Question

If z is a complex number of unit modulus and argument θ ,then arg(1+z/1+zbar) equals:

1) θ

2) pi- θ

3) - θ

4) 90- θ

Answer 1

Answer:

Step-by-step explanation:

4 one is the correct answer

Answer 2

Answer:

zz can be written as cosθ+isinθ⟹z¯=cosθ−isinθcos⁡θ+isin⁡θ⟹z¯=cos⁡θ−isin⁡θ

So,

1+z1+z¯=1+cosθ+isinθ1+cosθ−isinθ

1+z1+z¯=1+cos⁡θ+isin⁡θ1+cos⁡θ−isin⁡θ

=2cos2θ2+2icosθ2sinθ22cos2θ2−2icosθ2sinθ2

=2cos2⁡θ2+2icos⁡θ2sin⁡θ22cos2⁡θ2−2icos⁡θ2sin⁡θ2

=cosθ2+isinθ2cosθ2−isinθ2

=cos⁡θ2+isin⁡θ2cos⁡θ2−isin⁡θ2

=(cosθ2+isinθ2)2(cosθ2−isinθ2)(cosθ2+isinθ2)

=(cos⁡θ2+isin⁡θ2)2(cos⁡θ2−isin⁡θ2)(cos⁡θ2+isin⁡θ2)

assuming cosθ2≠0cos⁡θ2≠0 i.e., θ2≠(2n+1)π2θ2≠(2n+1)π2 i.e., θ≠(2n+1)πθ≠(2n+1)π where nn is any integer

as θ=(2n+1)π,1+cosθ±isinθ=0θ=(2n+1)π,1+cos⁡θ±isin⁡θ=0

1+z1+z¯=cosθ+isinθ using de Moivre's formula

1+z1+z¯=cos⁡θ+isin⁡θ using de Moivre's formula

⟹1+z1+z¯=z

⟹1+z1+z¯=z

⟹arg(1+z1+z¯)=arg(z)