Today in math class we hade a discussion about linear regression, which is all about finding the best (though not perfect) linear equation that passes through a countable set of points, and people then started wondering about having an arbitary polynomial $p(x)$ reaching every point "perfectly", i.e. $p(x_k) = y_k$ for any list $((x_0,y_0), \cdots (x_n,y_n))$. This usually requires a polynomial of degree $n-1$, and is rather hard to find by hand.
But I remember reading this post a while ago, and started to figure, whether an equation of the form $$y= \alpha \sin(ax) + \beta \sin(bx + \xi), \alpha, \beta,a,b \in \mathbb{R} , \xi \in \mathbb{R} \backslash \mathbb{Q}$$ (i.e. $\xi$ being irrational) would be able to pass through any countable list of points in $\mathbb{R}^2$, my motivation is the following
Since the function is nonperiodic we can scale it down such that it will be able to pass through points close to one another
Since the individual $\sin$-functions are periodic, we have an extra $2 \pi n$ when we take the $\arcsin$, which will give us a countable amount of equations which we can use to solve using the conditions in any given list.
My motivation may be a tad vague, but I hope you'll atleast be able to get the idea of what I'm after, a trigonometric function that passes through any point of our choise.
Thanks in advance