Q&A - Ask Doubts and Get Answers
Q

Help me solve this - Algebra - JEE Main

If z is a complex number of unit modulus and argument \theta ,then arg  (\frac{1+z}{1+\bar{z}})  equals:

  • Option 1)

    \pi - \theta

  • Option 2)

    -\theta

  • Option 3)

    \dfrac{\pi}{2}-\theta

  • Option 4)

    \theta

 
Answers (2)
133 Views
N neha

As we have learned

Euler's Form of a Complex number -

z=re^{i\theta}

- wherein

r denotes modulus of z and \theta denotes argument of z.

 

 

Polar Form of a Complex Number -

z=r(cos\theta+isin\theta)

- wherein

r= modulus of z and \theta is the argument of z

 

 |z| = 1

Arg (z)= \theta

\Rightarrow z = e^{i\theta }= \cos \theta + i \sin \theta

So, \frac{1+z}{1+z}= \frac{1+\cos \theta +i \sin \theta }{1+\cos \theta -i\sin \theta }

\frac{2 \cos^2\theta h+2 i\sin \theta h\cos \theta /2}{2\cos ^{2}\theta h-2i\sin \theta h\cos \theta }

=\frac{\cos \theta h+i\sin \theta h}{\cos \theta h-i\sin \theta h}

=\frac{e^{i\theta h}}{e^{-i\theta h}}= e^{i\theta }

\left ( \frac{1+z}{1+\bar{z}} \right )= \theta

 

 

 

 

 

 

 


Option 1)

\pi - \theta

Option 2)

-\theta

Option 3)

\dfrac{\pi}{2}-\theta

Option 4)

\theta

Exams
Articles
Questions