Question

Express the following complex numbers in polar form and exponential form: 

`-1 + sqrt(3)"i"`

Answer 1

Let z = `-1 + sqrt(3)"i"`

This is of the form a + bi, where a = – 1, b = `sqrt(3)`

∴ r = `sqrt("a"^2 + "b"^2)`

= `sqrt((-1)^2 + (sqrt(3))^2`

= `sqrt(1 + 3)`

= 2

Also, cos θ = `"a"/"r" = (-1)/2`

and sin θ = `"b"/"r" = sqrt(3)/2`

`∴ θ = 120°    ...[(cos 120° = cos(180° –  60°) = – cos 60°  = -1/2 and), (sin 120° = sin(180° –  60°) = sin 60° = sqrt(3)/2)]`

∴ the polar form of z = r (cos θ + i sin θ)

= 2(cos 120° + i sin 120°)

= `2(cos  (2pi)/3 + "i"sin  (2pi)/3)`

and the exponential form of z = re^iθ = `2"e"^(((2pi)/3)"i")`