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Question
Write in polar form of the following complex numbers
`2 + "i" 2sqrt(3)`
Solution
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 1st 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"`
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