Find the complex fourth roots of 81(cos(3pi/8) + i sin(3pi/8)) a. find the modulus for all of the fourth roots b. find the angle for each of the four roots c. find all of the fourth roots of this

By using De Moivre's theorem:

If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴
k= 0, 1 , 2, ..... , (n-1)

For The given complex number ⇒ z = 81(cos(3π/8) + i sin(3π/8))


Part (A) find the modulus for all of the fourth roots

∴ The modulus of the given complex number = l z l = 81

∴ The modulus of the fourth root =

Part (b) find the angle for each of the four roots

The angle of the given complex number =
There is four roots and the angle between each root =
The angle of the first root =
The angle of the second root =
The angle of the third root =
The angle of the  fourth root =

Part (C): find all of the fourth roots of this

The first root =
The second root =

The third root =
The fourth root =