# Proof that cos(1) is transcendental?

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?

• I believe it comes from the fact that $\cos(1) = (e^i+e^{-i})/2$, so you're looking for the proof that $e^i$ is transcendental. Feb 16, 2014 at 0:19
• See the Lindemann-Weierstrass theorem, in conjunction with Euler's formula. Feb 16, 2014 at 0:22

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.
Since $i$ is a non zero algebraic number then $\lbrace i , 2i ,0\rbrace$ is a set of distinct algebraic numbers in $\mathbb{C}$. By the Lindemann-Weierstrass theorem for any non-zero algebraic numbers $\beta_{1},\beta_{2},\beta_{3}$ we have $\beta_1 e^{i} + \beta_2 e^{2i}+\beta_3 e^{0}\ne 0$. To obtain a contradiction assume that $cos(1)$ is algebraic. Recall that $cos(1) =\dfrac{e^{i}+e^{-i}}{2}.$ Then $2cos(1) .e^{i}-e^{2i}-1=0$ and hence $(2cos(1) )e^{i}-1.e^{2i}-1.e^{0}=0$. But this is a contradiction with the Lindemann-Weierstrass theorem. Therefore $cos(1)$ is transcendental. $\square$