Find the fifth root of 32(cos 280° + i sin 280°).

Solution:

Let Z = 32(cos 280° + i sin 280°)

Taking fifth root on both sides

Z^1/5 = (8)^1/5 [cos 280° + i sin 280° ]^1/5

Using De- Moivers theorem

If z = r [cos θ + i sinθ] then zⁿ = rⁿ[cos nθ + i sin nθ]

z^1/5 = (2⁵)^1/5 [cos(280°/5) + i sin (280°/5)]

z^1/5 = 2[cos 56°+ i sin 56°]

Find the fifth root of 32(cos 280° + i sin 280°).

Summary:

The fifth roots of 32(cos 280° + i sin 280°) is 2[cos 56°+ i sin 56°]