You have not simplified anything. You have demonstrated half of DeMoivre's formula. Now move on to the other half. You should find two solutions separated by $\displaystyle \pi$.
What are $\displaystyle sin(\pi/6)$ and $\displaystyle cos(\pi/6)$? You should be able to give exact values. ($\displaystyle \pi/6$ radians is the same as 30degreees.)