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

Question

In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees.

The complex square roots of 25(cos 210° + i sin 210°)

Accepted Answer

Welcome back. Everyone. In this problem, we want to determine all the complex roots for the complex number. 36 multiplied by the cosine of 60 degrees plus I multiplied by the sign of 60 degrees. And we want to find its complex square roots writing them in the polar form using angles in degrees. That just means we're going to write them back in the same form. They already are for or answer choices. A is six multiplied by the cosine of 60 plus I sine 66. A cosine multiplied by the cosine of 210 plus I multiplied by sine 210. B is six multiplied by the cosine of 30 plus I multiplied by the sin of 36 multiplied by the cosine of 4 20 plus I sine 4 20 C is 36 multiplied by cost 30 plus I sine 36 multiplied by cost 2 10 plus I sine 2 and D is six multiplied by cost 30 plus I sine 36 multiplied by the cosine of 2 10 plus I sine 2 10. If you heard me say cost earlier, I was just referring to the co sign, I just had the chance to say it a little quicker just to get everything in right now. Let's give it a go. What do we already know? Well, we know that we want to find the complex square roots of our complex number and to find the complex square roots, recall the theorem for the nro and recall that, that theorem says that if we have a complex number that's in polar form, that is, it's written as R multiplied by the cosine of theta plus I multiplied by the sign of theta. Then to find the nth root of that complex number, then each of those numbers where K represents each case is going to be equal to the nth root of R. That is the nth root of the, the, the, the constant multiplied by the cosine of theta plus 360 degrees multiplied by K all divided by N plus I sine with the same angle theta plus 360 degrees multiplied by K all divided by N where K is equal to all the numbers up to N minus one. So what it's basically telling us is that for each route, this is how we can figure out what that route is going to be by putting our values into this formula. Now let's look at the equation that we already have the, the, the complex number that we already have. We know we're looking for the complex square roots. So already that tells us that N equals two because N represents the power then we know is the complex square roots of 36 multiplied by the cosine of 60 plus I sin 60. So 36 is our value for R and finally, 60 degrees is the angle that we're working with. So theta equals 60 degrees. So if we plug those values into our formula, then that means for each root of our complex number is going to be equal to the square root of all multiplied by the cosine of theater. Well, theta is 60 degrees. So that's the core set of 60 degrees plus 360 degrees multiplied by K all divided by N which we know it's two plus I multiply it by the sign of the same angle. Now, before we continue, I'm going to simplify this a little bit to make it easier when we start actually working them out because no, we can simplify the root of 36. Let me put it in a different color. Can simplify the square root of 36 which because that equals six and we can find we can simplify our angle for our cosine and sine no, 60 divided by two is 30 degrees and 360 degree multiplied by K, all divided by two will be 180 degrees multiplied by K. Likewise, that's the same thing that's going to happen. To, I multiplied by a sign. I'm just going to do one little thing here and it's going to help us further on. I'm going to separate that being multiplied by K. And yeah, while we're working, you're gonna see why I did that just to make things here flow a little faster. But don't worry, you'll, you'll be writing out and you'll be great in no time. So this formula is the formula that we're going to use to find the complex square roots of our complex number. So remember we said K is equal to all other values up to N minus one. So that means for both of our roots, we're going to substitute the values for K equals zero and K equals one. So let's do that. So let me start with K equals zero. OK? Now what I'm going to do, I'm going to just take my formula here. I, I don't think I want to write it out all the time. I'm going to put it right here and wherever I see K, I'm going to substitute for zero because that's what K is. OK. So it's going to be six multiplied by the cosine off 30 degrees plus 180 degrees multiplied by zero plus I multiplied by the sine of 30 degrees plus degrees plus multiplied by zero. Now, when we work that out, OK. First, inside the bracket, that will be six multiplied by the cosine of 30 degrees plus I multiplied by the sign of 30 degrees. And remember we want to leave it in polar form. That's pretty important. So I'm not gonna work it any further. I'm just going to leave the answer as six multiplied by the core sign of 30 plus I multiplied by the sign of degrees. So that's our first route for the next route. So let me get my other here and or when K equals one, just gonna do the same thing again. So I'm gonna take our formula, let me put it here. And now I'm going to replace my value of K with one. They always say work smarter, not harder, right? That's kind of what we're doing here. That's what I want you to do too and you're doing pretty great so far. Now, if we write that, oh, let me put that in red, we're going to have six multiplied by the cosine of 30 degrees plus 100 and 80 degrees, which is 210 degrees. Likewise, it will be the same for I multiplied by sine. So our complex roots for our equation, our complex square roots would have been six multiplied by the cosine of 30 plus I multiplied by the sine of multiplied by the cosine of 210 degrees plus I multiplied by the sine of 210 degrees. And when you look back in our answer choices, that would be answer choice. D thanks a lot for watching everyone. I hope this video helped.

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In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 25(cos 210° + i sin 210°)