Question

Answer the following:

Convert the complex numbers in polar form and also in exponential form.

z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`

Answer 1

z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`

= `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i") xx (5 - sqrt(3)"i")/(5 - sqrt(3)"i")`

= `(10 - 2sqrt(3)"i" + 30sqrt(3)"i" - 18"i"^2)/(25 - 3"i"^2)`

= `(10 + 28sqrt(3)"i" + 18)/(25 + 3)`  ...[∵ i² = – 1]

= `(28 + 28sqrt(3)"i")/28`

∴ 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

If θ is the amplitude, then cos θ = `"a"/"r" = 1/2`

and sin θ = `"b"/"r" = sqrt(3)/2`

∴ θ = `pi/3   ...[because cos  pi/3 = 1/2 and sin  pi/3 = sqrt(3)/2]`

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

= `2(cos  pi/3 + "i" sin  pi/3)`

and the exponential form of z = re^iθ

= `2"e"^("i"(pi/3))`

= `2"e"^(pi/3"i")`