Answer
$\sin{\left(\dfrac{3\pi}{2}+\theta\right)}=-\cos{\theta}$
Work Step by Step
Use a graphing utility to graph the given expression. (Refer to the graph below.)
Note that the graph is identical to the graph of $-\cos{\theta}$.
This means that $\sin{(\frac{3\pi}{2}+\theta)}=-\cos{\theta}$.
RECALL:
$\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}$
Use the identity above with $A=\frac{3\pi}{2}$ and $B=\theta$ to obtain:
\begin{align*}
\sin{\left(\frac{3\pi}{2}+\theta\right)}&=\sin{\frac{3\pi}{2}}\cos{\theta}+\cos{\frac{3\pi}{2}}\sin{\theta}\\\\
&=-1\cdot \cos{\theta}+0\cdot\sin{\theta}\\\\
&=-\cos{\theta}+0\\\\
&=-\cos{\theta}
\end{align*}
Therefore,
$$\sin{\left(\dfrac{3\pi}{2}+\theta\right)}=-\cos{\theta}$$
