Video: Multiplying Complex Numbers in Polar Form

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

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Video Transcript

Given that 𝑧 one is equal to five multiplied by cos two π‘Ž plus 𝑖 sin two π‘Ž and 𝑧 two is equal to a quarter multiplied by cos four π‘Ž plus 𝑖 sin four π‘Ž, 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 π‘Ÿ one and an argument of πœƒ one and 𝑧 two which has a modulus of π‘Ÿ two and an argument of πœƒ two, their products can be found by multiplying together their moduli and adding together their arguments. It’s π‘Ÿ one π‘Ÿ two multiplied by cos of πœƒ one plus πœƒ two plus 𝑖 sin of πœƒ one plus πœƒ 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 π‘Ž, and the argument of 𝑧 two is four π‘Ž. So, we find the sum of their arguments. That’s two π‘Ž plus four π‘Ž, which is six π‘Ž. 𝑧 one 𝑧 two is therefore given as five-quarters multiplied by cos six π‘Ž plus 𝑖 sin six π‘Ž.

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