Question

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

`(1 + 7"i")/(2 - "i")^2`

Answer 1

Let z = `(1 + 7"i")/(2 - "i")^2`

= `(1 + 7"i")/(4 - 4"i" + "i"^2)`

= `(1 + 7"i")/(4 - 4"i" - 1)`

= `(1 + 7"i")/(3 - 4"i")`

= `((1 + 7"i")(3 + 4"i"))/((3 - 4"i")(3 + 4"i"))`

= `(3 + 4"i" + 21"i" + 28"i"^2)/(9 - 16"i"^2)`

= `(25"i" + 3 + 28(-1))/(9 - 16(-1))`

= `(25"i" - 25)/25`

∴ z = – 1 + i

∴ a = – 1, b = 1

∴ |z| = r

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

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

= `sqrt(2)`

Here, (–1, 1) lies in 2^nd quadrant

θ = amp (z)

= `pi + tan^-1("b"/"a")`

= `pi + tan^-1(1/(-1))`

= π + tan^–1(–1)

= π – tan^–1(1)

= `pi - pi/4`

= `(3pi)/4`

∴ θ = 135° = `(3pi)/4`

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

= `sqrt(2) (cos 135^circ +  "i"  sin 135^circ)`

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

The exponential form of z = re^iθ = `sqrt(2)"e"^((3pi)/4"i"`