To match polar coordinates that represent the same point, we need to understand the principles of polar coordinates. Polar coordinates are in the form $(r, \theta)$, where $r$ is the radius (distance from the origin), and $\theta$ is the angle. Two points represent the same location if either:

1. The radius $r$ is the same, and the angle $\theta$ is coterminal (differ by multiples of $2\pi$).
2. The radius $r$ is negated, and the angle $\theta$ is shifted by $\pi$ (because negating the radius reverses the direction of the point).

Now let's analyze and match the given coordinates:

Given Coordinates:

1. $(3, \frac{5\pi}{4})$
2. $(-3, \frac{3\pi}{4})$
3. $(-3, \frac{5\pi}{2})$
4. $(-3, \frac{3\pi}{2})$

Coordinates to Match:

1. $(3, \frac{5\pi}{4})$
2. $(-3, \frac{3\pi}{2})$
3. $(-3, \frac{3\pi}{4})$
4. $(3, \frac{5\pi}{4})$

Solution:

Step-by-Step Matching:

1. $(3, \frac{5\pi}{4})$:

   • This has a positive radius $r = 3$ and angle $\frac{5\pi}{4}$. To match, look for:
     • Same $r = 3$, with a coterminal angle ($\frac{5\pi}{4} \pm 2n\pi$).
     • OR $r = -3$, with $\theta = \frac{5\pi}{4} + \pi$ ($\frac{9\pi}{4}$).
   • Match: $(3, \frac{5\pi}{4})$.

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

   • This has a negative radius $r = -3$ and angle $\frac{3\pi}{2}$. To match, look for:
     • Opposite radius $r = 3$ with $\theta = \frac{3\pi}{2} + \pi = \frac{7\pi}{2}$ or coterminal.
   • Match: $(-3, \frac{3\pi}{2})$.

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

   • This has a negative radius $r = -3$ and angle $\frac{3\pi}{4}$. To match, look for:
     • Positive radius $r = 3$, with $\theta = \frac{3\pi}{4} + \pi = \frac{7\pi}{4}$ or coterminal.
   • Match: $(-3, \frac{3\pi}{4})$.

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

   • This has a negative radius $r = -3$ and angle $\frac{5\pi}{2}$. To match, look for:
     • Positive radius $r = 3$, with $\theta = \frac{5\pi}{2} + \pi = \frac{9\pi}{2}$ or coterminal.
   • Match: $(-3, \frac{5\pi}{2})$.

Let me know if you need a more detailed breakdown! Here are some follow-up questions for you:

1. What are coterminal angles in polar coordinates?
2. How does the sign of the radius affect the angle in polar coordinates?
3. How do we convert polar coordinates to rectangular coordinates?
4. What are some practical applications of polar coordinates?
5. How do $\theta$ values beyond $2\pi$ impact matching points?

Tip: Remember that you can always add or subtract multiples of $2\pi$ to angles for coterminal points!