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Question
Find principal argument of `(1 + i sqrt(3))^2`.
Solution
Given that: `(1 + i sqrt(3))^2 = 1 + i^2 . 3 + 2sqrt(3) i`
= `1 - 3 + 2sqrt(3)i`
= `-2 + 2sqrt(3)i`
`tan alpha = |(2sqrt(3))/2|` ......`[because tan alpha = |("Img"(z))/("Re"(z))|]`
⇒ `tan alpha = |- sqrt(3)| = sqrt(3)`
⇒ `tan alpha = tan pi/3`
∴ `alpha = pi/3`
Now Re(z) < 0 and image(z) > 0.
∴ arg(z) = `pi - alpha`
= `pi - pi/3`
= `(2pi)/3`
Hence, the principal arg = `(2pi)/3`.
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