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

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) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$

Quizplus · Accepted Answer

$\cos 165^\circ$ can be expressed using the sum or difference identity for cosine: $\cos(120^\circ + 45^\circ)$. Using the identity $\cos(a + b) = \cos a \cos b - \sin a \sin b$, and knowing that $\cos 120^\circ = -\frac{1}{2}$ (since $120^\circ$ is in the second quadrant where cosine is negative) and $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$, we get $\cos 165^\circ = -\frac{1}{2} \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{-\sqrt{6} - \sqrt{2}}{4}$.

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

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