Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$

Sorry for the previous spam. I shall prove for $\cos$, cosine of any rational numbers except for 0 cannot get rational numbers.

By using polynomial argument, we shall only have to prove for integers.

Suppose that $m\in\mathbb{N}$, $\cos(m)\in\mathbb{Q}$.

For any fixed prime number $p>m$, consider polynomial $x\in(0,m)$ $$f(x) = \frac{(x-m)^{2p}(m^2-(x-m)^2)^{p-1}}{(p-1)!}$$

And $$F(x) = \sum_{n = 0}^{2p-1} (-1)^{n+1}f^{2n}(x)$$ Which satisfies $$(F'(x)\sin(x)-F(x)\cos(x))' = F''(x)\sin(x) +F(x)\sin(x) = f(x)\sin(x)$$ since the other terms cancelled. $$\int_0^mf(x)\sin(x)dx = F'(m)\sin(m)-F(m)\cos(m)+F(0)$$ Consider $f$ is a polynomial of $(x-m)^2$, thus $F'(m) = 0$, and we can see that $$f(m-x) = x^{2p}(m^2-x^2)^{p-1}/(p-1)!$$ By computing, $p|f^{(l)}(m)$ for every $l$. That means $F(m)$ is a multiple of $p$ by definition of $F$, say $pM$.

If $\cos(m) = s/t$, then $$t\int_0^m f(x)\sin(x)dx = -spM+tN$$ is an integer, While $$f\le \frac{m^{4p-2}}{(p-1)!} $$ thus $$t\int_0^mf(x)\sin(x)dx\le t\frac{m^{4p-2}}{(p-1)!}\cdot m <1 $$ when $p$ is large enough. Contradiction of it is an integer.

As a result of this, $\sin$ should also satisfy this.

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.