Arctan(-squareroot of3)

The unit circle is a handy thing to have on hand for these.

Take a look at tan(\frac{2\pi}{3}), \;\ tan(\frac{5\pi}{3})

Notice the coordinates on the circle. For instance, look at 120 degrees or \frac{2\pi}{3}

See the (-\frac{1}{2},\frac{\sqrt{3}}{2})?.

They come from the triangle since x=cos({\theta}), \;\ y=sin({\theta})

Therefore, tan^{-1}(\frac{-\sqrt{3}}{1})=\frac{2\pi}{3}=\frac{-\pi}{3}

Same for \frac{5\pi}{3}=\frac{-\pi}{3}

Hence, \fbox{tan(\frac{(3n-1){\pi}}{3})=-\sqrt{3}}