The polar coordinates $(2\sqrt{3}, \frac{7\pi}{6})$ can be converted to rectangular coordinates $(x, y)$ using the following formulas:

$x = r \cos \theta$ $y = r \sin \theta$

where $r = 2\sqrt{3}$ and $\theta = \frac{7\pi}{6}$.

Step 1: Calculate $x$

$x = 2\sqrt{3} \cdot \cos\left(\frac{7\pi}{6}\right)$ The angle $\frac{7\pi}{6}$ is in the third quadrant, where cosine is negative. The reference angle is $\frac{\pi}{6}$, and $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$. Therefore: $\cos\left(\frac{7\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ Now substitute: $x = 2\sqrt{3} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -3$

Step 2: Calculate $y$

$y = 2\sqrt{3} \cdot \sin\left(\frac{7\pi}{6}\right)$ In the third quadrant, sine is also negative. The reference angle is $\frac{\pi}{6}$, and $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$. Therefore: $\sin\left(\frac{7\pi}{6}\right) = -\frac{1}{2}$ Now substitute: $y = 2\sqrt{3} \cdot \left(-\frac{1}{2}\right) = -\sqrt{3}$

Final Answer:

The rectangular coordinates are $(-3, -\sqrt{3})$.

Do you need further details or explanations? Here are five related questions you could explore:

1. How are reference angles used when converting polar coordinates?
2. What are the signs of trigonometric functions in each quadrant?
3. How can you convert rectangular coordinates back to polar coordinates?
4. What are the unit circle values for common angles like $\frac{\pi}{6}$?
5. How do trigonometric functions change as the angle increases?

Tip: Always check which quadrant an angle lies in to determine the correct sign of sine and cosine.