Answer
$$\left( {8,\frac{{2\pi }}{3}} \right)$$ or $$\left( { - 8, - \frac{\pi }{3}} \right)$$
Work Step by Step
$$\eqalign{
& \left( { - 4,4\sqrt 3 } \right) \Rightarrow x = - 4{\text{ and }}y = 4\sqrt 3 \cr
& {\text{Converting to cartesian coordinates }}\left( {r,\theta } \right).{\text{ Where}} \cr
& r = \sqrt {{x^2} + {y^2}} {\text{ and }}\theta = {\tan ^{ - 1}}\left( {\frac{y}{x}} \right) \cr
& {\text{We obtain}}{\text{,}} \cr
& r = \sqrt {{{\left( { - 4} \right)}^2} + {{\left( {4\sqrt 3 } \right)}^2}} = 8 \cr
& {\text{Coordinate }}x{\text{ }}\left( {{\text{positive}}} \right){\text{, Coordinate }}y{\text{ }}\left( {{\text{positive}}} \right),{\text{ Quadrant II}} \cr
& \theta = {\tan ^{ - 1}}\left( {\frac{{4\sqrt 3 }}{{ - 4}}} \right) + \pi = - \frac{\pi }{3} + \pi = \frac{{2\pi }}{3} \cr
& {\text{The polar coordinates are:}} \cr
& \left( {r,\theta } \right) = \left( {8,\frac{{2\pi }}{3}} \right) \cr
& or \cr
& \left( { - 8, - \frac{\pi }{3}} \right) \cr} $$
