To find the rectangular coordinates $(x, y)$ from the polar coordinates given, use the following formulas:

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

From the image:

• $r = 2$
• $\theta = \frac{5\pi}{6}$

Now calculate:

1. $x$: $x = 2 \cos\left(\frac{5\pi}{6}\right)$ The cosine of $\frac{5\pi}{6}$ is $-\frac{\sqrt{3}}{2}$. So, $x = 2 \times -\frac{\sqrt{3}}{2} = -\sqrt{3}.$

2. $y$: $y = 2 \sin\left(\frac{5\pi}{6}\right)$ The sine of $\frac{5\pi}{6}$ is $\frac{1}{2}$. So, $y = 2 \times \frac{1}{2} = 1.$

Thus, the rectangular coordinates are: $(x, y) = (-\sqrt{3}, 1).$

Would you like a detailed explanation or assistance with related concepts?

Here are five related questions to explore:

1. How are polar and rectangular coordinates used in real-world applications?
2. Can polar coordinates be converted to rectangular coordinates if $\theta$ is negative?
3. What are the reverse formulas to convert rectangular to polar coordinates?
4. How can you interpret angles like $\frac{5\pi}{6}$ geometrically?
5. What are the key differences between polar and rectangular coordinate systems?

Tip: Always