Question

Write in polar form of the following complex numbers

– 2 – i2

Answer 1

– 2 – i2 = r(cos θ + i sin θ)

Let z = – 2 – i2 = r(cos θ + i sin θ)

Equating real and imaginary parts

r cos θ = – 2

r sin θ = – 2

r² cos²θ + r²  sin²θ = (– 2)² + (– 2)²

r² = 4 + 4 = 8

r² = 8

|z| = r = `2sqrt(2)`

cos θ = `(-2)/(2sqrt(2) = (-1)/sqrt(2)`

sin θ = `(-2)/(2sqrt(2)) = (-1)/sqrt(2)`

Since cos θ and sin θ both are in – ve so lies in III quadrant.

Argument = `2"k"pi - 3 pi/4`

As θ = `pi/4 - pi = - (3pi)/4`

∴ Polar from z = r(cos θ + i sin θ)

– 2 – i2 = `2sqrt(2) (cos(2"k"pi - (3pi)/4) + "i" sin(2"k"pi - 3/4)) "k" ∈ "z"`