​ Find the modulus of each of the following complex numbers and hence express each of them in polar form: 1+3i/1-2i

Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: \(\frac{1+3i}{1-2i}\)

Answer 1

= i - 1

Let Z = 1 - i = r(cosθ + isinθ)

Now, separating real and complex part , we get

-1 = rcosθ ……….eq.1

1 = 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,

\(\frac{rsin\theta}{rcos\theta}=\frac{1}{-1}\)

Tanθ = -1

Since cosθ = -1/√2, sinθ = 1/√2 and tanθ = -1.

Therefore the θ lies in second quadrant.

Tanθ = -1 , therefore θ = 3π/4

Representing the complex no. in its polar form will be

Z = √2{cos(3π/4)+i sin(3π/4)}