\sin \frac{\pi }{3}$ , $-1$ for $\cos \pi $ , $\frac{1}{2}$ for $\cos \frac{\pi }{3}$ and $-1$ for $\sin \frac{3\pi }{2}$.
$\begin{align}
& x=\frac{\sqrt{3}}{2}\left( -1 \right)-\left( \frac{1}{2} \right)\left( -1 \right) \\
& =-\frac{\sqrt{3}}{2}+\frac{1}{2} \\
& =\frac{-\sqrt{3}+1}{2} \\
& =\frac{1-\sqrt{3}}{2}
\end{align}$
The exact value of the expression is $\frac{1-\sqrt{3}}{2}$.
