What is the principal argument of the complex number $\displaystyle -1+i\sqrt{3}$?

My attempt:

$\displaystyle \begin{array}{rcrcrc}

tan\alpha=\frac{\sqrt{3}}{-1}\\

\alpha=\tan^{-1}(\frac{\sqrt{3}}{-1})\\

=\frac{-1}{3}\pi\\

arg(z)=\pi-\alpha\\

=\frac{4}{3}\pi

\end{array}$

However, I know this is wrong because it does not fit the inequality $\displaystyle -\pi<\theta\leq\pi$.