Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: (i²⁵)³

Answer 1

= i⁷⁵ = i⁴ⁿ⁺³ where n = 18

Since i⁴ⁿ⁺³ = -i

i⁷⁵ = -i

Let Z = - i = r(cosθ + i sinθ)

Now, separating real and complex part, we get

0 = rcosθ ……….eq.1

-1 = rsinθ …………eq.2

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

1 = r²

Since r is always a positive no., therefore,

r = 1,

Hence its modulus is 1.

Now , dividing eq.2 by eq.1, we get,

\(\frac{rsin\theta}{rcos\theta}=\frac{-1}{0}\)

tanθ = - ∞

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

therefore the θ lies in fourth quadrant.

Tanθ = - ∞, therefore θ = - π/2

Representing the complex no. in its polar form will be

Z = 1{cos(-π/2)+i sin(-π/2)}