Answer
$(4,\frac{4\pi}{3})$
Work Step by Step
Solve for $r$ using the formula $r=\sqrt {x^{2}+y^{2}}$ to obtain:
$$r=\sqrt {(-2)^{2}+(-2\sqrt {3})^{2}}=4$$
Solve for the angle using the formula $\alpha= \tan^{-1}(\frac{y}{x})$ to obtain:
$$\alpha=\tan^{-1}\left(\frac{-2\sqrt {3}}{-2}\right)=\frac{\pi}{3}$$
Since $(-2, -2\sqrt {3})$ is in the third quadrant, then the angle measurement is:
$$\theta=\alpha+\pi=\frac{\pi}{3}+\pi=\frac{4\pi}{3}$$
Thus, the polar coordinates for the point $(-2, -2\sqrt {3})$ can be given by $(4,\frac{4\pi}{3})$.
