How to construct a second independent equation

Question

Consider the following equation: $$Eq. 1\qquad f(t) = a(t) \sin(\omega t) + r(t)$$ where function $f(t)$ is known and functions $a(t)$ and $r(t)$ are unknowns. It is aimed to find these two unknown functions. Differentiating both side of above equation would result in following: $$Eq. 2\qquad f'(t) = a'(t)\sin(\omega t) + wa(t)\cos(\omega t) + r'(t)$$ Is it possible to consider $Eq. 2$ as an independent equation and consider both equations as a system to solve for unknown functions, either analytically or numerically?