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The image shows a problem asking to express a point in polar coordinates \( \left(5, \frac{\pi}{4} \right) \) in two other forms:

1. One with the same positive radius \( r > 0 \) and a different angle.
2. One with a negative radius \( r < 0 \).

### Explanation:

#### Case 1: \( r > 0 \)
For this case, we keep \( r = 5 \) and adjust the angle by adding \( 2\pi \) (or a multiple of \( 2\pi \)) to the given angle, as the angle in polar coordinates can be expressed by adding or subtracting full rotations (i.e., \( 2\pi \)).

So:
\[
\left( 5, \frac{\pi}{4} \right) = \left( 5, \frac{\pi}{4} + 2\pi \right) = \left( 5, \frac{9\pi}{4} \right)
\]

#### Case 2: \( r < 0 \)
For this case, we will express the point using a negative radius. When using a negative radius, we need to adjust the angle by adding \( \pi \) to move to the opposite direction along the same line.

So:
\[
\left( 5, \frac{\pi}{4} \right) = \left( -5, \frac{\pi}{4} + \pi \right) = \left( -5, \frac{5\pi}{4} \right)
\]

### Final Result:
1. \( \left( 5, \frac{9\pi}{4} \right) \)
2. \( \left( -5, \frac{5\pi}{4} \right) \)

Would you like more details on polar coordinate transformations or any further clarifications?

Here are 5 related questions to explore further:
1. How do we convert polar coordinates to Cartesian coordinates?
2. What happens when the angle in polar coordinates is negative?
3. How can we generalize the polar form for all rotations of a point?
4. Why is the radius negative, and how does that affect the plot of the point?
5. How can you find the distance between two points in polar coordinates?

**Tip:** Remember, you can add or subtract multiples of \( 2\pi \) to the angle in polar coordinates to get an equivalent representation of the same point.

We are given the point in polar coordinates (5, π/4). We are asked to find two other pairs of polar coordinates to represent the point, one with the radius r > 0 and one with r < 0.

Explore how to find two alternative polar coordinate representations for the point (5, π/4). This involves understanding the cyclic nature of trigonometric functions, working with positive and negative radii, and applying angle shifts.

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