To convert polar coordinates $(r, \theta)$ into rectangular coordinates $(x, y)$, the formulas are:

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

From the image:

• $r = 2$
• $\theta = \frac{5\pi}{6}$

Step-by-step Solution:

1. Calculate $x$: $x = r \cdot \cos\left(\frac{5\pi}{6}\right) = 2 \cdot \cos\left(\frac{5\pi}{6}\right)$ $\cos\left(\frac{5\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$ $x = 2 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3}$

2. Calculate $y$: $y = r \cdot \sin\left(\frac{5\pi}{6}\right) = 2 \cdot \sin\left(\frac{5\pi}{6}\right)$ $\sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$ $y = 2 \cdot \frac{1}{2} = 1$

Final Rectangular Coordinates:

$(x, y) = \left(-\sqrt{3}, 1\right)$

Would you like further clarifications or details on how polar-to-rectangular conversions work?

Five related questions:

1. What are polar coordinates, and how do they differ from rectangular coordinates?
2. How do we determine the quadrant of the rectangular coordinates from the polar angle?
3. What is the geometric interpretation of the conversion formulas?
4. Can we go from rectangular to polar coordinates? If so, how?
5. How do trigonometric identities help in simplifying conversions?

Tip: Always remember the unit circle values for common angles (like $\pi/6, \pi/4, \pi/3$) for quick computations in conversions!