Need help solving a trigonometric equation I am preparing for finals and there is one exercise in my book that i don`t know how to solve.
$$\frac{\sin a}{\sin \frac{a}{b}}=b$$
I just need to solve this for b. I tried wolfram alpha but it does not seem to work.
Any help would be appreciated. Thank you in advance.
 A: By taking $f(x)=\frac{\sin x}{x}$ your equation can be written as:
$$ f(a) = f\left(\frac{a}{b}\right) $$
so $b=\pm 1$ is always a solution ($f$ is an even function), and there may be other solutions (a finite amount, if $a\not\in\pi\mathbb{Z}$) if $a$ is sufficiently far from the origin, by just considering the behaviour of the graph of $f(x)$. For instance, if $a=3$ there are $14$ solutions:
$\hspace{2cm}$
In general, the number of solutions is bounded by the ratio between a fixed constant and the distance of $a$ from $\pi\mathbb{Z}$, since $f(x)$ is a Lipschitz-continuous function with bounded derivative.
A: On the (admittedly debatable) assumption that the question means

Find all real $b$ such that  $\frac{\sin a}{\sin \frac{a}{b}} = b$ for all real $a \neq 0$,

there are precisely two solutions: $b = \pm1$.
First, rewrite the condition
$$
\sin a = b\sin \tfrac{a}{b}\quad\text{for all real $a \neq 0$,}
$$
and note that $b \neq 0$ (since the left side is not identically zero). The left side is periodic in $a$ with smallest positive period $2\pi$; the right side is periodic in $a$ with smallest positive period $2\pi|b|$. It follows that $|b| = 1$.
