To convert polar coordinates $(-4, -2\pi)$ into rectangular coordinates $(x, y)$, we use the following formulas:

$x = r \cdot \cos(\theta)$ $y = r \cdot \sin(\theta)$ Where:

• $r = -4$ (the radial distance)
• $\theta = -2\pi$ (the angle in radians)

Step 1: Calculate $x$

$x = -4 \cdot \cos(-2\pi)$ Since $\cos(-2\pi) = \cos(2\pi) = 1$: $x = -4 \cdot 1 = -4$

Step 2: Calculate $y$

$y = -4 \cdot \sin(-2\pi)$ Since $\sin(-2\pi) = \sin(2\pi) = 0$: $y = -4 \cdot 0 = 0$

Final Rectangular Coordinates:

$(x, y) = (-4, 0)$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What happens if the angle is in degrees instead of radians?
2. How does a negative radius affect the rectangular coordinates?
3. Can you convert rectangular coordinates back to polar coordinates?
4. What are the rectangular coordinates of $(-4, \pi)$?
5. How does $\cos(\theta)$ and $\sin(\theta)$ behave for different values of $\theta$?

Tip: Remember that the cosine and sine of angles at multiples of $\pi$ often produce simple values like 0, 1, or -1.