Question

Express the following complex numbers in polar form and exponential form:

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

Answer 1

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

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

= `(1 + 3"i" + 2"i" + 6"i"^2)/(1 - 9"i"^2)`

= `(1 + 5"i" - 6)/(1 + 9)`    ...[∵ i² = – 1]

∴ z = `(-5 + 5"i")/10`

= `-1/2 + 1/2`i

This is of the form a + bi, where a = `-1/2`, b = `1/2`

∴ r = `sqrt("a"^2 + "b"^2)`

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

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

= `1/sqrt(2)`

Also, cos θ = `"a"/"r" = ((-1/2))/((1/sqrt(2))) = -1/sqrt(2)`

and sin θ = `"b"/"r" = ((1/2))/((1/sqrt(2))) = 1/sqrt(2)`

`∴ θ = (3pi)/4    ...[(because  cos  (3pi)/4 = cos (pi - pi/4) = -cos  pi/4 = -1/sqrt(2)),(and sin  (3pi)/4 = sin(pi - pi/4) = sin  pi/4 = 1/sqrt(2))]`

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

= `1/sqrt(2)(cos  (3pi)/4 + "i" sin  (3pi)/4)`

and the exponential form of z = re^iθ

= `1/sqrt(2)"e"^("i"((3pi)/4)`

= `1/sqrt(2)"e"^(((3pi)/4)"i"`