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?

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?

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 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?

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?