Question

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

Options

• π

• –π

• `-π/2`

• `π/2`

Answer 1

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

Explanation:

Given Arg(z) < 0

i.e. Arg(z) = –θ

Then = r[(cos(–θ) + isin(–θ)]

z = r[cosθ – isinθ]

–z = –r[cosθ – isinθ]

–z = r[–cosθ + isinθ]

–z = r[cos(π – θ) + isin(π – θ)]

∴ Arg(–z) = π – θ

∴ Arg(–z) – Arg(z) = π – θ – (–θ)

= π – θ + θ

= π