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If z is a complex number of unit modulus and argument θ, then arg((1 + z)/(1 + z̄)) equals:

1. −θ
2. π/2 − θ
3. θ
4. π − θ

Correct answer: θ

Solution

The argument of the quotient of two complex numbers is the difference of their arguments. Since z has unit modulus, its conjugate z̄ has the same modulus, and the argument of (1 + z) and (1 + z̄) can be shown to differ by θ, leading to the conclusion that arg((1 + z)/(1 + z̄)) equals θ.

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