Question

Im[(1 − i)e^iθ]. Your answer should include the term θ.

Answer 1

e^iθ = cosθ + i sinθ

so,
(1-i)e^iθ = cosθ + i sinθ - i cosθ - i^2 sinθ = (sinθ+cosθ) + (sinθ-cosθ) i

Answer 2

or, since 1-i = √2 e^(i π/4), you have

(1-i)e^iθ = √2 e^(i π/4) e^iθ = √2 e^(θ+π/4)i