Question

If z is a complex number of unit modulus and argument $\theta $ ,then find the value of $\arg (\dfrac{{1 + z}}{{1 + \overline z }})$
A. \[\dfrac{\pi }{2} - \theta \]
B. \[\theta \]
C. \[\pi - \theta \]
D. \[ - \theta \]

Answer 1

Hint: z is a complex number of unit modulus means $\left| z \right| = 1$ and argument is $\theta $ means arg z = $\theta $. Also, $\overline z = \dfrac{1}{z}$.

Complete step-by-step answer:
Let’s simplify $\arg (\dfrac{{1 + z}}{{1 + \overline z }})$ to find its value by substituting $\overline z = \dfrac{1}{z}$. Thus, our equation will become,
\[\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \arg \left( {\dfrac{{1 + z}}{{1 + \dfrac{1}{z}}}} \right)\]
Now we will cross multiply the denominator of RHS
$\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \arg \left( {\dfrac{{1 + z}}{{\dfrac{{z + 1}}{z}}}} \right)$
Now we will simplify the RHS of the equation
$\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \arg \left( {(1 + z) \div \dfrac{{1 + z}}{z}} \right)$
Now we will change the sign of division from RHS to multiplication and reverse the fraction $\dfrac{{1 + z}}{z}$ to $\dfrac{z}{{1 + z}}$ as per the rule
$\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \arg \left( {(1 + z) \times \dfrac{z}{{1 + z}}} \right)$
Now we will eliminate 1+z from the RHS
\[\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \arg \left( z \right)\]
And we know that arg z = $\theta $ so our equation will become $\arg \left( {\dfrac{{1 + z}}{{1 + \overline z }}} \right) = \theta $
Hence, option b is the correct answer.

Note: Don’t forget that there is a bar sign with z in the denominator. Also, $z \times \overline z = z \times \dfrac{1}{z} = 1$.