Write the complex number 3 cos π 3 + i sin π 3 in Cartesian form.

Given Polar form is 3 cos π 3 + i sin π 3 We know that Polar form r cos θ + i sin θ with an equivalent Cartesian form is given by x = r cos θ and y = r sin θ So, x = 3 cos π 3 = 3 2 and y = 3 sin π 3 = 3 3 2 Hence, the required Cartesian form is 3 2 + 3 3 2 i

Write the complex number 5 cos 3 π 8 + i sin 3 π 8 in Cartesian form.

We have, 5 cos 3 π 8 + i sin 3 π 8 = 5 sin ⁡ π 2 − 3 π 8 + i cos ⁡ π 2 − 3 π 8 ∵ cos ⁡ ( x ) = sin ⁡ π 2 − x , sin ⁡ ( x ) = cos ⁡ π 2 − x = 5 sin ⁡ π 8 + i cos ⁡ π 8 = 5 sin ⁡ 22 . 5 ° + i cos ⁡ 22 . 5 ° = 5 2 − 2 2 + i 2 + 2 2 ∵ sin ⁡ 22 . 5 ° = 2 − 2 2 , cos ⁡ 22 . 5 ° = 2 + 2 2 = 5 0.38 + i 0.92 = 1 . 9 + i 4 . 6 Hence, the required Cartesian form is 1 . 9 + i 4 . 6 .

Write the complex number e i π 2 in Cartesian form.

We have, e i π 2 = 1 2 cos π + i sin π ∵ e i θ = cos ⁡ θ + i sin ⁡ θ = 1 2 - 1 + i × 0 = 1 2 × - 1 = - 1 2 Hence, the required Cartesian form is - 1 2 .

Write the complex number 3 e - i π 4 in Cartesian form.

We have, 3 e - i π 4 = 3 { cos ⁡ - π 4 + i sin ⁡ - π 4 } ∵ e i θ = cos ⁡ θ + i sin ⁡ θ = 3 { 2 2 + i - 2 2 } = 3 2 2 + i - 3 2 2 = 3 2 2 − i 3 2 2 Hence, the Cartesian form is 3 2 2 − i 3 2 2 .