Simplifying complex radicals from trigonometric expression How do I evaluate $ sin(20)$ exactly? [in degrees]
I derived the relationship between $sin(x) $ and $sin(3x)$  where $x = sin(x)$ and $ y = sin(3x)$
http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3D+y%2C+solve+for+x
Now I am interested in subsituting $sin(60) $ and moving along but I am not sure which formula will result in me getting a real solution.
http://www.wolframalpha.com/input/?i=-4x%5E3+%2B+3x+%3D+%283%29%5E%281%2F2%29%2F2%2C+solve+for+x
I am curious how to de-nest this mess into something cleaner. I would really like to get rid of all the imaginary numbers but if that is not possible I still feel that this can definitely be de-nested into a simpler looking form even if the radical count does not go down. 
I have tried a couple attempts at substituting stuff back in but my answer seems to change whenever I move thing in and out of the cube root

You can't get an expression for this using real radicals: see http://en.wikipedia.org/wiki/Casus_irreducibilis
