To fill in the box for part (a), we need to find a new polar representation of the point $(-4, 3\pi)$ where $r > 0$ and $-2\pi \leq \theta < 0$.

Step-by-step solution:

1. Understand the current coordinates:

   • The point is $(-4, 3\pi)$, meaning the radius $r = -4$ and the angle $\theta = 3\pi$.

2. Convert to a positive radius:

   • If $r > 0$, we need to add $\pi$ to the angle to represent the same point with a positive $r$. Thus: $\theta = 3\pi - \pi = 2\pi$

   • However, $2\pi$ is not within the required interval $-2\pi \leq \theta < 0$. So, subtract $2\pi$ from the angle: $\theta = 2\pi - 2\pi = 0$

3. Verify the new coordinates:

   • With $r = 4$ and $\theta = 0$, the point is represented in polar form as $(4, 0)$.

Final Answer:

$(4, 0)$

If you need further clarification or want to verify additional conditions, feel free to ask!

Follow-Up Questions:

1. How do we know the angle adjustment keeps the point equivalent in polar coordinates?
2. Why is the negative radius important for certain polar representations?
3. Could there be other equivalent coordinates for this point?
4. How do periodic properties of angles in polar coordinates help in conversions?
5. What is the geometric reasoning behind changing $r$ and $\theta$?

Tip:

Always verify the angle is within the specified range when adjusting polar coordinates!