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 $f(x) = 0$.

Not an answer but an observation.

If $(a,x)$ is a solution for the equation: $$a \sin(x) = \sin(ax)$$

then so does $(\pm a,\pm x)$ and $(\pm a^{-1}, \pm ax)$. Ignoring the trivial case $a = 0$ or $\pm 1$ and $x = 0$, we can concentrate on the case where $a > 1$ and $x > 0$. We can rewrite the equation as:

$$\frac{\sin x}{x} = \frac{\sin(ax)}{ax}\quad\quad\text{(assume a > 1)}\tag{*1}$$

Ploting $\frac{\sin x}{x}$ vs $x$, one immediately see that $(*1)$ doesn't have any solution for $|x| <$ some $x_c \sim 2.777068336$. $x_c$ is a root of the equation:

$$\frac{\sin x}{x} = \frac{1}{\sqrt{1+\beta^2}} \sim 0.128374554$$ where $\beta \sim 7.725251837$ itself is a root of another equation $\tan \beta = \beta$.

Update

For $a > 0$, rational, express $a$ as a fraction $\frac{m}{n}$ in its lowest term. Let $x = n \theta$ and $d = \max(m,n)$. We can rewrite the equation once again as:

$$\begin{align} & a \sin(x) = \sin(a x)\\ \iff & m \sin(n\theta) - n \sin(m\theta) = 0\\ \iff & \left(m U_{n-1}(\cos\theta) - n U_{m-1}(\cos\theta)\right)\sin\theta = 0 \end{align}$$ where $U_k(t)$ is the Chebyshev's polynomial of the $2^{nd}$ kind. Asides from the trivial solutions:

$$\sin\theta = 0 \iff x = 0, \pm n\pi, \pm 2n\pi, \ldots$$

$ \cos\theta $ will be a root of a polynomial of degree $d-1$: $G_{m,n}(t) = m U_{n-1}(t) - n U_{m-1}(t)$.

Notice $U_k(1) = k+1$, $U_k(-1) = (-1)^k(k+1)$ and in general $U_k(-x) = (-1)^kU_k(x)$. We see

• when $m$ and $n$ have same parity, i.e. both of them are odd.

  • $G_{m,n}(1) = G_{m,n}(-1) = 0$
  • $G_{m,n}(t) = (t^2-1) P_{m,n}(t^2)$ for some polynomial $P_{m,n}(\cdot)$ of degree $\frac{d-3}{2}$.

• When $m$ and $n$ have different parity, i.e. one of them is odd, the other is even.

  • $G_{m,n}(1) = 0$
  • $G_{m,n}(t) = (t-1) Q_{m,n}(t)$ for some polynomial $Q_{m,n}(\cdot)$ of degree $d-2$.

This means when

$$m, n \le \begin{cases}6,& m \not\equiv n \pmod{2}\\11,& m \equiv n \pmod{2}\end{cases}$$

The root $ \cos\theta $ of $G_{m,n}(t)$ can be expressed in terms of radicals.

The simplest example is $\frac{m}{n} = \frac23$, we have:

$$\begin{align} &Q_{2,3}(t) = 8 t + 2 \\ \implies & \cos\theta = t = -\frac14\\ \implies & x = n\theta \stackrel{\text{can be}}{=} \pm3\cos^{-1}(-\frac14) + 6K\pi,\text{ where } K \in \mathbb{Z} \end{align}$$

Another examples is $\frac{m}{n} = \frac35$, we have:

$$\begin{align} &P_{3,5}(t) = 48 t - 8 \\ \implies & \cos\theta = \sqrt{t} = \pm\frac{1}{\sqrt{6}}\\ \implies & x = n\theta \stackrel{\text{can be}}{=} \pm 5\cos^{-1}(\pm\frac{1}{\sqrt{6}}) + 10K\pi,\text{ where } K \in \mathbb{Z}\\ \iff & x = n\theta \stackrel{\text{can be}}{=} \pm 5\cos^{-1}(\frac{1}{\sqrt{6}}) + 5K'\pi,\text{ where } K' \in \mathbb{Z} \end{align}$$

Other non-trivial solutions for small $m,n$ can be derived in similar manner.