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