Express the complex number $-1 + i \sqrt{3}$ in its polar form.
The given complex number is $\;$ $z = -1 + i \sqrt{3}$
$z$ $\;$ is of the form $\;$ $x + iy$ $\;$ where $\;$ $x = -1$, $\;$ $y = \sqrt{3}$
Modulus of $z = \left|z\right| = \sqrt{x^2 + y^2} = \sqrt{\left(-1\right)^2 + \left(\sqrt{3}\right)^2} = 2$
$\because$ $\;$ $x$ is negative and $y$ is positive, argument or amplitude $\theta$ of $z$ lies in the second quadrant in the complex plane.
Let $\;$ $\alpha = \tan^{-1} \left(\dfrac{\left|y\right|}{\left|x\right|}\right) = \tan^{-1} \left(\dfrac{\left|\sqrt{3}\right|}{\left|-1\right|}\right) = \tan^{-1} \left(\sqrt{3}\right) = \dfrac{\pi}{3}$
$\therefore$ $\;$ $\theta = \pi - \alpha = \pi - \dfrac{\pi}{3} = \dfrac{2 \pi}{3}$
$\therefore$ $\;$ Polar form of $z = r \left(\cos \theta + i \sin \theta\right) = r \; cis \; \theta = 2 \; cis \; \left(\dfrac{2\pi}{3}\right)$