Question

If z is a complex number of unit modulus and argument $\theta$, arg $\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

• A. - θ
• B. (π/2)- θ
• C. θ
• D. π- θ

Answer 1

The correct option is C

θ

We know that $|z{|}^2$ = z $\overline{z}$

Given |z| = 1 $\Rightarrow$ z $\overline{z}$ = 1

$\overline{z}$ = $\frac{1}{z}$

$\because arg\left(\frac{1+z}{1+\overline{z}}\right)$ = $arg\left(\frac{1+z}{1+\frac{1}{z}}\right)$

=$arg\left[\frac{1+z}{\left(\frac{1+z}{z}\right)}\right]$

= arg Z = $\theta$