In Exercises 65–68, find all the complex roots. Write roots in...

Question

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 27(cos 306° + i sin 306°)

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine all the complex ruts for the given complex number express the complex ruts in the polar form using angles in degrees. And what we're asked to find are the complex cube ruts of 729 multiplied by the quantity of cosine 225 degrees plus I sine 225 degrees. Our answer choices are answer choice. A nine multiplied by the quantity of cosine 75 degrees plus I sine 75 degrees and nine multiplied by the quantity of the cosine of 195 degrees plus I sine 195 degrees and nine multiplied by the quantity of cosine 315 degrees plus I sine 315 degrees. Answer choice B 81 multiplied by the quantity of cosine 75 degrees plus I sine 75 degrees. Nine multiplied by the cosine of the, the quantity of cosine 195 degrees plus I sine 195 degrees and nine multiplied by the quantity of cosine 315 degrees plus I sine 315 degrees answer choice. C 729 multiplied by the quantity of cosine 70 five degrees plus I sine 75 degrees nine, multiplied by the quantity of cosine 195 degrees plus I sine 195 degrees and nine multiplied by the quantity of cosine 315 degrees plus I sine 315 degrees. And answer choice D nine multiplied by the quantity of cosine 195 degrees plus I sine 195 degrees. Nine multiplied by the quantity of cosine 225 degrees plus I sine 225 degrees and nine multiplied by the quantity of cosine 315 degrees plus I sine 315 degrees. All right. So we have got this complex number given to us in polar form and we're asked to find the complex cube ruts. So recalling demo's theorem, we know that we can find the complex cube ruts of our number by determining our N our N here because it's the cubed rut is going to be three. And then we are going to take the nth rut. So the cubed rut or the third rut of our modulus, the modulus is the number in front of the parentheses. So here we're taking the cubed rut of 729 and we are going to divide the argument. The argument is our angle. So 225 degrees by that same end value by three. So both of these 225 degrees is, is are going to be divided by three and then that's going to get us our Z sub zero our first root. So if we do this work, the cube of 729 is nine and then that's multiplied by the quantity of the cosine of 225 divided by three degrees. That's 75. So we have the cosine of 75 degrees plus I sine of and again, it's 225 divided by three degrees. So again, 75 degrees. So there's our first one. Now we can find the next one, Z sub one. We're going to keep a lot of this the same. We bring it from the previous one, the Z sub zero. So it's going to be nine multiplied by the quantity of cosine of 75 degrees. But that cosine is not just of 75 degrees, it's of 75 degrees plus 360 degrees divided by our N which is three and then that'll be plus I multiplied by the sign of. And then very similarly, we're taking not just the sign of 75 degrees, it's the sign of 75 degrees plus 360 degrees divided by our N which is three. And so now we can do that simplification going off to the side for a moment, 360 degrees divided by three is degrees. So we're adding 75 plus 120 and that is 195. So we have for our Z sub one, it's nine multiplied by the quantity of the cosine of 120 plus 75 degrees is 195 degrees. Then plus I sine of again, 75 plus 120 is degrees. All right. And then for our Z sub two, we just continue with our Z sub one. We take nine multiplied by the cosine of 195 degrees. And again, we're going to add that same 360 degrees divided by three which we know now is 120 degrees and then bring down plus I sign of. And then again, it's taking 195 degrees plus the 120 degrees in 195 degrees plus 120 degrees is degrees. So it's nine multiplied by the quantity of the cosine of 315 degrees plus I sine of 315 degrees. And this makes sense for our cube ruts, we should have three answers and we do nine multiple it by the quantity of cosine 75 plus I sine 75 9 multiplied by the quantity of cosine 195 plus I sine and nine multiplied by the quantity of cosine 315 degrees plus I sine of 315 degrees. 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 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 27(cos 306° + i sin 306°)

Key Concepts

Complex Numbers in Polar Form

De Moivre's Theorem

Finding Complex Roots

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