Given point M(6, -6squareroot3), which is located on the rectangular coordinate plane, what are the representations of the point in polar form, where -2pi less than equal to theta greater than equal to 2pi

To convert the point \( M(6, -6\sqrt{3}) \) from rectangular (Cartesian) coordinates to polar coordinates \((r, \theta)\), we use the following formulas:

1. **Calculate the radius \( r \):**
   \[
   r = \sqrt{x^2 + y^2}
   \]
   where \( x = 6 \) and \( y = -6\sqrt{3} \).

   \[
   r = \sqrt{6^2 + (-6\sqrt{3})^2} = \sqrt{36 + 108} = \sqrt{144} = 12
   \]

2. **Calculate the angle \( \theta \):**
   \[
   \theta = \tan^{-1}\left(\frac{y}{x}\right)
   \]
   \[
   \theta = \tan^{-1}\left(\frac{-6\sqrt{3}}{6}\right) = \tan^{-1}(-\sqrt{3})
   \]

   The angle whose tangent is \(-\sqrt{3}\) is \(\theta = -\frac{\pi}{3}\). 

3. **Determine the correct quadrant:**
   The point \( M(6, -6\sqrt{3}) \) is in the fourth quadrant because \( x > 0 \) and \( y < 0 \). The angle \(\theta = -\frac{\pi}{3}\) correctly places the point in the fourth quadrant.

Therefore, the polar coordinates of the point \( M(6, -6\sqrt{3}) \) are:
\[
\boxed{(12, -\frac{\pi}{3})}
\]
where \( r = 12 \) and \( \theta = -\frac{\pi}{3} \).

### Do you want further details or have any questions?
Here are 5 related questions to expand your understanding:

1. How would you convert the polar coordinates \((12, -\frac{\pi}{3})\) back to rectangular coordinates?
2. What would be the polar form of the point if it were located in the second quadrant?
3. How does the sign of the angle \(\theta\) affect the location of the point on the polar coordinate plane?
4. What are the possible values of \(\theta\) for the same point \( M \) in different quadrants?
5. How do polar coordinates help in understanding the symmetry of points on the coordinate plane?

**Tip:** When converting between rectangular and polar coordinates, always check the quadrant of the point to ensure that the angle \(\theta\) is correctly placed.

Learn how to convert the point (6, -6√3) from rectangular coordinates to polar coordinates (r, θ). This problem involves using the distance formula and arctan function to determine the radius and angle of the point. Suitable for high school students studying coordinate geometry.

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