Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2i

Answer 1

Let Z = 2i = r(cosθ + isinθ)

Now, separating real and complex part, we get

0 = rcosθ ……….eq.1

2 = 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,

\(\frac{rsin\theta}{rcos\theta}=\frac{2}{0}\)

Tanθ = ∞

Since cosθ = 0, sinθ = 1 and tanθ = ∞.

Therefore the θ lies in first quadrant.

tanθ = ∞, therefore θ = π/2

Representing the complex no. in its polar form will be

Z = 2{cos(π/2)+i sin(π/2)}