# Why does LW theorem imply $\sin(\alpha)$ is transcendental when $\alpha \neq 0$ is algebraic?

I am looking for a more explicit explanation of how the Lindemann–Weierstrass theorem implies that $\sin(\alpha)$ is transcendental when $\alpha \neq 0$ is algebraic than is given here:

Thanks much.

$i \alpha$ is also algebraic. $$e^{i \alpha} = \cos \alpha + i \sin \alpha$$ If $\sin \alpha$ were algebraic, then so would be $\cos \alpha = \pm \sqrt {1 - \sin^2 \alpha}$ and finally so would be $e^{i \alpha}$