How do you multiply two complex numbers in trigonometric form?

When multiplying two complex numbers in trigonometric form, we use the concept of their magnitudes and angles. The general form of a complex number in trigonometric (or polar) form is expressed as ( r(\cos(x) + i\sin(x)) ), where ( r ) is the magnitude and ( x ) is the argument (angle) of the complex number.

To multiply two complex numbers ( z_1 ) and ( z_2 ), we combine their magnitudes and add their angles. Specifically, if we have:

[ z_1 = r_1 (\cos(x_1) + i \sin(x_1)) ]

[ z_2 = r_2 (\cos(x_2) + i \sin(x_2)) ]

The product ( z_1 z_2 ) can be calculated as follows:

1. The magnitudes multiply: ( r_1 \cdot r_2 ).

2. The angles add: ( x_1 + x_2 ).

Therefore, the product can be expressed as:

[ z_1 z_2 = r_1 r_2 \left( \cos(x_1 + x_2) +