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4.36 Multiplying complex numbers in polar form

Polar products

Let $z=re^{i\theta}$ and $w=se^{i\phi}$ be non-zero complex numbers in polar form.

(i) Then to multiply complex numbers, multiply the moduli and add the arguments

zw=rse^{i(\theta+\phi)}.

(ii) Likewise the quotient is

{{z}\over{w}}={{r}\over{s}}e^{i(\theta-\phi)}.

(i) Consider the formula

e^{i(\theta+\phi)}=e^{i\theta}e^{i\phi}=(\cos\theta+i\sin\theta)(\cos\phi+i% \sin\phi),

multiply this out, and use trigonometric addition rules.