To solve the given problem, we need to convert the polar coordinates $(r, \theta)$ into rectangular coordinates $(x, y)$ using the formulas:

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

Given:

$r = 5, \quad \theta = 30^\circ$

Step 1: Compute $x$:

$x = r \cos \theta = 5 \cos 30^\circ$

Using the trigonometric value: $\cos 30^\circ = \frac{\sqrt{3}}{2}$, so: $x = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}$

Step 2: Compute $y$:

$y = r \sin \theta = 5 \sin 30^\circ$

Using the trigonometric value: $\sin 30^\circ = \frac{1}{2}$, so: $y = 5 \cdot \frac{1}{2} = \frac{5}{2}$

Final Rectangular Coordinates:

$\left(\frac{5\sqrt{3}}{2}, \frac{5}{2}\right)$

Match with Answer Choices:

The correct answer is: $\boxed{\left(\frac{5\sqrt{3}}{2}, \frac{5}{2}\right)}$

Let me know if you would like further details or explanations!

Related Questions:

1. What are polar coordinates, and how do they relate to rectangular coordinates?
2. What are the sine and cosine values for standard angles like $30^\circ, 45^\circ, 60^\circ$?
3. How do you find the angle $\theta$ in polar coordinates given rectangular coordinates?
4. Can this process be applied for negative values of $r$?
5. How would the solution change if $\theta$ were given in radians instead of degrees?

Tip:

Always memorize the sine and cosine values for common angles like $0^\circ, 30^\circ, 45^\circ, 60^\circ,$ and $90^\circ$. It makes conversions much quicker!