I want to prove (or disprove) that $(\mathbb{Q}\times\mathbb{Q})\cap S^1$ is dense in $S^1$, where $S^1$ is the unit circle in $\mathbb{R}^2$ centered at the origin, and we want to use the definition of dense subset as the following: for $A\subset B\subset X$, $A$ is dense in $B$ if for every open set $O\subset X$, $O\cap B\neq\varnothing\implies O\cap A\neq\varnothing$. I am also trying to use the fact that it is sufficient to show the above condition for every open ball in $S^1$ by using the Euclidean metric of $\mathbb{R}^2$. However, moving from 'grids' to a circle is nontrivial to me. For instance, to satisfy the condition $$(x,y)\in B_\epsilon((a,b))$$ where $(x,y)$ is the point we are looking for and $B_\epsilon((a,b))$ is a ball of radius $\epsilon$ centered at a point $(a,b)\in \mathbb{Q}\times\mathbb{Q} \cap S^1$, by direct substitution of the definition of $S^1$, it is sufficient to satisfy $$r^2-\epsilon^2/2<ax+by.$$ However, to show that there are such $(x,y)$ that satisfy this condition and $x^2+y^2=1$ simultaneously seems particularly nontrivial for me.
Would there be a better way to show that $(\mathbb{Q}\times\mathbb{Q})\cap S^1$ is dense in $S^1$?
