Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: \(\cfrac{(1-i)}{(cos\frac{\pi}{3}+isin\frac{\pi}{3})}\)

Answer 1

Now, separating real and complex part , we get

\(\frac{1-\sqrt3}{2}\) = rcosθ……….eq.1

\(\frac{1-\sqrt3}{2}\) = rsinθ…………eq.2

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

2 = r²

Since r is always a positive no., therefore,

r = √2,

Hence its modulus is √2.

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

Therefore the θ lies in second quadrant.

As Tanθ = \(\frac{1+\sqrt3}{1-\sqrt3}\), therefore θ = 7π/12

Representing the complex no. in its polar form will be

Z = √2{cos(7π /12)+i sin(7π /12)}