The problem asks you to find two other pairs of polar coordinates that represent the same point as $(9, \frac{\pi}{4})$, one with $r > 0$ and the other with $r < 0$.

Step 1: Finding another pair with $r > 0$

The radius $r = 9$ stays the same, but you can adjust the angle by adding or subtracting any multiple of $2\pi$. In this case, to maintain $r > 0$, you can add $2\pi$ to the angle $\frac{\pi}{4}$.

$\theta = \frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4}$

Thus, the second set of coordinates is: $(9, \frac{9\pi}{4})$

Step 2: Finding another pair with $r < 0$

When $r < 0$, you must add or subtract an odd multiple of $\pi$ to the angle to point in the opposite direction. So, you subtract $\pi$ from the angle:

$\theta = \frac{\pi}{4} - \pi = \frac{\pi}{4} - \frac{4\pi}{4} = -\frac{3\pi}{4}$

With $r = -9$, the third set of coordinates is: $(-9, -\frac{3\pi}{4})$

Thus, the two additional polar coordinates are:

1. $(9, \frac{9\pi}{4})$
2. $(-9, -\frac{3\pi}{4})$

Would you like a more detailed explanation or help with another step?

Here are 5 related questions to deepen your understanding:

1. What is the relationship between adding $2\pi$ to the angle and the position of the point?
2. How does reversing the radius change the angle in polar coordinates?
3. Why is $r = 9$ unchanged when adding multiples of $2\pi$ to the angle?
4. How would the Cartesian coordinates change for each polar pair?
5. What would be the next polar pair if you subtracted $2\pi$ instead of adding it?

Tip: In polar coordinates, angles are periodic, so adding or subtracting multiples of $2\pi$ represents the same direction.