Can you define a sin function given two points on its graph?

Question

In other words I`m searching for a function of the form $f(x) = a*sin(bx)$ with real constants a,b. Can you determine (a,b) given two points $(x_1, y_1), (x_2, y_2)$ on its graph? (I think this is solvable because you have two variables of freedom in the function and two restraints yielding a single possibility.)
It leads to the following system of equations: $$ a*sin(b*x_1) = y_1 \\ a*sin(b*x_2) = y_2 $$ From the second I extract a (extracting b makes things worse it seems) : $ a = \frac{y_2}{sin(b*x_2)}$ Which when substituting into the first equation yields: $y_2 * \frac{sin(b*x_1)}{sin(b*x_2)} = y_1$ I am stuck here atm and cannot find a way to solve for b in this situation. (Is there a simplification for the division of sines in general?) Any help would be appreciated.

Answer 1

The periodicity of $\sin$ obstructs there being a unique solution to this in general. For example, if we're given $(0, 0), (\pi/2, 1)$, then $a = 1, b = 4n + 1$ works for all $n \in \mathbb{Z}^+$. Scaling can make this problem carry over to a large set of values. Note also that no solution exists in the special case of a point $(0, y)$ with $y \neq 0$, as well. To have hope of a unique solution, one needs to bring in extra constraints (such as that $a \sin(bx)$ be monotone on $(x_0, x_1)$ for instance, which then makes $\arcsin$ a helpful tool).