Question

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

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

Answer 1

The correct option is C $\theta$

$z\overline{z}=1$ [ Since, $|z|=1$ ]

$\overline{z}=\frac{1}{z}$ ---- ( 1 )

Now,

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

$\Rightarrow$ $arg\left(\frac{1+z}{1+\frac{1}{z}}\right)$

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

$\Rightarrow$ $arg\left(\frac{1+z}{\frac{1+z}{z}}\right)$

$\Rightarrow$ $arg\left(z\right)$

$\Rightarrow$ $\theta$