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

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 80° + i sin 80°)]³

Accepted Answer

Hello. Today we're going to be using Demo's theorem to simplify the following complex number. So there are a few things that Demo's theorem states. Let's suppose we have a complex number in polar form which can be written as Z is equal to R multiplied by cosine of data plus I multiplied by sign of the. Let's suppose we want to find the value of this complex number raised to a certain exponent N as long as N is a positive integer, then Z raced to the power of N can be rewritten as R raced 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 Demos theorem, we first need to identify our integer N, our value of R and the angle theta of the given complex number. So the complex number given to us is five multiplied by the quantity cosine of 75 plus I multiplied by sine of 75. And we want to simplify form of this complex number race to the power of four. Let's start by identifying the positive integer N. Now the positive integer N is the exponent that the complex number is raised to. And in this case, that exponent is going to be four. So N will equal to four. Next, we're going to identify the coefficient of R R is the coefficient in front of the complex number. And in this case, that value is going to be five. So R will equal to five. Next, we'll need to identify our angle theta and the is the angle within S sign and cosine which in this case will be 75 degrees. So theta is equal to 75. Now that we have the values N R and the, we can plug this into duals theorem in order to find a simplified form of the complex number. So using Demov theorem, we can write that Z raised to the power of four is equal to five raised to the power of four multiplied by the quantity cosine of N multiplied by theta, which is four multiplied by 75 plus I multiplied by sine of four multiplied by 75. We'll need to do a bit of simplification. Five raised to the power of four will give us the value of 625 and four multiplied by 75 will give us the value of 300. So we have cosine of 300 plus I multiplied by sine of 300. Next, we'll need to identify the values of cosine 300 sine of 300. And in order to do this, we can use a unit circle. So the first thing we'll need to do is we'll need to identify where the angle 300 lies on the unit circle. 300 lies in the fourth quadrant of the unit circle. And the terminal point at this angle is given to us as one half comma negative square root 3/2. And recall that any X value of the terminal point is defined as cosine and any Y value of the terminal point is defined as sign. So that means cosine of 300 is the X value of the terminal point which in this case is one half. So cosine of 300 can be rewritten as one half and sin of 300 can be rewritten as negative square root 3/2. So we have 625 multiplied by one half plus I multiplied by the negative square root 3/2. Next, we're going to want to distribute 625 into this complex number. 625 multiplied by a half will give us the value of 6 25/2 and multiplied by negative square root 3/2 will leave us with 125 multiplied by the square root three all over two multiplied by I. And this will be the simplified form of the complex number using demo's theorem. And with that being said, the answer to this problem is going to be D. 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. [2(cos 80° + i sin 80°)]³

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Trigonometric Functions

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