### Instruction: Convert the point $( -2, -2 \sqrt{3}, -1)$ in rectangular coordinates to cylindrical coordinates.  Enter your answer in the form $(r,\theta,z),$ where $r > 0$ and $0 \le \theta < 2 \pi.$.### Response: Rectangular coordinates are in the form $(x,y,z)$.
Cylindrical coordinates are of the form $(r,\theta,z)$.
We are given the rectangular coordinate point $(-2,-2\sqrt{3},-1)$.
We see that $x=-2$, $y=-2\sqrt{3}$, and $z=-1$.
We can find $r$ by doing $\sqrt{x^2+y^2}$.
So $r=\sqrt{(-2)^2+(-2\sqrt{3})^2}=\sqrt{4+12}=\sqrt{16}=4$.
We can find $\theta$ by doing $\tan^{-1}(\frac{y}{x})$.
So $\theta=\tan^{-1}(\frac{-2\sqrt{3}}{-2})=\tan^{-1}(\sqrt{3})=\pi/3$.
We already know that $z=-1$.