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

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 = 20(cos 75° + i sin 75°)
z₂ = 4(cos 25° + i sin 25°)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to divide the following complex numbers dividing Z sub one by Z sub two and express the final answer in polar form sub one. So the numerator is 14 multiplied by the quantity of the cosine of 65 degrees plus I sign degrees. And that's divided by Z sub two which is seven multiplied by the quantity of the cosine of degrees plus I sine 35 degrees. Our answer choices are answer choice A two multiply by the quantity of cosine 30 degrees minus I sine 30 degrees. Answer choice B two multiplied by the quantity of cosine degrees plus I sine 100 degrees. Answer choice C two multiplied by the quantity of cosine degrees plus I sine 30 degrees and answer choice D two multiplied by the quantity of cosine 100 degrees minus I sine 100 degrees. All right. Both of these complex numbers are already in polar form, which is great because we recall from previous lessons in the textbook that when we're dividing complex numbers in polar form, we essentially just need to do two things. We need to divide the module. The modular are the numbers on the outside of the parentheses. And so for the numerator, the modulus is 14 and we're going to take 14 and divide it by the modulus of the denominator which is 7, 14, divided by seven is a positive two. So our new modulus is two and that's multiplied by the quantity of. Now. The second thing that we need to do is to subtract our arguments. Our arguments are theta, that's the degrees. So in the numerator, the argument is 65 degrees. And we're gonna take that and subtract the argument of the denominator which is 35 degrees, 65 degrees minus 35 degrees is a positive 30 degrees. So inside of those parentheses, we have two multiplied by the quantity of. Now in the parentheses is the cosine of our new argument, 30 degrees plus I sign up 30 degrees. And that's simplified. And that's our answer. In polar form, we look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

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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 = 20(cos 75° + i sin 75°) z₂ = 4(cos 25° + i sin 25°)