The problem is asking to find the rectangular coordinates (Cartesian coordinates) of the point given in polar coordinates $(4, -\frac{2\pi}{3})$.

Conversion from Polar to Rectangular Coordinates:

The formulas to convert from polar to rectangular coordinates are: $x = r \cdot \cos(\theta)$ $y = r \cdot \sin(\theta)$ where $r$ is the radial distance and $\theta$ is the angle in radians.

Given:

• $r = 4$
• $\theta = -\frac{2\pi}{3}$

1. Calculating $x$:

$x = 4 \cdot \cos\left(-\frac{2\pi}{3}\right)$ Since $\cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}$, we have: $x = 4 \cdot \left(-\frac{1}{2}\right) = -2$

2. Calculating $y$:

$y = 4 \cdot \sin\left(-\frac{2\pi}{3}\right)$ Since $\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}$, we have: $y = 4 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}$

Thus, the rectangular coordinates are: $\boxed{(-2, -2\sqrt{3})}$

Would you like further details or have any questions?

Here are five related questions to explore:

1. How can we convert rectangular coordinates to polar coordinates?
2. What are the common values of sine and cosine for angles in radians?
3. How can you interpret negative angles in polar coordinates geometrically?
4. Can polar coordinates have negative radial values?
5. What is the significance of the angle $-\frac{2\pi}{3}$ in the unit circle?

Tip: When converting polar to rectangular coordinates, ensure the angle is in radians for trigonometric calculations unless otherwise specified.