Question

Write in polar form of the following complex numbers

`("i" - 1)/(cos  pi/3 + "i" sin  pi/3)`

Answer 1

`("i" - 1)/(1/2 + "i"  sqrt(3)/2)`

= `(2("i" - 1))/(1 + "i"sqrt(3)) xx (1 - "i"sqrt(3))/(1 - "i"sqrt(3)`

= `((sqrt(3) - 1)/2) + "i" ((sqrt(3) + 1)/2)`

Let z = `((sqrt(3) - 1)/2) + "i" ((sqrt(3) + 1)/2)`

= r(cos θ + i sin θ)

Equating real and imaginary parts

r cos θ = `(sqrt(3) - 1)/2 (+  "ve")`

r sin θ = `(sqrt(3) + 1)/2 (+  "ve")`

r² cos²θ + r² sin²θ = `((sqrt(3) - 1)/2)^2 + ((sqrt(3) + 1)/2)^2`

r² = `8/4` = 2

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

Argument (or) Amplitude θ = `tan^-1 (y/x)`

= `tan^-1 ((sqrt(3) + 1)/(sqrt(3) - 1))`

= `tan^-1 ((1 + 1/sqrt(3))/(1 - 1/sqrt(3)))`

= `(5pi)/12`

∵ `(1 + 1/sqrt(3))/(1 - 1/sqrt(3)) = (1 + 1/sqrt(3))/(1 - 1 1/sqrt(3))`

= `(tan  pi/4 + tan  pi/6)/(1 - tan  pi/4  tan  pi/6)`

= `tan (pi/4 + pi/6)`

= `tan  (5pi)/12`

= `tan^-1 (tan  (5pi)/12)`

= `(5pi)/12`

Argument z = `2"k"pi + (5pi)/12`

∴ Polar form z = r(cos θ + i sin θ)

`((sqrt(3) - 1)/2) + "i" ((sqrt(3) + 1)/2) = sqrt(2) (cos(2"k"pi + (5pi)/12) + "i" sin(2"k"pi + (5pi)/12)), "k" ∈ "z"`