# Video: Multiplying Complex Numbers in Polar Form

Given that π§β = 5(cos 2π + π sin 2π) and π§β = 1/4 (cos 4π + π sin 4π), find π§β π§β.

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### Video Transcript

Given that π§ one is equal to five multiplied by cos two π plus π sin two π and π§ two is equal to a quarter multiplied by cos four π plus π sin four π, find π§ one π§ two.

Recall the product formula. This says that, for two complex numbers expressed in polar form, π§ one has a modulus of π one and an argument of π one and π§ two which has a modulus of π two and an argument of π two, their products can be found by multiplying together their moduli and adding together their arguments. Itβs π one π two multiplied by cos of π one plus π two plus π sin of π one plus π two.

In our question, the modulus of π§ one is five and the modulus of π§ two is one-quarter. This means we can find the modulus of π§ one π§ two by multiplying five by one-quarter, which is five-quarters. The argument of π§ one is two π, and the argument of π§ two is four π. So, we find the sum of their arguments. Thatβs two π plus four π, which is six π. π§ one π§ two is therefore given as five-quarters multiplied by cos six π plus π sin six π.