Proving the irrationality of the number $\cos\frac\pi9$ The question is if there exists a standard way of proving that 
$$ \cos{ π \over 9}$$ is an irrational number.

Observe that $\cos(\frac{\pi}{3}) = \dfrac{1}{2}$. Thus using the identity: $\cos (3x) = 4\cos^3 x - 3\cos x$, $\cos(\frac{\pi}{9})$ is a zero of the equation: $4x^3 - 3x = \dfrac{1}{2}$,and using the irrational root test, one concludes that this equation has no rational root, hence $\cos(\frac{\pi}{9})$ must be irrational.
