Discussion
The imaginary part of \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\) is
Answer: D
The imaginary part is \(\frac{-\sqrt{3}}{2}\).
Multiply numerator and denominator by the conjugate of the denominator, \(1 - i\sqrt{3}\):
\[ \begin{aligned} \frac{(1 - i\sqrt{3})^{2}}{1 + 3} &= \frac{1 - 2i\sqrt{3} - 3}{4} \\ &= \frac{-2 - 2i\sqrt{3}}{4} \\ &= -\frac{1}{2} - \frac{\sqrt{3}}{2}i \end{aligned} \]
The imaginary part is the coefficient of \(i\), \(-\frac{\sqrt{3}}{2}\). \(-\frac{1}{2}\) is the real part, and \(\frac{\sqrt{3}}{2}\) has the wrong sign.
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