​ Find the modulus of each of the following complex numbers and hence express each of them in polar form: √1+i/1-i

Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: \(\sqrt{\frac{1+i}{1-i}}\)

Answer 1

Now, separating real and complex part , we get

1/√2 = rcosθ……….eq.1

1/√2 = rsinθ…………eq.2

Squaring and adding eq.1 and eq.2, we get

1 = r²

Since r is always a positive no., therefore,

r = 1,

hence its modulus is 1.

now, dividing eq.2 by eq.1, we get,

\(\frac{rsin\theta}{rcos\theta}=\cfrac{\frac{i}{\sqrt2}}{\frac{i}{\sqrt2}}\)

tanθ = 1

Since, cosθ = 1/√2 sinθ = 1/√2 and tanθ =1. therefore the θ lies in first quadrant.

Tanθ = 1, therefore θ = π/4

Representing the complex no. in its polar form will be

Z = 1{cos(π/4)+i sin(π/4)}