Consider the complex number $w=2\left( \cos \frac{3\pi }{4}+\mathrm{i}\sin \frac{3\pi }{4} \right){{

Question

Consider the complex number $w=2\left( \cos \frac{3\pi }{4}+\mathrm{i}\sin \frac{3\pi }{4} \right){{\left( \cos \frac{\pi }{6}-\mathrm{i}\sin \frac{\pi }{6} \right)}^{3}}$. Find the exact values of the modulus and argument of $w$. Hence, express $w$ in modulus-argument form. [3] Find the possible value(s) of ${{w}^{n}}$, when $n$ is a multiple of $4$. [2]

Accepted Answer

(i) $w=2e^{i\pi/4}$; (ii) $n=4k$, $k\in\mathbb Z$, and $w^n=\pm2^n$

2010 SAJC P2 Q3

2010 SAJC P2 Q3

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