The given complex number is in polar form $z = r (\cos \theta + i \sin \theta)$. To convert it into the standard rectangular form $a + bi$, follow these steps:

Given:

• $r = 8$
• $\theta = 30^\circ$

Step 1: Use the trigonometric values of $\cos 30^\circ$ and $\sin 30^\circ$

• $\cos 30^\circ = \frac{\sqrt{3}}{2}$
• $\sin 30^\circ = \frac{1}{2}$

Step 2: Expand using $r \cdot (\cos \theta + i \sin \theta)$

$z = 8 \cdot \left( \cos 30^\circ + i \sin 30^\circ \right)$ $z = 8 \cdot \frac{\sqrt{3}}{2} + 8 \cdot i \cdot \frac{1}{2}$

Step 3: Simplify the terms

$z = 4\sqrt{3} + 4i$

Final Answer:

$z = 4\sqrt{3} + 4i$

Would you like me to explain any part of the process or provide further examples?

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Related Questions:

1. How do you convert a complex number from rectangular form to polar form?
2. What are the trigonometric values for other common angles, such as $45^\circ$ and $60^\circ$?
3. How is the magnitude of a complex number calculated?
4. What are the applications of polar form in electrical engineering or physics?
5. How do you compute powers of complex numbers in polar form?

Tip:

When working with polar forms, memorize the basic trigonometric values for angles like $30^\circ, 45^\circ,$ and $60^\circ$. It simplifies many calculations!