To solve this question, we need to convert the given rectangular coordinates $(-4\sqrt{3}, -4)$ into polar coordinates.

Steps to convert rectangular coordinates to polar coordinates:

1. Find the radius $r$: The radius $r$ is given by: $r = \sqrt{x^2 + y^2}$ Here, $x = -4\sqrt{3}$ and $y = -4$, so: $r = \sqrt{(-4\sqrt{3})^2 + (-4)^2} = \sqrt{48 + 16} = \sqrt{64} = 8$

2. Find the angle $\theta$: The angle $\theta$ is given by: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ Substituting $x = -4\sqrt{3}$ and $y = -4$: $\theta = \tan^{-1}\left(\frac{-4}{-4\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)$ From trigonometry, $\tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$. Since both $x$ and $y$ are negative, the point is in the third quadrant, so: $\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$

3. Write the polar coordinates: The polar coordinates are $(r, \theta) = (8, \frac{7\pi}{6})$. Alternatively, if $r$ is negative, the angle shifts by $\pi$. So $(-r, \theta - \pi)$ is equivalent: $(-8, \frac{\pi}{6})$

Matching with the options:

The correct polar coordinates corresponding to $(-4\sqrt{3}, -4)$ are $(-8, \frac{\pi}{6})$, which is Option C.

Let me know if you'd like any part of this calculation clarified!

Follow-up Questions:

1. How are polar and rectangular coordinate systems related geometrically?
2. Why does the angle shift by $\pi$ when the radius is negative?
3. Can a point have multiple polar coordinate representations?
4. How can you verify polar coordinates by converting them back to rectangular form?
5. What is the significance of quadrants in determining the angle?

Tip:

Always ensure that the angle $\theta$ matches the quadrant of the rectangular coordinates when converting!