When does $a \cdot\sin(x) = \sin(a \cdot x)$?

Question

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 $a$?

Answer 1

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 $f(x) = 0$.

Answer 2

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.

Answer 3

If $a = 0, \pm 1$ there are infinitely many solutions. If $a$ is an integer, the solution is $\pi n, n \in \mathbb{Z}$. Otherwise, the solution is $2\pi dn, n \in \mathbb{Z}$, where $d$ is twice the denominator of the fraction.

Answer 4

There will be no explicit solutions in general for 'a' being Real except those special solutions in the answer above. You need to solve the equation numerically to arbitrary precision for x given 'a' using a computer or by hand if you like Newton's method. Since sine is periodic you will find that it has infinitely many solutions.

https://en.wikipedia.org/wiki/Newton%27s_method

Answer 5

This is not intended to really answer the question, but I think it is appropriate to add as an answer.

An implicit plot of the values for $a$ and $x$ which satisfy the given relation is shown below. Clearly, there are solutions for rational $a$ values, but a closed form solution for $x$ is likely difficult for arbitrary $a$, given the previous answers to this question.