To plot the given polar coordinate $(-1, \frac{5\pi}{4})$:

1. Understand the polar coordinate:

   • The radius $r = -1$: A negative radius means the point is in the opposite direction of the angle.
   • The angle $\theta = \frac{5\pi}{4}$: This is in the third quadrant (measured counterclockwise from the positive x-axis).

2. Adjust for the negative radius:

   • A negative radius inverts the point to the direction $\theta - \pi$. For $\theta = \frac{5\pi}{4}$, subtract $\pi$ to get $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$.

3. Final position:

   • The equivalent positive-radius point is at $r = 1$, $\theta = \frac{\pi}{4}$, which is in the first quadrant.

4. Plotting:

   • Locate $\frac{\pi}{4}$ in the first quadrant.
   • Move a distance of 1 unit along the line at this angle. This corresponds to where the original point $(-1, \frac{5\pi}{4})$ lies.

Would you like further clarification or a breakdown of these steps?

Here are 5 related questions:

1. How do you convert between polar and Cartesian coordinates?
2. What does a negative radius imply in polar coordinates?
3. How can angles greater than $2\pi$ or less than $0$ be normalized in polar systems?
4. How do polar coordinates differ in representing the same Cartesian point?
5. What are the applications of polar coordinates in real-world problems?

Tip: To avoid confusion with negative radii, you can always add $\pi$ to the angle to