Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-tan 165°
A) $- 2 + \sqrt { 3 }$
B) $2 + \sqrt { 3 }$
C) $- 2 - \sqrt { 3 }$
D) $2 - \sqrt { 3 }$

Question

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
-tan 165° 
A) $- 2 + \sqrt { 3 }$
B) $2 + \sqrt { 3 }$
C) $- 2 - \sqrt { 3 }$
D) $2 - \sqrt { 3 }$

Quizplus · Accepted Answer

The tangent function can be expressed in terms of sine and cosine as $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. For $\tan(165°)$, we can use the sum formula for tangent, recognizing that $165° = 120° + 45°$, where $\tan(120°) = \sqrt{3}$ and $\tan(45°) = 1$. The sum formula for tangent is $\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$, which gives $\frac{\sqrt{3} + 1}{1 - \sqrt{3}}$, simplifying to $-2 + \sqrt{3}$.

Write the Expression in Terms of Sine and Cosine, and Simplify