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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.

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    $\begingroup$ It doesn't seem to be a usual trigonometric equation...and neither an easy one. Are you sure that's the expression? Aren't there any conditions on the angles $\;a,b\;$ ? $\endgroup$ – DonAntonio Jun 26 '16 at 14:01
  • $\begingroup$ No, exercise asks to solve this for b and does not say anything about a and b. $\endgroup$ – Radon Jun 26 '16 at 14:08
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    $\begingroup$ @Radon Is this a high school math problem? If so, are you sure you copied it down correctly? I don't know if there's some kind of trick to this problem, but this seems way too hard for high school. $\endgroup$ – Noble Mushtak Jun 26 '16 at 14:09
  • $\begingroup$ what is the original problem? $\endgroup$ – Dr. Sonnhard Graubner Jun 26 '16 at 14:10
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    $\begingroup$ What graph are you people talking about?! $\endgroup$ – DonAntonio Jun 26 '16 at 14:16
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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:

$\hspace{2cm}$enter image description here

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.

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  • $\begingroup$ I need a solution for b. b = f(a) = ... . Thats what solution for b is. letter b is on the right side of equation and the everything else is on the left side of equation. I think i would manage to get b values for a given a, but i need a generic solution. $\endgroup$ – Radon Jun 26 '16 at 16:17
  • $\begingroup$ @Radon: $b=\pm 1$ is a generic solution and we cannot provide much more: you cannot ask for a function that takes $14$ different values at $a=3$. $\endgroup$ – Jack D'Aurizio Jun 26 '16 at 16:19
  • $\begingroup$ @JackD'Aurizio Yes, but for example polynomial equation, you can get there it is equal y and get all those x values. I just wondering is there a way to get all b values with a given a. I just want to know if there is a way to get all points there f(a) = f(a/b) with a given a. For example if a=3 is there a way to get all those 14 values? $\endgroup$ – Radon Jun 26 '16 at 16:33
  • $\begingroup$ @Radon: $f(x)=\frac{\sin x}{x}$ is way different from a polynomial. We cannot even give an explicit form to the solutions of $\frac{\sin x}{x}=\frac{1}{7}$, for instance. The associated equation is essentially Kepler's equation (en.wikipedia.org/wiki/Kepler%27s_equation). $\endgroup$ – Jack D'Aurizio Jun 26 '16 at 16:36
  • $\begingroup$ Physicists would be glad to have a closed form solution of Kepler's equation, but sadly, there isn't. However, Kepler's equation can be efficiently solved numerically through Newton's method. $\endgroup$ – Jack D'Aurizio Jun 26 '16 at 16:38
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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 $|b| = 1$.

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