TITLE: Is $\sin(1)$ algebraic over $\mathbb{Q}$?
QUESTION [9 upvotes]: Is $\sin(1)$ algebraic over $\mathbb{Q}$? 
At the moment I have no idea how to proceed. Could you tell me how to solve it?

REPLY [9 votes]: As explained in usersujo’s answer, the answer is YES if $1$ means $1$ degree.
The answer is NO, however, if $\theta=1$ means $1$ radian.
Because then $\sin(\theta)=\sqrt{1-\cos^2(\theta)}$ would be algebraic also, and hence
$e^{i\theta}=\cos(\theta)+i\sin(\theta)$ would be algebraic. But we know $e^{i\theta}$ is not algebraic
by the Lindemann–Weierstrass theorem