# Video: Multiplying Complex Numbers in Polar Form

Given that 𝑧₁ = 5(cos 2𝑎 + 𝑖 sin 2𝑎) and 𝑧₂ = 1/4 (cos 4𝑎 + 𝑖 sin 4𝑎), find 𝑧₁ 𝑧₂.

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### Video Transcript

Given that 𝑧 one is equal to five multiplied by cos two 𝑎 plus 𝑖 sin two 𝑎 and 𝑧 two is equal to a quarter multiplied by cos four 𝑎 plus 𝑖 sin four 𝑎, find 𝑧 one 𝑧 two.

Recall the product formula. This says that, for two complex numbers expressed in polar form, 𝑧 one has a modulus of 𝑟 one and an argument of 𝜃 one and 𝑧 two which has a modulus of 𝑟 two and an argument of 𝜃 two, their products can be found by multiplying together their moduli and adding together their arguments. It’s 𝑟 one 𝑟 two multiplied by cos of 𝜃 one plus 𝜃 two plus 𝑖 sin of 𝜃 one plus 𝜃 two.

In our question, the modulus of 𝑧 one is five and the modulus of 𝑧 two is one-quarter. This means we can find the modulus of 𝑧 one 𝑧 two by multiplying five by one-quarter, which is five-quarters. The argument of 𝑧 one is two 𝑎, and the argument of 𝑧 two is four 𝑎. So, we find the sum of their arguments. That’s two 𝑎 plus four 𝑎, which is six 𝑎. 𝑧 one 𝑧 two is therefore given as five-quarters multiplied by cos six 𝑎 plus 𝑖 sin six 𝑎.