Question

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

• A. $\frac{\pi }{4}$
• B. $\theta$
• C. $\frac{\pi }{2}-\theta$
• D. $2\theta$

Answer 1

The correct option is B $\theta$

$|z|=1\Rightarrow z\overline{z}=1\Rightarrow \overline{z}=\frac{1}{z}$
Now, $\frac{1+z}{1+\overline{z}}=\frac{1+z}{1+\frac{1}{z}}=z$
$\therefore \arg \left(\frac{1+z}{1+\overline{z}}\right)=\arg \left(z\right)$
and we are given that $\arg \left(z\right)=\theta$