In Exercises 45–52, find the quotient z₁/z₂ of the complex num...

Question

In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.

z₁ = cos 80° + i sin 80°

z₂ = cos 200° + i sin 200°

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. Ensure that for the final answer, theta is between zero and 360 degrees inclusive of zero and 360 degrees. Our Z sub one. So the numerator is the cosine of 65 degrees plus I sign 65 degrees. And that's divided by Z sub two, Z sub two is the cosine of 315 degrees plus I sine 315 degrees. And our answer choices are answer choice. A cosine 250 degrees plus I sign 250 degrees. Answer choice B cosine 250 degrees minus I sign 250 degrees. Answer choice C cosine 110 degrees minus I sign 200 or excuse me, 110 degrees and answer choice. D cosine 110 degrees plus I sign 110 degrees. All right. So in order to divide complex numbers, we want them to be in polar form. And these don't quite look like they're in polar form, but they actually, are we recall from previous lessons that polar form is when we have R multiplied by the quantity of the cosine of theta plus I sine of theta. And we've got most of that, we've got the cosine of in the numerator. The theta is 65 degrees plus I sign of 65 degrees. But there's no R in the numerator. But there is really, here's how there is. So in the numerator, we could put a parentheses around cosine 65 degrees plus I sine 65 degrees. And then the number in front of it that we'd multiply is just a positive one. So R equals a positive one. And it's the same thing for the denominator. The R value, there is a positive one, the theta for the denominator is 315 degrees. And so when we've got numbers in polar form, in order to divide, in order to find the quotient, we need to do two things. We are going to divide the moduli. Those are the RS what's in front being multiplied by the parentheses. So dividing one by one. Well, that's still one. And so we don't need to write that for our answer choice. It's just a one for our modulus. The second thing that we need to do is subtract our arguments. The arguments are the, theta. So we take the one in the numerator and subtract the one in the denominator, we have 65 degrees minus 315 degrees, 65 degrees minus 315 degrees is a negative 250 degrees. And this is why they have in the directions. Ensure that for the answer that theta is in between zero and 360 degrees because negative 250 degrees is not. So we add one full rotation, we'll add 360 degrees to our negative 250 degrees. And that breaks brings us up to a positive 110 degrees. So there is our new argument. So we write this simplified as the cosine of our new argument, 110 degrees plus I sine of 110 degrees. And there is our answer simplified. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 45–52, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form. In Exercises 49–50, express the argument as an angle between 0° and 360°.
z₁ = cos 80° + i sin 80°
z₂ = cos 200° + i sin 200°

Key Concepts

Complex Numbers in Polar Form

Division of Complex Numbers in Polar Form

Argument of a Complex Number and Angle Normalization

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