Find the modulus of each of the following complex numbers and hence ex

1.	Find the modulus of each of the following complex numbers and hence express each of them in polar form: 2i
Answer» 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)}

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

Answer»

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)}