# How do I multiply complex numbers in polar form?

To explain this, I will name two generic complex.
${c}_{1} = a \cdot c i s \left(\alpha\right)$ and ${c}_{2} = b \cdot c i s \left(\beta\right)$

The product between ${c}_{1}$ and ${c}_{2}$ is:
$a b \cdot c i s \left(\alpha\right) c i s \left(\beta\right) =$
$a b \cdot \left(\cos \left(\alpha\right) + i \sin \left(\alpha\right)\right) \left(\cos \left(\beta\right) + i \sin \left(\beta\right)\right) =$
ab*({cos(alpha)cos(beta)-sin(alpha)sin(beta)} +
$\left\{i \left(\sin \left(\alpha\right) \sin \left(\beta\right) + \cos \left(\alpha\right) \sin \left(\beta\right)\right\}\right) =$
$a b \cdot \left\{\cos \left(a + b\right) + i \sin \left(a + b\right)\right\}$//

Therefore, we can assume that the product of the two complex numbers ${c}_{1}$ and ${c}_{2}$ can be generaly given by the form above.

Ex.:
$\left(2 \cdot c i s \left(\pi\right)\right) \cdot \left(3 \cdot c i s \left(2 \pi\right)\right) = 6 \cdot c i s \left(3 \pi\right) = 6 \cdot c i s \left(\pi\right)$

Hope it helps.