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 40° + i sin 40°)]³

Accepted Answer

Welcome back. Everyone in this problem, we want to find seven multiplied by the cosine of 100 and 10 degrees plus I multiplied by the sine of 110 degrees. Always to the third part using the M theorem. And we want to express our angle in rectangular form. That is the standard form for complex numbers for our answer choices. A is 343 multiplied by the skirt of three divided by two minus 343 divided by two. IB is 343 divided by two plus 343. I multiplied by the skirt of two, all divided by two C is 343 multiplied by the skirt of two, divided by two minus 343 divided by two, I and D is 343 divided by two minus 343. I multiplied by the skirt of three divided by two. Now, what do we already know? Well, we have a complex number that we're trying to raise to a poor and that's why we're going to use the More theorem. Recall that the marvel the tells us that if we have a complex number Z, which is written in the form R multiplied by the cosine of theta plus I multiplied by the sine of theta. Then if we raise that complex number to a power, we're going to raise our constant R to a power and we're going to multiply theta or angle by the value of that power. So it would become R raised to the power of N multiplied by the cosine of N theta plus I multiplied by the sine of N theta. So here already, just by looking at what we have notice that the value of R is seven and we're raising it to the third part. So R equals seven and N equals three which tells us that no, we can simplify, we can substitute rather our values into our expression to help us to write our answer. So let me put that down here. Let me see. So we're raising our complex number to the third part. So now we're going to rewrite it as seven raise to the third part multiplied by the cosine of our angle which is 110 multiplied by the value of that power. So the cosine of three multiplied by 110 plus I multiplied by the sine of three multiplied by 110 degrees. Now seven to the third power costs 343. And the co, the cosine of three multiplied by 100 and 10, 3 multiplied by 100 and 10 is 330 degrees. The same thing would apply for a sign. So now to move forward, all we need to do is to figure out our values of the cosine of 330 the sine of 330. Well, that means we can go back to our unit circle and recall that from our unit circle, the cosine of 330 degrees equals the square root of three divided by two. Also recall that the sine of 330 degrees equals negative a half. So now we can substitute those values into our equation because that means a complex number is now going to be 343 multiplied by the cosine of 330 which we said was the square root of three divided by two plus. I am multiplied by the sine of 330 which was negative a half by distributing our value of 343 343 multiplied by the square of three divided by two. Well, well, obviously be 343 multiplied by the square of three divided by two. I don't know why I said it first and distributing again 343 multiplied by negative half, multiplied by I becomes negative 343 divided by two. I, so that means our answer for this question is answer choice. A thanks for watching everyone. I hope this video helped. Have a great day.

In Exercises 53–64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 40° + i sin 40°)]³

Key Concepts

DeMoivre's Theorem

Polar and Rectangular Forms of Complex Numbers

Trigonometric Identities for Powers and Angles

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