Find the value of the principal argument of the complex number $z = \frac{{(1 + i \sqrt{3})}^{2}}{{(1 - i)}^{3}}$

- \frac{17 π}{12}

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\frac{17 π}{2}

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\frac{7 π}{12}

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\frac{17 π}{12}

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Solution

The correct option is B $\frac{17 π}{12}$
$= 1 + i \sqrt{3} = 2 e^{i π / 3}$
$= 1 - i = \sqrt{2} e^{- i π / 4}$
$= z = \frac{(2 e^{i π / 3})^{2}}{(\sqrt{2} e^{- i π / 4})^{3}} = \frac{\sqrt{2} 4 e^{i 2 π / 3}}{2 \sqrt{2} e^{- i 3 π / 4}} = \sqrt{2} e^{i ⎛ ⎝ \frac{2 π}{3} + \frac{3 π}{4} ⎞ ⎠}$
$= \sqrt{2} e^{i 17 π / 12}$
$= \frac{17 π}{12}$

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