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. [1/2 (cos π/12 + i sin π/12)]⁶

Accepted Answer

Hello. Today we're going to be using Demo's theorem to simplify the given complex number. So there are a few things that Demos theorem states. First, let's suppose that we have a complex number in polar form, which can be written as Z is equal to R multiplied by cosine of theta plus I multiplied by sine of theta. If we want to raise this complex number to a certain degree N and N is a positive integer, then Demo's theorem allows us to rewrite the number Z raised to the power of N as R raised to the power of N multiplied by the quantity cosine of N multiplied by theta plus I multiplied by sine of N multiplied by theta. So in order to use Demo's theorem, we're going to need to identify our R value, the end value and our angle data. So let's take a look at the given complex number. So we are given one third multiplied by the quantity cosine three pi over 10 plus I multiplied by sine of three pi over 10. And this entire complex number is raised in the power of five. So let's start by identifying our end value Now again, the end value is the positive integer that the complex number is raised to. And in this case, that number is going to be five. So N will equal to five ne. Next, let's identify our A R value and R is the coefficient in front of the complex number which in this case is one third. So R will equal to one third. Finally, let's identify our angle theta and theta is the angle inside of sine and cosine, which is going to be three pi over 10. So theta is equal to three pi over 10. Now that we have these three values, if we use Demo's theorem, demo's theorem allows us to state that Z race to the power of five is equal to one third race to the power of five multiplied by the quantity cosine of N multiplied by theta, which is five, multiplied by three pi over 10 plus I multiplied by sine of five multiplied by three pi over 10. So what we'll need to do is we'll need to perform a little bit of simplification. One third race to the power of five will give us the value of one over 243 and inside the cosine and sine values five multiplied by three pi over 10 will give us the value of three pi over two. So we have cosine three pi over two plus I multiplied by sine of three pi over two. Now what we'll need to do is we'll need to evaluate cosine three pi over two and sign a three pi over two. And in order to do that, we're going to need to use a unit circle. So the first thing we'll need to do is we'll need to identify where three pi over two lies on the unit circle. And three pi over two will be the 270 degree mark of the unit circle. And three pi over two has a terminal point of zero comma negative one. Now recall that the X values of any terminal point are defined as cosine and the Y values of any terminal point are defined as sine. So that means we can rewrite cosine of three pi over two as the X value of the terminal point which is zero plus I multiplied by sine of three pi over two, which is the Y value of the terminal point which will be negative one finally zero plus I multiplied by negative one will leave us with the value of negative I. So we have one over 243 multiplied by negative I and multiplying these two values together will leave us with negative one over 243 multiplied by I. So using Demo's theorem, Z race of the power of five is equal to negative one over 243 multiplied by I. And with that being said, the answer to this problem is going to be b so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [1/2 (cos π/12 + i sin π/12)]⁶

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Trigonometric Identities for Cosine and Sine

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