Find the exact value by using a half-angle identity.
-$\cos 165 ^ { \circ }$
A) $- \frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$
B) $\frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$
C) $- \frac { 1 } { 2 } \sqrt { 2 + \sqrt { 3 } }$
D) $\frac { 1 } { 2 } \sqrt { 2 - \sqrt { 3 } }$

Question

Quizplus · Accepted Answer

The half-angle identity for cosine is $\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}$. For $\theta = 330^\circ$, $\cos 330^\circ = \frac{\sqrt{3}}{2}$. Thus, $\cos 165^\circ = \cos \frac{330^\circ}{2} = -\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = -\frac{1}{2}\sqrt{2 + \sqrt{3}}$.

Find the Exact Value by Using a Half-Angle Identity cos⁡165∘\cos 165 ^ { \circ }cos165∘

Find the Exact Value by Using a Half-Angle Identity $\cos 165 ^ { \circ }$