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

$\sin 165^\circ$ can be expressed using the sum formula $\sin(a + b) = \sin a \cos b + \cos a \sin b$ as $\sin(120^\circ + 45^\circ)$. Using the values $\sin 120^\circ = \frac{\sqrt{3}}{2}$, $\cos 120^\circ = -\frac{1}{2}$, $\sin 45^\circ = \frac{\sqrt{2}}{2}$, and $\cos 45^\circ = \frac{\sqrt{2}}{2}$, we get $\sin 165^\circ = \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}}$.

Find the Exact Value of the Given Expression Using a Sum