We need to find the value of $\tan \frac{4\pi}{3}$.

1. Problem Description

We need to find the value of $\tan \frac{4\pi}{3}$.

2. Solution Steps

First, we can rewrite $\frac{4\pi}{3}$ as $\pi + \frac{\pi}{3}$. Therefore, we have:

$\tan \frac{4\pi}{3} = \tan(\pi + \frac{\pi}{3})$

Since the tangent function has a period of $\pi$, we have

$\tan(\pi + x) = \tan x$.

Therefore,

$\tan (\pi + \frac{\pi}{3}) = \tan \frac{\pi}{3}$.

We know that $\tan \frac{\pi}{3} = \tan 60^\circ = \frac{\sin 60^\circ}{\cos 60^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$.

3. Final Answer

$\sqrt{3}$