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Video: Finding the Quotient of Two Complex Numbers in Polar Form

Alex Cutbill

Given that 𝑧₁ = 20 (cos (𝜋/2) + 𝑖 sin (𝜋/2)) and 𝑧₂ = 4 (cos (𝜋/6) + 𝑖 sin (𝜋/6)), find 𝑧₁/𝑧₂ in polar form.

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Video Transcript

Given that 𝑧 one equals 20 times cos 𝜋 by two plus 𝑖 sin 𝜋 by two and 𝑧 two equals four times cos 𝜋 by six plus 𝑖 sin 𝜋 by six, find 𝑧 one over 𝑧 two in polar form.

We have two complex numbers, both given in polar form, and we have to find their quotient, also in polar form. Writing numbers in polar form makes it very easy to find their product or quotient. To find their product, we’d simply have to multiply their moduluses together to find the modulus of the product and add their arguments together to find the argument of the product.

But as we want to find the quotient, we’ll have to divide one modulus by the other and subtract one argument from the other. So let’s write this as 20 times cos 𝜋 by two plus 𝑖 sin 𝜋 by two over four times cos 𝜋 by six plus 𝑖 sin 𝜋 by six. And we want to express this in polar form, so that means 𝑟 times cos 𝜃 plus 𝑖 sin 𝜃.

We find the value of 𝑟, which is the modulus of our answer, by dividing the modulus in the numerator, 20, by the modulus in the denominator, four. This gives us a modulus of five.

Now we simply have to find the argument of our answer, 𝜃. We do this by finding the argument in the numerator and subtracting the argument in the denominator. So 𝜃 is equal to 𝜋 by two minus 𝜋 by six, which is 𝜋 by three.

So our answer is five times cos 𝜋 by three plus 𝑖 sin 𝜋 by three. And notice how having all three numbers written in polar form made finding this quotient very easy.