Question

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

(1 − i)⁵ 

Answer 1

Let z = 1 − i

This is of the form a + bi, where a = 1, b = −1

∴ r = `sqrt("a"^2 + "b"^2)`

= `sqrt(1^2 + (-1)^2)`

= `sqrt(1 + 1)`

= `sqrt(2)`

Also, cos θ = `"a"/"r" = 1/sqrt(2)` and sin θ = `"b"/"r" = (-1)/sqrt(2)`

∴ θ lies in the fourth quadrant.

∴ tan θ = `sintheta/costheta = (((-1)/sqrt(2)))/((1/sqrt(2))` = −1

= `- tan  pi/4`

= `tan(2pi - pi/4)`

∴ θ = `(7pi)/4 = tan  (7pi)/4`

∴ z = r(cos θ + i  sin θ) = `sqrt(2)(cos  (7pi)/4 + "i" sin  (7pi)/4)`

∴ z⁵ = (1 − i)⁵ = `[sqrt(2)(cos  (7pi)/4 + "i" sin  (7pi)/4)]^5`

= `(sqrt(2))^5[cos(2pi - pi/4) + "i" sin(2pi - pi/4)]^5`

= `4sqrt(2)(cos  pi/4 - "i" sin  pi/4)^5`

= `4sqrt(2)(cos  (5pi)/4 - "i" sin  (5pi)/4)`  ...[∵ (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ]

= `4sqrt(2)[cos(pi + pi/4) - "i" sin(pi + pi/4)]`

= `4sqrt(2)(-cos  pi/4 + "i" sin  pi/4)`

= `4sqrt(2)(- 1/sqrt(2) + "i" xx 1/sqrt(2))`

= – 4 + 4i