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

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

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Answer 1 · 2017-05-02T12:34:03

Simplify into a + bi form first

Explanation:

$z = \frac{1}{1 - i} = \frac{1}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{1 + i}{2} = \frac{1}{2} + \frac{i}{2}$ $z = 1/(1-i) = 1/(1-i)*(1+i)/(1+i) = (1+i)/2 = 1/2 + i/2$

Now find $r = | z |$ $r = | z |$ .

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

In Quadrant 1, $θ = arctan (\frac{b}{a}) = arctan (1) = \frac{π}{4}$ $theta = arctan(b/a) = arctan(1) = pi/4$ .

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

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

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