When does $a \cdot\sin(x) = \sin(a \cdot x)$? I am examining the expression $a \cdot \sin(x) =\sin(a \cdot x)$ where $a$ is a rational constant. Is there a way to determine which values of $x$ would be valid? Does it only hold true for certain values of $a$?

If $a$ is not $0$, $1$ or $-1$, $\sin(ax)/\sin(x)$ is a non-constant meromorphic function, so there will be at most a discrete set of solutions for $x$.  If $a = m/n$ with $m$ and $n$ relatively prime integers, writing $x = nt$ you want to solve $f(t) = n \sin(mt) - m \sin(nt) = 0$.  This is periodic with period $2 \pi$, and is $0$ at multiples of $\pi$.  The interesting question is whether there are other real solutions. It appears that there always are unless $m$ or $n$ is $1$.
WLOG assume $1 < m < n$.  Note that $f(k\pi/n) = n \sin(km\pi/n)$ for 
integers $k$.  The points $x_k = k m\pi/n$ for $k = 0, 1, \ldots,  n$ are separated by a distance $< \pi$, and since $x_{n} - x_0 = m \pi \ge 2\pi$
there must be at least one $x_j$ in the interval $(\pi, 2 \pi)$ where 
$\sin(x_j) < 0$, i.e. $f(x_j/m) < 0$ and at least one $x_k$ in the interval $(0, \pi)$ where $\sin(x_k) > 0$, i.e. $f(x_k/m) > 0$.  By the Intermediate Value Theorem, between $x_k/m$ and $x_j/m$ there is some $x$ with $f(x) = 0$.
