If z is a non-real complex number, then the minimum value of I m z 5 I m z 5 is ( I m z = Imaginary part of z )

Step-by-step solution

Let, z = r ( cos θ + i sin θ ) z 5 = r 5 ( cos 5 θ + i sin 5 θ ) I m ( z 5 ) = r 5 sin ⁡ 5 θ and I m z 5 = r 5 sin 5 ⁡ θ I m ( z 5 ) I m z 5 = sin ⁡ 5 θ sin 5 ⁡ θ = A (Let) ⇒ d A d θ = 5 sin 5 ⁡ θ cos ⁡ 5 θ - 5 sin ⁡ 5 θ sin 4 ⁡ θ cos ⁡ θ sin 5 ⁡ θ 2 ⇒ d A d θ = 5 sin 4 ⁡ θ sin ⁡ θ cos ⁡ 5 θ - sin ⁡ 5 θ cos ⁡ θ sin 10 ⁡ θ ⇒ d A d θ = 5 sin 6 θ sin − 4 θ = 0 ⇒ θ = π 4 The minimum value of A will be at θ = π 4 . ⇒ sin ⁡ 5 π 4 sin ⁡ π 4 5 = - 1 2 1 2 5 = - 2 4 = - 4