# What is the formula for multiplying complex numbers in trigonometric form?

In trigonometric form, a complex number looks like this:
$a + b i = c \cdot c i s \left(\theta\right)$
where $a$, $b$ and $c$ are scalars.

Let two complex numbers:
$\to {k}_{1} = {c}_{1} \cdot c i s \left(\alpha\right)$
$\to {k}_{2} = {c}_{2} \cdot c i s \left(\beta\right)$
${k}_{1} \cdot {k}_{2} = {c}_{1} \cdot {c}_{2} \cdot c i s \left(\alpha\right) \cdot c i s \left(\beta\right) =$
$= {c}_{1} \cdot {c}_{2} \cdot \left(\cos \left(\alpha\right) + i \cdot \sin \left(\alpha\right)\right) \cdot \left(\cos \left(\beta\right) + i \cdot \sin \left(\beta\right)\right)$

This product will end up leading to the expression
${k}_{1} \cdot {k}_{2} =$
$= {c}_{1} \cdot {c}_{2} \cdot \left(\cos \left(\alpha + \beta\right) + i \cdot \sin \left(\alpha + \beta\right)\right) =$
$= {c}_{1} \cdot {c}_{2} \cdot c i s \left(\alpha + \beta\right)$

By analyzing the steps above, we can infer that, for having used generic terms ${c}_{1}$, ${c}_{2}$, $\alpha$ and $\beta$, the formula of the product of two complex numbers in trigonometric form is:
$\left({c}_{1} \cdot c i s \left(\alpha\right)\right) \cdot \left({c}_{2} \cdot c i s \left(\beta\right)\right) = {c}_{1} \cdot {c}_{2} \cdot c i s \left(\alpha + \beta\right)$

Hope it helps.