# 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

https://brainmass.com/math/complex-analysis/complex-numbers-polar-form-demoivres-theorem-70522

#### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

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)

ïƒ° 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.