(A) Let $Z = \frac{1}{2} + i \frac{\sqrt{3}}{2}$.We can write $Z$ in polar form as $Z = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} = e^{i \frac{\pi}{3}

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$\operatorname{cis} \frac{\pi}{12}$

(A) Let $Z = \frac{1}{2} + i \frac{\sqrt{3}}{2}$.We can write $Z$ in polar form as $Z = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} = e^{i \frac{\pi}{3}}$.To find the fourth root,we calculate $Z^{1/4} = (e^{i \frac{\pi}{3}})^{1/4} = e^{i \frac{\pi}{12}}$.Using the definition $\operatorname{cis} \theta = \cos \theta + i \sin \theta = e^{i \theta}$,we get $e^{i \frac{\pi}{12}} = \operatorname{cis} \frac{\pi}{12}$.Thus,one of the fourth roots is $\operatorname{cis} \frac{\pi}{12}$.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers De Moivre's theorem and Roots of unity

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Which of the following is a fourth root of $\frac{1}{2} + \frac{i \sqrt{3}}{2}$?

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A
$\operatorname{cis} \frac{\pi}{12}$
B
$\operatorname{cis} \frac{\pi}{4}$
C
$\operatorname{cis} \frac{\pi}{6}$
D
$\operatorname{cis} \frac{\pi}{3}$

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1 Complex Numbers and Quadratic Equations: Complex Numbers

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