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:

http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem .

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 $e^{i \alpha}$

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