Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: -√3 - i

Answer 1

Let Z = - i - √3 = r(cosθ + i sinθ)

Now, separating real and complex part, we get

-√3 = rcosθ ……….eq.1

-1 = rsinθ …………eq.2

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

4 = 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 third quadrant.

tanθ = 1/√3, therefore θ = -5π/6

 Representing the complex no. in its polar form will be

Z = 2{cos(-5π/6)+i sin(-5π/6)}