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Transforming the complex number $z=-\sqrt{3}+3i$ into polar form will bring me to the problem to solve this two equations to find the angle $\phi$: $\cos{\phi}=\frac{\Re z}{|z|}$ and $\sin{\phi}=\frac{\Im z}{|z|}$.

For $z$ the solutions are $\cos{\phi}=-0,5$ and $\sin{\phi}=-0,5*\sqrt{3}$. Using Wolfram Alpha or my calculator I can get $\phi=\frac{2\pi}{3}$ as solution. But using a calculator is forbidden in my examination.

Do you know any (cool) ways to get the angle without any other help?

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    Draw a picture and measure the angle!2012-07-16
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    There are **very few** instances where the answer will be "nice." One needs to learn them. And you probably did at some stage. Maybe the reason you did not recognize it is writing $0.5$ instead of $\frac{1}{2}$. If you had been looking for $\cos \phi=-\frac{1}{2}$, $\sin\phi=\frac{\sqrt{3}}{2}$ you might have remembered.2012-07-16

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