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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)

best_answer

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)

$\theta$

Posted by

Himanshu

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