Consider a sine function parameterized by amplitude, frequency, phase, and constant offset. In other words,
$h(x) = a\sin(fx + p) + c\tag*{}$
where $a$, $f$, $p$, and $o$ are arbitrary real numbers. Suppose we are also given a set of points $\{\,(x_1, y_1),$ $(x_2, y_2),$ $\ldots,$ $(x_n, y_n)\,\}.$ The only constraint on the points is that the $x_i$ are distinct. That is, $i\ne j \implies x_i \ne x_j.$ What is the maximum value $N$ for $n$ such that there always exist choices for $a$, $f$, $p$, and $c$ so that $h(x_i) = y_i$ for all integers $i \in [1, n]\text ?$
A degrees-of-freedom argument suggests that $N \le 4.$ Is $N=4\text ?$ If not, what is it?
An analogous problem for the exponential function suggests that degrees of freedom isn't always an achievable bound. For instance, if $g(x) = ae^{bx} + c,$ then a degree-of-freedom argument suggests that $g(x)$ can always choose $a,$ $b,$ and $c$ such that $g(x)$ interpolates three given points. However, the exponential function is monotonic, so we could never fit a point set that has a local extremum (say, $\{\,(1,0),\ (2,1),\ (3,0)\,\}).$ I haven't been able to find an analogous argument for the sine function, but I'm unconvinced that $N=4.$