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In the image, the problem involves finding the polar coordinates of a given point based on parametric conditions. Let's break it down:

1. **Step 1**: You are given that \( r^2 = 36 \). To find \( r \), take the square root of both sides:
   \[
   r = \sqrt{36} = 6
   \]
   So, \( r = 6 \), as you can see in the provided solution.

2. **Step 2**: You are also given that:
   \[
   \tan(\theta) = \frac{3\sqrt{3}}{3} = \sqrt{3}
   \]
   This is the tangent value for \( \theta = \frac{\pi}{3} \), a known trigonometric value from the unit circle for the first quadrant (since the tangent is positive).

3. **Step 3**: The problem asks to ensure that \( \theta \) is within \( 0 \leq \theta < 2\pi \), and since the point \( (3, 3\sqrt{3}) \) lies in the first quadrant, you choose \( \theta = \frac{\pi}{3} \).

4. **Step 4**: However, in the last step, the coordinates given as \( (r, \theta) = (-6, \theta) \) seem incorrect. Based on the positive square root, \( r \) should remain positive, i.e., \( r = 6 \), not \( -6 \).

Thus, the final polar coordinates should be \( (6, \frac{\pi}{3}) \), not \( (-6, \frac{\pi}{3}) \).

Would you like further details or clarification on any of these steps?

Here are 5 related questions to expand your understanding:
1. Why is \( r \) considered positive in polar coordinates by default?
2. What happens if we allow \( r \) to be negative in polar coordinates?
3. How does \( \tan(\theta) \) help in determining the quadrant?
4. What is the relationship between Cartesian and polar coordinates?
5. Can there be multiple values of \( \theta \) corresponding to the same point?

**Tip**: When working with polar coordinates, remember that \( r \) can sometimes be negative, but the angle \( \theta \) adjusts to maintain the correct point on the plane.

Find the polar coordinates of the given point based on parametric conditions, with r^2 = 36 and tan(θ) = 3√3/3.

Learn how to find polar coordinates for a given point using parametric conditions where r^2 = 36 and tan(θ) = 3√3/3. The solution covers trigonometric concepts and the relationship between Cartesian and polar coordinates. Suitable for high school students in Grades 10-12.

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