Zalgorithm

Multiplying complex numbers in polar form

This is not a math tutorial. See Why am I writing about math?

Related to notes / Multiplying complex numbers

The solution:

r1r2[cos(θ1+θ2)+isin⁡(θ1+θ2)] r_1r_2[cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]

Working it out, starting from:

[r1(cos⁡θ1+isin⁡θ1)]×[r2(cos⁡θ2+isin⁡θ2)] [r_1(\cos\theta_1 + i \sin \theta_1)] \times[r_2(\cos\theta_2 + i \sin\theta_2)] r1r2[(cos⁡θ1+isin⁡θ1)(cos⁡θ2+isin⁡θ2)] r_1r_2[(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2)] r1r2[cos⁡θ1cos⁡θ2+icos⁡θ1sin⁡θ2+isin⁡θ1cos⁡θ2+i2sin⁡θ1sin⁡θ2] r_1r_2[\cos\theta_1\cos\theta_2 + i\cos\theta_1\sin\theta_2 + i\sin\theta_1\cos\theta_2 + i^2\sin\theta_1\sin\theta_2] r1r2[cos⁡θ1cos⁡θ2−sin⁡θ1sin⁡θ2+i(sin⁡θ1cos⁡θ2+cos⁡θ1sin⁡θ2)] r_1r_2[\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 + i(\sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2)]

What’s significant in the last form of the expression is that it contains the angle addition formulas from trigonometry :

cos⁡(θ1+θ2)=cos⁡θ1cos⁡θ2−sin⁡θ1sin⁡θ2 \cos(\theta_1 + \theta_2) = \cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 sin⁡(θ1+θ2)=sin⁡θ1cos⁡θ2+cos⁡θ1sin⁡θ2 \sin(\theta_1 + \theta_2) = \sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2

That means that the formula for multiplying complex numbers in the polar form can be simplified to

r1r2[cos⁡(θ1+θ2)+isin⁡(θ1+θ2)] r_1r_2[\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]
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