To find which polar coordinates represent the same point as $(4, -\frac{\pi}{2})$, we need to remember the properties of polar coordinates:

• The point $(r, \theta)$ can be represented in multiple ways by modifying either $r$ or $\theta$, while keeping the position of the point unchanged.
• If $r > 0$, we can add or subtract multiples of $2\pi$ to the angle $\theta$ to find equivalent coordinates.
• If $r < 0$, we flip the direction of the radius and shift the angle by $\pi$.

Given the point $(4, -\frac{\pi}{2})$, let's evaluate the provided options:

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

   • For $r = -4$, the angle would be $-\frac{3\pi}{2} + \pi = -\frac{\pi}{2}$, which matches the angle of $(4, -\frac{\pi}{2})$. Therefore, this represents the same point.

2. $(4, \frac{5\pi}{2})$:

   • Since $\frac{5\pi}{2} - 2\pi = -\frac{\pi}{2}$, this is the same point as $(4, -\frac{\pi}{2})$.

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

   • For $r = -4$, you need to shift the angle by $\pi$. So the corresponding angle would be $-\frac{\pi}{2} + \pi = \frac{\pi}{2}$, which is not the same angle.

4. $(-4, \frac{\pi}{2})$:

   • This represents a point in the opposite direction. Since $r = -4$, we shift the angle by $\pi$, giving $\frac{\pi}{2} - \pi = -\frac{\pi}{2}$, which represents the same point.

5. $(4, \frac{7\pi}{2})$:

   • Since $\frac{7\pi}{2} - 2\pi = \frac{3\pi}{2}$, this is not the same as $-\frac{\pi}{2}$.

6. $(4, -\frac{5\pi}{2})$:

   • Since $-\frac{5\pi}{2} + 2\pi = -\frac{\pi}{2}$, this represents the same point.

Correct Answers:

• $(-4, -\frac{3\pi}{2})$
• $(4, \frac{5\pi}{2})$
• $(-4, \frac{\pi}{2})$
• $(4, -\frac{5\pi}{2})$

Would you like further details, or do you have any other questions?

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Here are 5 related questions:

1. What is the formula to convert polar coordinates to Cartesian coordinates?
2. How do you convert from Cartesian coordinates to polar coordinates?
3. Can two different radii represent the same point in polar coordinates?
4. How does the angle in polar coordinates relate to the unit circle?
5. What are the implications of negative radii in polar coordinates?

Tip: Always reduce angles to a standard form (between $0$ and $2\pi$) when working with polar coordinates for easier comparison.