Could anyone please help me walk through the steps. I understand the underlying concepts through the use of deMoivre's Formula, and that it is to be written in the form of Euler's equation. I would like some guidance on the exact steps taken to get the result.
This is what I have:
$$w = \operatorname{cis}\left(\frac\pi9+\frac{2\pi k}{3}\right) \quad k=0, 1, 2$$
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Simon S
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What do you mean by $cis$ and $rcis$? – Jack D'Aurizio Jul 12 '15 at 12:40
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1@Jack, cis is $\cos\theta+i\sin\theta$, and rcis is $r(\cos\theta+i\sin\theta)$. – Gerry Myerson Jul 12 '15 at 12:49
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If you'd been solving $$w^3=\frac12\left(1-i\sqrt3\right),$$ then your solution would be correct. – Cameron Buie Jul 12 '15 at 13:01
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Write $w=r\operatorname{cis}\theta$. As $z=\operatorname{cis}\Bigl(\dfrac\pi6\Bigr)$, we have $$r=1,\quad 3\theta\equiv \frac\pi6\mod 2\pi,\quad\text{whence}\quad \theta\equiv \frac\pi{18}\mod \frac{2\pi}3$$ so that $$w=\operatorname{cis}\Bigl( \frac\pi{18}+\frac{2k\pi}3\Bigr),\quad k=0,1,2.$$
Bernard
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