# How do you evaluate tan ((2pi)/3)?

Dec 4, 2016

$\tan \left(\frac{2 \pi}{3}\right) = - \sqrt{3}$

#### Explanation:

$\tan \left(\frac{2 \pi}{3}\right)$

Recall the identity $\tan \theta = \sin \frac{\theta}{\cos} \theta$

According to the unit circle,

$\sin \left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2}$ and $\cos \left(\frac{2 \pi}{3}\right) = - \frac{1}{2}$

$\tan \left(\frac{2 \pi}{3}\right) = \frac{\sin \left(\frac{2 \pi}{3}\right)}{\cos \left(\frac{2 \pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$

$= \frac{\sqrt{3}}{2} \cdot - \frac{2}{1} = \frac{\sqrt{3}}{\cancel{2}} \cdot - \frac{\cancel{2}}{1} = - \sqrt{3}$