Question

Find argument of 1/1-i

Answer 1

1/1-i=1+i/2
=1/2+i/2⇒pi/4 (in graph its an isoceles triangle 45 degree)

Answer 2

Answer:

The argument of z is $z=\frac{\pi}{4}$

Step-by-step explanation:

Given : $z=\frac{1}{1-i}$

To find : The argument?

Solution :

First we solve the given expression by rationalizing,

$z=\frac{1}{1-i}\times \frac{1+i}{1+i}$

$z=\frac{1+i}{1^2-i^2}$

$z=\frac{1+i}{1-(-1)}$

$z=\frac{1+i}{2}$

$z=\frac{1}{2}+\frac{i}{2}$

We know, $z=r\cos\theta+ir\sin\theta$

Where, $r=\sqrt{a^2+b^2} \\r=\sqrt{(\frac{1}{2})^2+(\frac{1}{2})^2} \\r=\sqrt{\frac{2}{4}} \\r=\sqrt{\frac{1}{2}}$

On comparing,

$r\cos\theta=\frac{1}{2}$

Put $r=\sqrt{\frac{1}{2}}$

$\frac{1}{\sqrt{2}}\cos\theta=\frac{1}{2}$

$\cos\theta=\frac{\sqrt{2}}{2}$

$\cos\theta=\cos(\frac{\pi}{4})$

$\theta=\frac{\pi}{4}$

On comparing,

$r\sin\theta=\frac{1}{2}$

Put $r=\sqrt{\frac{1}{2}}$

$\frac{1}{\sqrt{2}}\sin\theta=\frac{1}{2}$

$\sin\theta=\frac{\sqrt{2}}{2}$

$\sin\theta=\sin(\frac{\pi}{4})$

$\theta=\frac{\pi}{4}$

Both are positive so argument lies in first quadrant.

Therefore, The argument of z is $z=\frac{\pi}{4}$