To multiply the two complex numbers given in polar form:

Given:

• $z_1 = [3, 52^\circ]$
• $z_2 = [2, 15^\circ]$

The product of two complex numbers in polar form is given by: $z_1 \times z_2 = [r_1 \times r_2, \theta_1 + \theta_2]$

Where:

• $r_1$ and $r_2$ are the magnitudes
• $\theta_1$ and $\theta_2$ are the angles

Step 1: Multiply the magnitudes: $r_1 \times r_2 = 3 \times 2 = 6$

Step 2: Add the angles: $\theta_1 + \theta_2 = 52^\circ + 15^\circ = 67^\circ$

So, the product is: $z_1 \times z_2 = [6, 67^\circ]$

Result:

$z_1 \times z_2 = [6, 67^\circ]$

Would you like more details or have any questions?

Here are five related questions:

1. How would you convert a complex number in polar form to rectangular form?
2. What is the inverse operation of multiplying two complex numbers in polar form?
3. Can you multiply two complex numbers if one is in polar form and the other is in rectangular form?
4. How does multiplying the magnitudes and adding the angles correspond to geometric transformations in the complex plane?
5. What is the significance of the angle in the polar form of a complex number?

Tip: When multiplying complex numbers in polar form, always ensure that the angles are measured in the same units (degrees or radians).