Complex numbers can be expressed in polar form, which simplifies operations like multiplication. When multiplying two complex numbers in polar form, the process is straightforward: multiply the magnitudes (r values) and add the angles (θ values). This method avoids the need to convert to rectangular form, making calculations more efficient.

For example, consider two complex numbers represented as \( z_1 = r_1 \text{cis} \theta_1 \) and \( z_2 = r_2 \text{cis} \theta_2 \). The product of these two complex numbers can be calculated using the formula:

\[ z_1 \cdot z_2 = (r_1 \cdot r_2) \text{cis} (\theta_1 + \theta_2) \]

To illustrate, if \( z_1 = 3 \text{cis} 15^\circ \) and \( z_2 = 2 \text{cis} 30^\circ \), the multiplication proceeds as follows:

1. Multiply the r values: \( 3 \times 2 = 6 \).

2. Add the angles: \( 15^\circ + 30^\circ = 45^\circ \).

3. Combine the results: \( z_1 \cdot z_2 = 6 \text{cis} 45^\circ \).

This notation, where \( \text{cis} \theta \) stands for \( \cos \theta + i \sin \theta \), provides a compact way to express complex numbers in polar form.

In another example, if \( z_1 = 4 \text{cis} \frac{\pi}{6} \) and \( z_2 = 5 \text{cis} \frac{\pi}{3} \), the multiplication would be:

1. Multiply the r values: \( 4 \times 5 = 20 \).

2. Add the angles: \( \frac{\pi}{6} + \frac{\pi}{3} \). To add these, convert \( \frac{\pi}{3} \) to a common denominator: \( \frac{\pi}{3} = \frac{2\pi}{6} \). Thus, \( \frac{\pi}{6} + \frac{2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \).

3. Combine the results: \( z_1 \cdot z_2 = 20 \text{cis} \frac{\pi}{2} \).

In summary, when multiplying complex numbers in polar form, remember to multiply the magnitudes and add the angles. Using the \( \text{cis} \) notation streamlines the process, making it easier to work with complex numbers in various mathematical contexts.