What is the value of $\cos\left(\frac{2\pi}{7}\right)$? What is the value of $\cos\left(\frac{2\pi}{7}\right)$ ?
I don't know how to calculate it.

I suggest you have a look at http://mathworld.wolfram.com/TrigonometryAnglesPi7.html which clearly explains the problem.
As you will see, $\cos\left(\frac{2\pi}{7}\right)$ is the solution of $$8 x^3+4 x^2-4 x-1=0$$ Using Cardano, you will get $$\cos\left(\frac{2\pi}{7}\right)=\frac{1}{6} \left(-1+\frac{7^{2/3}}{\sqrt[3]{\frac{1}{2} \left(1+3 i
   \sqrt{3}\right)}}+\sqrt[3]{\frac{7}{2} \left(1+3 i \sqrt{3}\right)}\right)$$
If you want to approximate it, even very accurately, you could expand $\cos(x)$ as a Taylor series at $x=\frac{\pi}{3}$ which gives $$\cos(x)=\frac{1}{2}-\frac{1}{2} \sqrt{3} \left(x-\frac{\pi }{3}\right)-\frac{1}{4}
   \left(x-\frac{\pi }{3}\right)^2+\frac{\left(x-\frac{\pi }{3}\right)^3}{4
   \sqrt{3}}+\frac{1}{48} \left(x-\frac{\pi }{3}\right)^4+\cdots$$ 
Using the first term, you will get $0.6295570974$, and adding terms $0.6239620836$, $0.6234788344$, $0.6234892692$, $0.6234898099$, $0.6234898021$ for an exact value equal to $0.6234898019$.
I cannot resist to provide the approximation $\cos(x)=\frac{\pi^2-4x^2}{\pi^2+x^2}$ which would give an estimate of $\frac{33}{53}$.
