Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\tan 165 ^ { \circ }$
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 $\tan(\frac{\theta}{2}) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$. For $\tan(165^\circ)$, we can use $\theta = 330^\circ$ so that $\frac{\theta}{2} = 165^\circ$. $\cos(330^\circ) = \cos(-30^\circ) = \sqrt{3}/2$. Plugging into the formula gives $\tan(165^\circ) = \sqrt{\frac{1 - \sqrt{3}/2}{1 + \sqrt{3}/2}} = \sqrt{\frac{2 - \sqrt{3}}{2 + \sqrt{3}}} \cdot \sqrt{\frac{2 + \sqrt{3}}{2 + \sqrt{3}}} = \sqrt{\frac{4 - 2\sqrt{3}}{4}} = -2 + \sqrt{3}$.

Use the Half-Angle Formulas to Find the Exact Value of the Trigonometric