Find the fifth roots of the complex number:

Question

Find the fifth roots of the complex number:
cos(5pi/4)+isin(5pi/4)

Answer 1

Using DeMoivre:

(cos 5π/4 + i sin 5π/4)^1/5

1. cos π/4 + i sin π/4

2. cos 7π/20 + i sin 7π/20

3. cos 9π/20 + i sin 9π/20

4. cos 11π/20 + i sin 11π/20

5. cos 13π/20 + i sin 13π/20

Answer 2

Express the number in polar form, take the fifth root of the polar form, and convert back to rectangular form.

 

z = cos(5π/4) + i sin(5π/4) = re^iφ

 

r = √[cos²(5π/4) + sin²(5π/4)] = √1 = 1

 

φ = 5π/4

 

z = 1e^i5π/4 = e^i5π/4

 

z^1/5 = (e^i5π/4)^1/5 = e^iπ/4 = cos(π/4) + i sin(π/4)

 

[This is the principal branch only.]