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Question
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) - "i"`
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Solution
Let z = `sqrt(3) - "i"`
∴ a = `sqrt(3)`, b = – 1, i.e. a > 0, b < 0
∴ |z| = `sqrt("a"^2 + "b"^2)`
= `sqrt((sqrt(3))^2 + (-1)^2`
= `sqrt(3 + 1)`
= `sqrt(4)`
= 2
Here, `(sqrt(3), -1)` lies in 4th quadrant.
∴ amp (z) = `tan^-1("b"/"a") + 2pi`
= `tan^-1((-1)/sqrt(3)) + 2pi`
= `-tan^-1(1/sqrt(3)) + 2pi`
= `(-pi)/6 + 2pi`
= `(11pi)/6`
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