Question

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

Answer 1

By using De Moivre's theorem:

If we have the complex number ⇒ z = a ( cos θ + i sin θ)
∴
`\sqrt[n]{z} = \sqrt[n]{a} \ (cos \ (\theta + 360K)/(n) + i \ sin \ (\theta +360k)/(n) )`
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 =
`\sqrt[4]{z} = \sqrt[4]{81} = 3`

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

The angle of the given complex number =
`(3 \pi)/(8)`
There is four roots and the angle between each root =
`(2 \pi)/(4) = (\pi)/(2)`
The angle of the first root =
`( (3 \pi)/(8) )/(4) = (3 \pi)/(32)`
The angle of the second root =
`(3\pi)/(32) + (\pi)/(2) = (19\pi)/(32)`
The angle of the third root =
`(19\pi)/(32) + (\pi)/(2) = (35\pi)/(32)`
The angle of the fourth root =
`(35\pi)/(32) + (\pi)/(2) = (51\pi)/(32)`

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

The first root =
`z_(1) = 3 ( cos \ (3\pi)/(32) + i \ sin \ (3\pi)/(32))`
The second root =
`z_(2) = 3 ( cos \ (19\pi)/(32) + i \ sin \ (19\pi)/(32))`

The third root =
`z_(3) = 3 ( cos \ (35\pi)/(32) + i \ sin \ (35\pi)/(32))`
The fourth root =
`z_(4) = 3 ( cos \ (51\pi)/(32) + i \ sin \ (51\pi)/(32))`