Method to find $\sin (2\pi/7)$ I just thought a way to find $\sin\frac{2π}{7}$. 
Considering the equation $x^7=1$
$⇒(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$
$⇒(x-1)[(x+\frac1 x)^3+(x+\frac1 x)^2-2(x+\frac1 x)-1]=0$
We can then get the 7 solutions of x, but the steps will be very complicated, especially when solving cubic equation, and expressing x as a+bi. The imaginary part of the second root of x will be $\sin\frac{2π}{7}$. 
Besides this troublesome way, are any other approach? Thank you.

Using this or Point#$24$ of this ,
$\sin 7x=7s-56s^3+112s^5-64s^7$ where $s=\sin x$
If $\sin 7x=0,7x=n\pi$ where $n$ is any integer. 
So, $x=\frac{n\pi}7$ where $n=0,1,2,3,4,5,6$
So, the roots of $7s-56s^3+112s^5-64s^7=0--->(1)$ are $\sin\frac{n\pi}7$ where $n=0,1,2,\cdots 5,6$
So, the roots of $64s^6-112s^4+56s^2-7=0--->(2)$ are $\sin\frac{n\pi}7$ where $n=1,2,\cdots 5,6$
So, the roots of $64t^3-112t^2+56t-7=0 --->(3)$ are $\sin^2\frac{n\pi}7$ where $n=1,2,4$ or $3,5,6$ as $\sin \frac{(7-r)\pi}7=\sin(\pi-\frac{r\pi}7)=\sin\frac{r\pi}7$
If we choose $n=1,2,4$ observe that $\sin^2\frac{4\pi}7-\sin^2\frac{2\pi}7=2\sin\frac{\pi}7\cos\frac{3\pi}7>0$ (Using $\sin^2A-\sin^2B=\sin(A+B)\sin(A-B)$)
Similarly, $\sin^2\frac{2\pi}7-\sin^2\frac{\pi}7>0$
So, $\sin^2\frac{4\pi}7>\sin^2\frac{2\pi}7>\sin^2\frac{\pi}7$
Using Cardano's method, we can solve the Cubic equation $(3)$
