How to express z= 1/(1-i) in polar form?

May 2, 2017

Simplify into a + bi form first

Explanation:

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

Now find $r = | z |$.

$| z | = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{2}{4}} = \frac{\sqrt{2}}{2}$

In Quadrant 1, $\theta = \arctan \left(\frac{b}{a}\right) = \arctan \left(1\right) = \frac{\pi}{4}$.

Therefore, $z = \left(\frac{\sqrt{2}}{2}\right) \left(\cos \left(\frac{\pi}{4}\right) + i \sin \left(\frac{\pi}{4}\right)\right)$.