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!