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,

tanθ = 1

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

Therefore the θ lies in third 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)}