But it's transcendental!

I will outline a proof, but there is a piece missing at the end. Let's start with the cyclotomic polynomial $\Phi_9(x)=x^6-x^3+1$ ; it follows that the minimal polynomial of $2\cos(\pi/9)$ is $x^3-3x+1$ . Using the double angle formulas twice, we see that $\cos(\pi/18)$ and $\sin(\pi/36)$ are roots of polynomials of degree 6 and 12 respectively (and these polynomials are even, so that $f(1)f(-1)=(f(1))^2$ ). It follows that $\sin(43\pi/36)=\sin(2015^o)$ is a root of a polynomial of degree 12 as well... but how can we show with elementary tools that this polynomial is irreducible? I can show it with Galois theory...