​ Find the modulus of each of the following complex numbers and hence express each of them in polar form: 5- i/2- 3i

Question

Find the modulus of each of the following complex numbers and hence express each of them in polar form: \(\frac{5-i}{2-3i}\)

Answer 1

= i + 1

Let Z = 1 + i = r(cosθ + i sinθ)

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,

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

Tanθ = 1

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

Therefore the θ lies in first quadrant.

Tanθ = 1, therefore θ = π/4

Representing the complex no. in its polar form will be

Z = √2{cos(π/4)+i sin(π/4)}