From Förberedande kurs i matematik 1

If we add \displaystyle 2\pi to \displaystyle -\frac{5\pi }{3}, we get a new angle in the first quadrant which corresponds to the same point on the unit circle as the old angle \displaystyle -\frac{5\pi }{3} and consequently has the same tangent value:

\displaystyle \begin{align} & \tan \left( -\frac{5\pi }{3} \right)=\tan \left( -\frac{5\pi }{3}+2\pi \right)=\tan \frac{\pi }{3} \\ & =\frac{\sin \frac{\pi }{3}}{\cos \frac{\pi }{3}}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3} \\ \end{align}