What is the characterization of $\omega$ and $\phi$ so that $\sin(\omega t_1+\phi)=\sin(\omega t_2+\phi)=\sin(\omega t_3+\phi)$ for three arbitrary $t_1<t_2<t_3$? Thanks!
I have tried to use that $\sin(A+B)=\sin(A)\cos(B)+\sin(B)\cos(A)$ in each of the three expressions, reducing the problem to two implicit equations in $\phi$, $\omega$. One of the resulting equations contains $\omega$ only, in terms $\sin(\omega tj)$ and $\cos(\omega tj)$ for $j=1,2,3$ and no $\phi$. So one can, in principle, solve for $\omega$ using this equation. The other equation contains a similar expression on ω and the ratio $\cos(\phi)/\sin(\phi)$. One can obtain $\phi$ from the second equation, given $\omega$. Yet the expressions don't suggest a clear characterization to me.
Obviously $\omega=0$ is a trivial case. If not, it must hold that either $$\omega t_i+\phi=\omega t_j+\phi+2k\pi\quad,\quad 1\le i\ne j\le 3$$(which doesn't hold) or $$\omega t_i+\phi=2k\pi +\pi-\omega t_j-\phi\quad,\quad 1\le i\ne j\le 3$$ which also doesn't hold. Therefore the case where $\omega=0$ is the only case