Answer
$$\left( {2,\frac{{4\pi }}{3}} \right){\text{ and }}\left( { - 2,\frac{\pi }{3}} \right)$$
Work Step by Step
$$\eqalign{
& {\text{We have the rectangular coordinates }}\left( {x,y} \right) = \left( { - 1, - \sqrt 3 } \right) \cr
& {\text{The polar coordinates are:}} \cr
& r = \sqrt {{x^2} + {y^2}} \cr
& r = \sqrt {{{\left( { - 1} \right)}^2} + {{\left( { - \sqrt 3 } \right)}^2}} \cr
& r = 2 \cr
& \tan \theta = \frac{{ - \sqrt 3 }}{{ - 1}} \cr
& \theta = {\tan ^{ - 1}}\left( {\sqrt 3 } \right) = \frac{\pi }{3} \cr
& \theta = {\tan ^{ - 1}}\left( {\sqrt 3 } \right) + \pi = \frac{4}{3}\pi \cr
& {\text{Therefore, the polar coordinates are:}} \cr
& \left( {2,\frac{{4\pi }}{3}} \right){\text{ and }}\left( { - 2,\frac{\pi }{3}} \right) \cr} $$
