Question

use reference angles to find the exact value. rationalize the denominator of your answer, if needed. select \undefined\ if applicable. $\tan\frac{2pi}{3}=$

Explanation:

Step1: Determine the reference angle

The angle $\frac{2\pi}{3}$ is in the second - quadrant. The reference angle $\theta'$ for $\theta=\frac{2\pi}{3}$ is $\pi-\frac{2\pi}{3}=\frac{\pi}{3}$.

Step2: Recall the tangent function properties

We know that $\tan\theta=\frac{\sin\theta}{\cos\theta}$, and in the second - quadrant, $\sin\theta>0$ and $\cos\theta < 0$, so $\tan\theta<0$. Also, $\tan\frac{\pi}{3}=\sqrt{3}$. So, $\tan\frac{2\pi}{3}=-\tan\frac{\pi}{3}=-\sqrt{3}$.

Answer:

$-\sqrt{3}$