Recently I was thinking on the following problem: find all $\alpha\in R$ such that $\sin\frac{\alpha}{3}$ has an algebraic expression involving only rational powers over rational constants and "$\sin\alpha$". This question is related to the one asked by Hugo Chapdelaine and discussed here (a question about algebraic expressions for $\sqrt[n]{a+ib}$).

For example, it can be shown that $\sin\frac{\pi}{9}$ can't be expressed in real radicals, therefore it can't be expressed in terms of $\sin\frac{\pi}{3}=\frac{\sqrt3}{2}$ too. Using the well-known formula for $\sin(3\alpha)$ we can formulate this question in the following way: find all $\alpha$ such that the equation $3x-4x^3 =\sin\alpha$ has a root of form $x=f(\sin\alpha)$, where function $f$ is a composition of rational powers and algebraic operations. Maybe someone came along with such problems? For example, if $\alpha=1$, can we obtain such a representation for $\sin\frac{1}{3}$, or can we somehow prove that it does not exist?

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    $\begingroup$ A cubic equation can be solved in radicals! (Sometimes, they will involve $\sqrt{-1}$, which does not seem to be a very big deal.) So, your statement on $\sin(\pi/3)$ looks doubtful. $\endgroup$ – Alex Degtyarev Jan 10 '15 at 22:02
  • $\begingroup$ More generally, $\sin(\pi q)$ is solvable for all rational $q$, being $1/(2i)$ times the difference of two roots of unity. Presumably the question intends to only consider real radicals? $\endgroup$ – Emil Jeřábek Jan 10 '15 at 22:26
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    $\begingroup$ Yes, of course I meant real radicals: in the paper that I mentioned it is proved that $\sin\displaystyle\Big(\frac{\pi}{n}\Big)$ can be expressed in real radicals over rational numbers iff $\varphi(n)=2^m$ where $\varphi$ is Euler's totient function, so there is no such expression for $n=9$. But my question is a bit more complicated, because I allow not only rational numbers, but "$\sin\alpha$" as an argument too! The easiest example: can $\sin(1/3)$ be expressed as $f(\sin 1)$ where $f$ is a composition of rational powers and algebraic operations? $\endgroup$ – Alexander Jan 11 '15 at 4:21
  • $\begingroup$ This question is equivalent to the following: when is it possible to express the real and imaginary part of $\sqrt[3]{a+ib}$ as $f(a,b)$, where $f$ is a "real radical" function? $\endgroup$ – Alexander Jan 11 '15 at 4:37

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