Question

If 0 < arg (z) < π, then arg (z) – arg (–z) = ____________.

Answer 1

For 0 < arg z < π
Let z = r(cosθ, i sinθ)
i.e arg z = θ
Then –z = – r(cosθ + i sinθ)
= – r (+ cosθ + i (+ sinθ))

= (–1) re^iθ
= e^iπ re^iθ
= re^{i(θ + π)}

i.e arg (–z) = θ + π
⇒ arg z – arg(–z) = θ – θ – π
= – π