suppose a point has polar coordinates (-3,5pi/6)  find two additonal polar representatioons of the point

The given polar coordinates are \((-3, \frac{5\pi}{6})\). In polar coordinates, we have the radius \(r\) and the angle \(\theta\), and the point can be represented in multiple ways.

### To find two additional representations:

1. **Adjust the angle by \(2\pi\) (adding or subtracting full rotations):**

    - The polar coordinates are periodic with a period of \(2\pi\), so we can add or subtract multiples of \(2\pi\) to \(\theta\) to find equivalent points.
    - If we add \(2\pi\) to \(\frac{5\pi}{6}\), we get:

      \[
      \theta = \frac{5\pi}{6} + 2\pi = \frac{5\pi}{6} + \frac{12\pi}{6} = \frac{17\pi}{6}
      \]

      So, one representation is:

      \[
      \left(-3, \frac{17\pi}{6}\right)
      \]

2. **Change the sign of \(r\) and adjust \(\theta\) by \(\pi\):**

    - If we negate \(r\), we reverse the direction of the point, but we need to adjust the angle by adding \(\pi\) to account for the change in direction. This will give another valid representation.
    - If we change \(r = -3\) to \(r = 3\), then we add \(\pi\) to the angle \(\frac{5\pi}{6}\):

      \[
      \theta = \frac{5\pi}{6} + \pi = \frac{5\pi}{6} + \frac{6\pi}{6} = \frac{11\pi}{6}
      \]

      So, another representation is:

      \[
      (3, \frac{11\pi}{6})
      \]

### Final Answer:

- \((-3, \frac{17\pi}{6})\)
- \((3, \frac{11\pi}{6})\)

Let me know if you'd like more clarification!

### Related Questions:
1. How do you convert between rectangular and polar coordinates?
2. How does changing the angle affect the location of a point in polar coordinates?
3. What are the general rules for finding equivalent polar coordinates?
4. What is the significance of negative \(r\) in polar coordinates?
5. How do you handle angles outside the typical range of \([0, 2\pi)\)?

**Tip:** When working with polar coordinates, always remember that adding \(2\pi\) or subtracting it from the angle \(\theta\) results in an equivalent point, but you must consider the sign of \(r\) when adjusting the angle!

Suppose a point has polar coordinates (-3, 5pi/6). Find two additional polar representations of the point.

Learn how to find multiple polar coordinate representations for a given point. This solution explains how to adjust the angle by adding 2π and reverse the radius to find equivalent points in polar coordinates. Suitable for students in Grades 9-12 studying trigonometry and coordinate geometry.

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