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

Question

Quizplus · Accepted Answer

The half-angle formula for tangent is $\tan \left( \frac{\theta}{2} \right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$. For $\theta = 330^\circ$ (since $165^\circ$ is half of $330^\circ$), $\cos 330^\circ = \cos(-30^\circ) = \sqrt{3}/2$. Plugging this into the formula gives $\tan 165^\circ = \sqrt{\frac{1 - \sqrt{3}/2}{1 + \sqrt{3}/2}} = -2 + \sqrt{3}$.

Find the Exact Value by Using a Half-Angle Identity tan⁡165∘\tan 165 ^ { \circ }tan165∘

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