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Complex Numbers : Polar Form and DeMoivre's Theorem

6. Please explain step by step
Apply DeMoivre's Theorem to find (-1+i)^6
* change to polar form first
You will recognize the angle,
so put in the correct value of sine
and cosine to reduce back to simple complex form

7. Find the fourth roots of 16(cos pi/4 + i sine pi/4) ; n=4
Please explain in detail

9 solve for x: x^3 = 1-i
Please explian in detail

4. Convert to polar form: -4+ 3i

3. Convert to polar form 1-i

1. graph the number 4(3-2i)(-2+i)
Please explain in complete detail

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6. (-1 +I)^6

Solution:

First express (-1+I) in the modulus and amplitude form.
Let -1+I= r (cos(theta)+isin(theta)). Equating real and imaginary parts, we get

Rcos(theta)= -1 , rsin(theta) = 1

Therefore, r = sqrt((-1)^2+ (1)^2)

R= sqrt(1+1) => sqrt(2)

Hence, cos(theta) = -1/sqrt(2)

Sin(theta)= 1/sqrt(2)

 theta = 45 degrees.

-1 +I = sqrt(2) { cos(-45) + isin(45)} [ cos(-theta) = cos(theta)]

So, -1+I = sqrt(2) { cos(45)+isin(45)}

Here we need to find (-1+I)^6

(-1+I)^6 = [sqrt(2)]^6 {cos(45) +isin(45)}^6 [ 45 degrees = pi/4]

(-1+I)^6 = 8 { cos6(pi/4) +isin6(pi/4)}

(-1+I)^6 = 8 { cos(3pi/2) +isin(3pi/2)}

That's it.

6. Find the fourth roots of 16(cos pie/4 + i sine pie/4) ; n=4
Please explain in detail

General ...

Solution Summary

Complex Numbers, Polar Form and DeMoivre's Theorem are investigated. The change to polar form which will recognize the angles are determined.

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