Find the exact value of the given expression using a sum or difference formula.
$\sin 165 ^ { \circ }$
A) $\frac { - \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
B) $\frac { - \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$
C) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
D) $\frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$

Question

Quizplus · Accepted Answer

We can rewrite $\sin 165^\circ$ as $\sin (45^\circ + 120^\circ)$ using the sum formula for sine. Then, we have:

\begin{align*}
\sin(45^\circ + 120^\circ) &= \sin 45^\circ \cos 120^\circ + \cos 45^\circ \sin 120^\circ \
&= \frac{1}{\sqrt{2}} \cdot \left(-\frac{1}{2}ight) + \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} \
&= \frac{\sqrt{3}-1}{2\sqrt{2}}
\end{align*}

Therefore, the answer is choice C.

Find the Exact Value of the Given Expression Using a Sum