To plot the point and find three equivalent expressions for the point given in polar coordinates, let's analyze the point:

Given Point:
$(r, \theta) = (-5, \frac{7\pi}{2})$

The negative radius $r = -5$ indicates that the point is actually in the opposite direction from $\theta = \frac{7\pi}{2}$, meaning we can convert this to a positive radius by adjusting $\theta$ by $\pi$ radians.

Step-by-Step Solutions

Plotting the Point

1. Convert $\theta = \frac{7\pi}{2}$ into an equivalent angle between $0$ and $2\pi$: $\frac{7\pi}{2} = 2\pi + \frac{3\pi}{2}$ So, the direction of this point is equivalent to $\theta = \frac{3\pi}{2}$ with a radius of $-5$, pointing 5 units in the opposite direction from $\theta = \frac{3\pi}{2}$.

Finding Equivalent Expressions

We will rewrite this point in three different forms as requested.

1. Form (a): $r < 0$ and $0 \leq \theta < 2\pi$

   • $r = -5$ and $\theta = \frac{3\pi}{2}$ satisfy this condition.
   • Answer for (a): $(r, \theta) = (-5, \frac{3\pi}{2})$

2. Form (b): $r > 0$ and $-2\pi \leq \theta < 0$

   • To make $r$ positive, we take $r = 5$.
   • Since $\theta = \frac{3\pi}{2}$ in the opposite direction gives a positive radius of 5, we subtract $2\pi$ to bring it into the range $[-2\pi, 0)$: $\theta = -\frac{\pi}{2}$
   • Answer for (b): $(r, \theta) = (5, -\frac{\pi}{2})$

3. Form (c): $r > 0$ and $2\pi \leq \theta < 4\pi$

   • Use $r = 5$ and add $2\pi$ to $\theta = \frac{3\pi}{2}$ to get an equivalent angle in the range $2\pi \leq \theta < 4\pi$: $\theta = \frac{7\pi}{2}$
   • Answer for (c): $(r, \theta) = (5, \frac{7\pi}{2})$

Summary of Equivalent Points:

• (a) $(r, \theta) = (-5, \frac{3\pi}{2})$
• (b) $(r, \theta) = (5, -\frac{\pi}{2})$
• (c) $(r, \theta) = (5, \frac{7\pi}{2})$

Would you like further details on polar coordinate transformations?

Here are some related questions to explore:

1. How do you convert between polar and Cartesian coordinates?
2. Why is it useful to have multiple representations of polar coordinates?
3. What is the effect of changing $r$ and $\theta$ on the plotted position?
4. How can you visualize negative angles in polar coordinates?
5. What are other applications of polar coordinates in mathematics?

Tip: In polar coordinates, a negative radius reflects the point across the origin in the opposite direction of the angle, making conversions to positive radii straightforward with angle adjustments.