TITLE: Proof that cos(1) is transcendental?
QUESTION [15 upvotes]: So, I was playing around on Wolfram|Alpha (as we nerds like to do) and it said cos(1) was transcendental. Could someone provide me with the proof that cos(1) is transcendental?

REPLY [8 votes]: If $\cos1$ is algebraic, so is $\sin1=\sqrt{1-\cos^21}$. Thus $e^i=\cos1+i\sin1$ is algebraic. But, by Lindemann's theorem, $e^\alpha$ is transcendental whenever $\alpha$ is a nonzero algebraic number.