In Exercises 53–64, use DeMoivre's Theorem to find the indicat...

Question

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

Accepted Answer

Welcome back. Everyone. In this problem, we want to find or multiplied by the cosine of plus M multiplied by the sin of 42 degrees. All way to the fifth power. Using the theorem, we want to express our angle in rectangular form. And our answer choices are a is negative 512 multiplied by the square root of two plus 1024 I B is 1024 plus 512. IC is negative 512 multiplied by the square root of three minus 512 I and B is negative 1024 minus 512. I multiplied by the square root of three. Now, what do we already know? Well, we know that here we have a complex number that we're raising to a port. And when we do that, we can use the theorem recall that the theorem tells us that for a complex number written in the form are multiplied by the cosine of theta plus I multiplied by the sign of theater. If we raise that complex number to a power, then we can raise our constant R to that power. And we can multiply our angles by the value of that power. So raising Z to the N spore equals R to the N spore multiplied by the core sign of N theta plus I multiplied by the sign of N theater. Now, by looking at what we already have notice that we know our value of R equals four. We also know that we're raising our complex number to the fifth power. So N equals five and we know that our angle that we're using is 42 degrees. So we can substitute these values into the Morse theorem that forms our equation to help us to solve for our answer. So let's do just that. So raising our complex number to the fifth power means that we are going to write four to the fifth four multiplied by the cosine of N theta which is five multiplied by 42 degrees plus I multiplied by the sign of five, multiplied by 42 degrees. Oh, what do we know? Well, four raised to the fifth power is 1024. Also five multiplied by 42 is 210. So no, my answer becomes, why do I keep writing that? My answer becomes 1024 multiplied by the cosine of plus I multiplied by the sign of 210 and that's all degrees. So if we can figure out our values of the core sign of 210 degrees and the sign of 210 degrees, then we will be able to solve. Well, we can use our unit circle for that, right? Because recall that on the unit circle, the exact value of the coin of 210 degrees is a negative spirit of three divided by two. Also recall that the sign of 210 degrees is a negative half. So by using those values we can solve because no, our answer becomes 1024 multiplied by the core set of 210 degrees, which is a negative square of three divided by two plus I multiplied by the sign of 210 which is negative a half. No, if we expand our brackets, that will be 1024 multiplied by the square root. The negative ski of three divided by two, that's the negative 1024 divided by the square of three divided by two plus 1024 multiplied by negative or half multiplied by I. So actually, it would be minus divided by two. I, we can simplify that because we can factor two from which leaves us with 512. So our answer will be negative 512 multiplied by the square root of three minus 512. I. And if we go back up, that shows our answer choice is C but the correct answer is c thanks a lot for watching everyone. I hope this video helped.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 10° + i sin 10°)]³

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Conversion from Polar to Rectangular Form

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