Express the complex number 1 - i cos π 3 + i sin π 3 in the form r ( cos θ + i sin θ ) .

Given, 1 - i cos π 3 + i sin π 3 = 1 - i 1 + i 3 2 = 2 1 - i 1 + i 3 By rationalising, 2 1 - i 1 + i 3 = 2 1 - i 1 + i 3 × 1 - i 3 1 - i 3 = 2 1 - i 1 - i 3 1 + 3 = 1 - i - i 3 - 3 2 1 - 3 - i 1 + 3 2 Let, z = 1 - 3 - i 1 + 3 2 Then, z = 1 - 3 2 + - 1 - 3 2 2 z = 1 + 3 - 2 3 + 1 + 3 + 2 3 2 = 2 Let θ be the argument of z and α be the acute angle, then tan α = Im ( z ) | Re ( z ) | tan α = 1 + 3 1 - 3 = tan 45 + tan 60 1 - tan 45 tan 60 ⇒ tan α = tan 45 + 60 ⇒ α = 105 Hence, the required argument is ⇒ θ = 7 π 12 Therefore, polar form is 2 cos 7 π 12 + i sin 7 π 12 .

Express sin π 5 + i 1 - cos π 5 in polar form.

Let, z = sin π 5 + i 1 - cos π 5 ⇒ | z | = sin π 5 2 + 1 - cos π 5 2 = sin 2 π 5 + 1 + cos 2 π 5 - 2 cos π 5 = 2 - 2 cos π 5 = 2 2 sin 2 π 10 = 2 sin π 10 Let β be an acute angle given by tan β = | lm ( z ) | | Re ( z ) | Then, tan β = 1 - cos π 5 sin π 5 | = 2 sin 2 π 10 2 sin π 10 cos π 10 = tan π 10 ⇒ β = π 10 Clearly, Z lies in the first quadrant. Therefore, arg z = π 10 Hence, the polar form of z is 2 sin π 10 cos θ + i sin θ where θ = π 10 .