# Product of Complex Numbers in Polar Form

## Theorem

Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form.

Then:

$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$

### General Result

Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.

Let $z_j = \polar {r_j, \theta_j}$ be $z_j$ expressed in polar form for each $j \in \set {1, 2, \ldots, n}$.

Then:

$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$

## Proof

 $\ds z_1 z_2$ $=$ $\ds r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2}$ Definition of Polar Form of Complex Number $\ds$ $=$ $\ds r_1 r_2 \paren {\paren {\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2} + i \paren {\cos \theta_1 \sin \theta_2 + \sin \theta_1 \cos \theta_2} }$ Definition of Complex Multiplication $\ds$ $=$ $\ds r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \paren {\cos \theta_1 \sin \theta_2 + \sin \theta_1 \cos \theta_2} }$ Cosine of Sum $\ds$ $=$ $\ds r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$ Sine of Sum

$\blacksquare$

## Examples

### Example: $3 \cis 40 \degrees \times 4 \cis 80 \degrees$

$3 \cis 40 \degrees \times 4 \cis 80 \degrees = -6 + 6 \sqrt 3 i$

### Example: $5 \cis 20 \degrees \times 3 \cis 40 \degrees$

$5 \cis 20 \degrees \times 3 \cis 40 \degrees = \dfrac {15} 2 + \dfrac {15 \sqrt 3} 2 i$
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