In Exercises 32–35, find all the complex roots. Write roots in re...

Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form.

The complex cube roots of −1

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine all of the complex cube roots of negative 343 express the roots of the complex number in rectangular form. All right, when we're looking for complex ruts of a number, we are going to use Demore's theorem. We recall from previous lessons that demos theorem gives us the equation. And we're going to state in radiance for this Z sub K equals the NTH root of R multiplied by the quantity of the cosine of in the numerator theta plus two pi K all divided by N plus I sign of the same thing theta plus two pi K all divided by N. And so we just need to determine a few things. We need to know our N, our K values, our R and our theta N, we can figure out pretty quickly we're taking the cr so N is going to be the crit which is the third rut. So N is three, K is closely related to NK is the series from zero all the way up to N minus one. Well, N is 33 minus one is two. So our K values are going to be 01 and two. We're going to run demos theorem three times using each of the K values to find our complex cris. Now, our, our value and our theta, we need to figure out a little bit. We were given negative 343. That's the same as negative 343 plus zero I, that would be in rectangular form in order to find R and theta, we need to be in polar form. So we recognize that rectangular form is A plus B I. So our A value here is negative 343. Then our B value here in front of the I is zero. And so we recall from previous lessons that R is equal to the square root of A squared plus B squared. So we can just plug in what we know A squared. That's negative 343 squared plus B squared. B is zero, that's zero squared. Well, zero squared is nothing. And we know that when we square negative 343 that's going to now be a positive. So this is the same as the square root of 343 squared, which is going to be 343. So our R value is 343. Now, for theta, we recall from previous lessons that the tangent of theta equals B divided by A. So B is zero divided by A is ne negative 343 but that's just zero. So the tangent of theta is zero. Where is this true? We recall from previous lessons and from the tables in the textbooks that this is true when theta equals either zero ra radians or pi radians. But let's think about where we are. If we draw a quick sketch of a complex plane, we have a vertical imaginary axis and a horizontal real axis. Our a value is where we're moving along the real axis. So we're moving to the left and we're going neither up nor down because our imaginary value is zero. So we're right on pie. So that's our negative part of the real axis. We're on pi so we can disregard the zero radiance. Our theta value is pi radiance. Now it's just plugging things in. So we've got our first K value is going to be zero. So Z sub zero equals the N thre N is three, the cubed thread of RR is 343 multiplied by the quantity of the cosine of in the numerator. Theta is pi plus two pi multiplied by KK here zero, divided by N and is three plus I sine of the same thing pi plus two pi multiplied by zero, all divided by three. Now simplifying the crit of 343 is seven multiplied by the quantity of the cosine of. I'm going to simplify what's in our cosine and sign very quickly if you need to pause the video and use a calculator or do by that by hand, that's fine. Pi plus two pi multiplied by zero, all divided by three is going to be pi divided by three plus I sign of the same thing pi divided by three. Now, our next step is going to be finding exact values. So it's still seven multiplied by the quantity of. But now the cosine of pi divided by three, we can look at the tables in the textbook and that's going to be one half plus the sine of pi divided by three is going to be square root three divided by two. Don't forget the I last step is just distributing the seven. So seven multiplied by one half is seven halves plus seven multiplied by square root three divided by two is seven square root three divided by two. And don't forget the I. So there's our Z sub zero, we got one down and two to go. All right, our Z sub one is our next K value. Still taking the cub rot of 343 multiplying that by the quantity of the cosine of still pi plus two pi multiplied by our new K value of one all divided by three plus I sign of same thing pi plus two pi multiplied by one, all divided by three. Simplifying crit of 343 remains seven multiplied by the quantity of the cosine of pi plus two, pi multiplied by one divided by three is pi plus I sign of the same thing P So this is equal to seven multiplied by the quantity of the exact value of the cosine of pi. From the tables in the textbook is negative one plus the exact value of the sine of pi is zero. So zero is now distributing the +77 multiplied by negative one is negative +77, multiplied by zero, I is nothing. So Rz sub one is equal to negative seven chew down one to go. All right, our final one, Z sub two or K values two still taking the cub Drut of 343 multiplying that by the quantity of the cosine of pi plus two pi. Now two pi is multiplied by our new K value of two all divided by three plus I sign of the same thing pi plus two pi multiplied by two, all divided by three. The cub of 343 is still seven multiplied by the quantity of the cosine of pi plus two pi multiplied by two all divided by three. Simplifies to be five pi divided by three plus I sign of the same thing. Five pi divided by three equals seven multiplied by the quantity of. So for five pi divided by three, we need to think about that. For a second five pi divided by three. If you look at the unit circle that's in quadrant four. So in quadrant four, our cosine is going to be positive, but our sine is going to be negative. The reference angle for five pi divided by three is pi divided by three, which is what we were using earlier. So we know that the cosine of pi divided by three is one half and the sine of pi divided by three is square root three divided by two. So for the exact values for the cosine of five pi divided by three in quadrant four, that's going to be a positive one half. And then the sine of five pi divided by three in quadrant four is going to be a minus square root three divided by two. Don't forget the I and this is equal to, we'll distribute the 77 multiplied by one half is seven halves and seven multiplied by negative square root three, divided by two is going to be minus seven square root three divided by two I. And that's our Z sub two and there are all of the complex Q RTs of negative 343. Well done. We'll catch you on the next one.

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In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of −1