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**Maxim** I'm trying to express cos(pi/7) with radicals - that is, basic operations (addition, division, etc) and square and cube roots (or even higher roots if necessary).

I got this far:

If $\displaystyle s = 84\sqrt{3} \cdot i - 28$ and $\displaystyle c = \sqrt[3]{s}$, then $\displaystyle x = \frac{2+c}{12} + \frac{7}{3c} = \cos(\pi/7)$

Although this seems a complex expression, the imaginary part of x is actually zero, so x is in fact a real number (obviously, since cos(pi/7) is real). But I can't extract the real part as a "really real" radical expression (without i, that is).