Show that $\sin 10^\circ$ is irrational So, this is the problem I am working on. 
Show that $\sin 10^\circ$ is irrational. 
The solution to the problem is $$1/2 = \sin 30^\circ = 3 \sin 10^\circ - 4\sin^3 10^\circ .$$
Let $$x = 2\sin 10^\circ.$$ 
Then we have, $$x^3 - 3x + 1 = 0.$$ 
And, we have to work on this to find out the roots. But, what I don't understand is that why do I have to subtract $4\sin^3 10^\circ$ from $3\sin 10^\circ$. And,  how did they come up with $x^3 - 3x+1 = 0?$ I am confused. Can someone please explain this in details and is there any other way we can do this problem?

identity: $\sin(3a)=3\sin(a)-4\sin^3(a)$ By using this identity,
$$1/2 = \sin 30^\circ = 3 \sin 10^\circ - 4\sin^3 10^\circ$$
$$1=2\sin 30^\circ = 6 \sin 10^\circ - 8\sin^3 10^\circ$$ Then if you set $x=2\sin(10)$ you will get $$x^3 - 3x+1 = 0.$$
