In Exercises 22–24, find the quotient z₁/z₂ of the complex num...

Question

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.

z₁ = 5 (cos 4π/3 + i sin 4π/3)

z₂ = 10 (cos π/3 + i sin π/3)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to divide the following complex numbers. Z sub one divided by Z sub two and express the final answer in polar form our Zs sub one. So in the numerator will have nine multiplied by the quantity of the cosine of five pi divided by six plus I sine five pi divided by six. And that is being divided by our Z sub two which is 45 multiplied by the quantity of the cosine of pi divided by six plus I sine pi divided by six. Our answer choices are answer choice. A 1/5 multiplied by the quantity of the cosine of two pi divided by three plus I sign of two pi divided by three answer. Choice B five multiplied by the quantity of the cosine of two pi divided by three plus I sign of two pi divided by three answer choice C 1/ multiplied by the quantity of the cosine of pi plus I sign of pi. And answer choice D five multiplied by the quantity of the cosine of pi plus I sign of pi. All right. So we are going to divide these, this is great because they're already in polar form. And we recall from previous lessons that when we are finding the quotient of two complex numbers in polar form, we're going to do two things. We are going to divide the module. The modular are the numbers on the outside being multiplied by the set of parentheses. So here the module are nine and 45. So we're taking nine divided by 45 that's going to simplify to be 1/5. And then we are going to multiply that modulus. So that 1/5 is being multiplied by the quantity of. And now inside of the parentheses, we are going to subtract our arguments. Our arguments are the angles. So five pi divided by six in the numerator and pi divided by six in the denominator five pi divided by six minus pi divided by six is four pi divided by six which simplifies to two pi divided by three. So that 1/5 is multiplied by the quantity of now the cosine of the new argument, two pi divided by three plus I sign of two pi divided by three. And that is still in polar form. So that one is done, we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.
z₁ = 5 (cos 4π/3 + i sin 4π/3)
z₂ = 10 (cos π/3 + i sin π/3)

Key Concepts

Polar Form of Complex Numbers

Division of Complex Numbers in Polar Form

Trigonometric Identities for Cosine and Sine

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