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Question
Express the following complex numbers in polar form and exponential form:
`(1 + 7"i")/(2 - "i")^2`
Solution
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 2nd 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 = reiθ = `sqrt(2)"e"^((3pi)/4"i"`
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