Question

Write in polar form of the following complex numbers

`2 + "i" 2sqrt(3)`

Answer 1

Let `2 + "i" 2sqrt(3)` = r(cos θ + i sin θ)

Equating real and imaginary parts

r cos θ = 2(+ve)

r sin θ = `2sqrt(3)` (+ve)

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

r² = 4 + 12 = 16

|z| = r = 4

Since cos θ and sin θ are positive ‘θ’ lies in 1^st quadrant.

cos θ = `1/2`

sin θ =  `sqrt(3)/2`

∴ θ = sin θ = `pi3`

or

θ = `tan^-1 |y/x|`

= `tan^-1 |(2sqrt(3))/2|`

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

∴ Argument = `2"k"pi + pi/3`

∴ Polar form is z = r(cos θ + i sin θ)

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