TITLE: Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$
QUESTION [17 upvotes]: Is there a way to show, that the only solution of 
$$\sin(x)=y$$ 
is $x=y=0$ with $x,y\in \mathbb{Q}$.  
I am seaching a way to prove it with the things you learn in linear algebra and analysis 1+2 (with the knowledge of a second semester student).

REPLY [8 votes]: One can hit it with a really heavy-duty theorem, the Lindemann-Weierstrass theorem. It is a consequence of this that if $x$ is non-zero algebraic, then $\sin x$ is transcendental.
Ivan Niven, in his book Rational and Irrational numbers, has a nice proof, "elementary" but not entirely simple, of the fact that the sine of a non-zero rational is irrational. That proof does not make any real use of linear algebra.