Take the three points $(10,52)$, $(20,38)$ and $(50,-53)$
How would you calculate the sine regression line of the form: $$f(x)=A\ \sin{\frac{x+B}{C}}$$
In other words how would you calculate the constants $A$, $B$ and $C$ ?
Using the first two points I got as far as: $52\sin{\frac{20+B}{C}}=38\sin{\frac{10+B}{C}}$
I can't see what the best approach is.
All tips appreciated.
Better write the relation as
$$px+q=\arcsin(ry).$$
Then if you hypothesize some value of $r$, you get a system of $3$ equations in two unknowns, with the compatibility condition
$$\begin{vmatrix}x_0&1&\arcsin(ry_0)\\x_1&1&\arcsin(ry_1)\\x_2&1&\arcsin(ry_2)\end{vmatrix}=0.$$
Hence you have reduced the problem to the search for the zeroes of a univariate function.
Beware anyway that $2k\pi+\arcsin(ry)$ and $(2k+1)\pi-\arcsin(ry)$ are also solutions...