Question

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

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

Answer 1

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

∴ a = `1/2`, b = `sqrt(3)/2`, a, b > 0

∴ |z| = r

= `sqrt("a"^2 + "b"^2)`

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

= `sqrt(1/4 + 3/4)`

= 1

Here `(1/2, sqrt(3)/2)` lies in 1^st quadrant

amp (z) = θ = `tan^-1("b"/"a")`

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

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

= `pi/3`

∴ θ = 60° = `pi/3`

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

= 1(cos 60° + i sin 60°)

= `1(cos  pi/3 + "i" sin  pi/3)`