Let r and θ respectively be the modulus and amplitude of the complex number z = 2 - i 2 tan 5 π 8 , then ( r , θ ) is equal to

Step-by-step solution

Given: z = 2 - i 2 tan 5 π 8 ⇒ z = 2 1 - i tan 5 π 8 ⇒ z = 2 1 - i sin 5 π 8 cos 5 π 8 ⇒ z = 2 cos 5 π 8 cos 5 π 8 - i sin 5 π 8 ⇒ z = 2 cos π - 3 π 8 cos π - 3 π 8 - i sin π - 3 π 8 ⇒ z = 2 - cos 3 π 8 - cos 3 π 8 - i sin 3 π 8 ⇒ z = 2 cos 3 π 8 cos 3 π 8 + i sin 3 π 8 ⇒ z = 2 sec 3 π 8 e i · 3 π 8 Now on comparing with z = z e i θ we get, ⇒ θ = 3 π 8 , r = 2 sec 3 π 8