Degrees are not the only units we use to measure angles. We also use radians. Just as there are $360^\circ$ in a circle, there are $2\pi$ radians in a circle. Compute $\tan \frac{8 \pi}{3},$ where the angle $\frac{8 \pi}{3}$ is in radians.
Converting to degrees,
\[\frac{8 \pi}{3} \cdot \frac{180^\circ}{\pi} = 480^\circ.\]Then by the repeated use of cofunction identities,
\[\tan \frac{8 \pi}{3} = -\tan (480^\circ - 360^\circ) = -\tan 120^\circ.\]Since the tangent function has period $180^\circ,$ $\tan 120^\circ = \tan (120^\circ - 2 \cdot 180^\circ) = \tan (-240^\circ).$
Since $\tan (-x) = -\tan x,$ $\tan (-240^\circ) = -\tan 240^\circ.$ Then since $\tan 240^\circ = \tan (240^\circ - 2 \cdot 180^\circ) = \tan (-120^\circ),$
\begin{align*}
\tan \frac{8 \pi}{3} &= -\tan 120^\circ \\
&= -\tan (-240^\circ) \\
&= -\tan 240^\circ \\
&= -\tan (-120^\circ) \\
&= -\tan 120^\circ \\
&= \boxed{-\sqrt{3}}.
\end{align*}