Question

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

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

Answer 1

The correct option is C $\theta$

Given, $|z_1|=1$, arg $z=\theta \therefore z=e^{-i\theta }$
But $\overline{z}=\frac{1}{z}\therefore arg\left(\frac{1+z}{1+\frac{1}{z}}\right)=arg\left(z\right)=\theta$