Algebraic set in $\Bbb R^2$

Question

Why $C=\{(x,\sin x): x\in\mathbb R\}\subset\mathbb R^2$ not is algebraic set?

i.e. $C$ are the zeros of $S\subset \mathbb R[X,Y]$.

Thank you, for any suggestion.

Answer 1

Hint: Consider the intersection of $C$ with ${\bf R}\times \{0\}$.