Product of Complex Numbers in Polar Form

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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

\(\displaystyle z_1 z_2\) \(=\) \(\displaystyle r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2}\) Definition of Polar Form of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle r_1 r_2 \paren {\paren {\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2} + i \paren {\cos \theta_1 \sin \theta_2 + \sin \theta_1 \cos \theta_2} }\) Definition of Complex Multiplication
\(\displaystyle \) \(=\) \(\displaystyle r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \paren {\cos \theta_1 \sin \theta_2 + \sin \theta_1 \cos \theta_2} }\) Cosine of Sum
\(\displaystyle \) \(=\) \(\displaystyle r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \, \map \sin {\theta_1 + \theta_2} }\) 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: $5 \cis 20 \degrees \times 3 \cis 40 \degrees$

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


Sources