Use Identities to find the exact value.
-$\cos 165 ^ { \circ }$
A) $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$
B) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
C) $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$
D) $\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }$

Question

Quizplus · Accepted Answer

$\cos 165^\circ$ can be expressed using the sum formula for cosine as $\cos(120^\circ + 45^\circ)$. Using the identity $\cos(a + b) = \cos a \cos b - \sin a \sin b$, we get $\cos 165^\circ = \cos 120^\circ \cos 45^\circ - \sin 120^\circ \sin 45^\circ$. Since $\cos 120^\circ = -\frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 120^\circ = \frac{\sqrt{3}}{2}$, and $\sin 45^\circ = \frac{\sqrt{2}}{2}$, substituting these values gives $\cos 165^\circ = -\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{-\sqrt{2} - \sqrt{6}}{4}$.

Use Identities to Find the Exact Value cos⁡165∘\cos 165 ^ { \circ }cos165∘

Use Identities to Find the Exact Value $\cos 165 ^ { \circ }$