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?
Best Answer
memorize sin/cos for angles $0,{\pi \over 6},{\pi \over 4},{\pi \over 3},{\pi \over 2}$ and in your examination look at the unit circle to figure out what is going on