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

1 Answer
Bernardo Russo G. Ferreira
Oct 2, 2014

In trigonometric form, a complex number looks like this:
a + bi = c*cis(theta)
where a, b and c are scalars.

Let two complex numbers:
-> k_(1) = c_(1)*cis(alpha)
-> k_(2) = c_(2)*cis(beta)
k_(1)*k_(2) = c_(1) * c_(2) * cis(alpha) * cis(beta) =
= c_(1) * c_(2) * (cos(alpha) + i*sin(alpha)) * (cos(beta) + i*sin(beta))

This product will end up leading to the expression
k_(1)*k_(2) =
= c_(1)*c_(2)*(cos(alpha + beta) + i*sin(alpha + beta)) =
= c_(1)*c_(2)*cis(alpha+beta)

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:
(c_(1) * cis(alpha)) * (c_(2) * cis(beta)) = c_(1)*c_(2)*cis(alpha+beta)

Hope it helps.

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