# Solutions of equation $\sin \pi x_1\sin \pi x_2=\sin \pi x_3\sin \pi x_4$ [closed]

I am interested in finding all the solution $$(x_1,x_2,x_3,x_4)\in \mathbb{R}^4$$ of equations:

$$\sin \pi x_1\sin\pi x_2=\sin \pi x_3\sin\pi x_4.$$

I have found out a paper: Rational products of sines of rational angles --- GERALD MYERSON, which answers my question partially.

Here is my interpretation:

Suppose $$\alpha,\beta,\gamma,\delta\in \left(0,\dfrac{\pi}{2}\right]\cap \mathbb{Q}\pi$$ and $$\sin \alpha\sin \beta=\sin\gamma\sin\delta$$.

Then $$(\alpha,\beta,\gamma,\delta)=$$ $$(x,y,x,y)$$ or $$(x,y,y,x)$$

or $$\left(\dfrac{\pi}{6},\phi,\dfrac{\phi}{2},\dfrac{\pi}{2}-\dfrac{\phi}{2}\right)$$ or $$\left(\dfrac{\pi}{6},\phi,\dfrac{\pi}{2}-\dfrac{\phi}{2},\dfrac{\phi}{2}\right)$$ or $$\left(\phi,\dfrac{\pi}{6},\dfrac{\phi}{2},\dfrac{\pi}{2}-\dfrac{\phi}{2}\right)$$ or $$\left(\phi,\dfrac{\pi}{6},\dfrac{\pi}{2}-\dfrac{\phi}{2},\dfrac{\phi}{2}\right)$$

or $$\left(\dfrac{\pi}{2}-\dfrac{\phi}{2},\dfrac{\phi}{2},\phi,\dfrac{\pi}{6}\right)$$ or $$\left(\dfrac{\phi}{2},\dfrac{\pi}{2}-\dfrac{\phi}{2},\phi,\dfrac{\pi}{6}\right)$$ or $$\left(\dfrac{\pi}{2}-\dfrac{\phi}{2},\dfrac{\phi}{2},\dfrac{\pi}{6},\phi\right)$$ or $$\left(\dfrac{\phi}{2},\dfrac{\pi}{2}-\dfrac{\phi}{2},\dfrac{\pi}{6},\phi\right)$$

or $$\left(x_1^0,x_2^0,x_3^0,x_4^0\right)$$ or $$\left(x_1^0,x_2^0,x_4^0,x_3^0\right)$$ or $$\left(x_2^0,x_1^0,x_3^0,x_4^0\right)$$ or $$\left(x_2^0,x_1^0,x_4^0,x_3^0\right)$$

or $$\left(x_4^0,x_3^0,x_2^0,x_1^0\right)$$ or $$\left(x_3^0,x_4^0,x_2^0,x_1^0\right)$$ or $$\left(x_4^0,x_3^0,x_1^0,x_2^0\right)$$ or $$\left(x_3^0,x_4^0,x_1^0,x_2^0\right)$$, where $$x,y,\phi\in \left(0,\dfrac{\pi}{2}\right]$$ and $$(x_1^0,x_2^0,x_3^0,x_4^0)\in \left\{ \left( \dfrac{\pi}{21}, \dfrac{8\pi}{21}, \dfrac{\pi}{14}, \dfrac{3\pi}{14} \right), \left( \dfrac{\pi}{14}, \dfrac{5\pi}{14}, \dfrac{2\pi}{21}, \dfrac{5\pi}{21} \right), \left( \dfrac{4\pi}{21}, \dfrac{10\pi}{21}, \dfrac{3\pi}{14}, \dfrac{5\pi}{14} \right), \left( \dfrac{\pi}{20}, \dfrac{9\pi}{20}, \dfrac{\pi}{15}, \dfrac{4\pi}{15} \right), \left( \dfrac{2\pi}{15}, \dfrac{7\pi}{15}, \dfrac{3\pi}{20}, \dfrac{7\pi}{20} \right), \left( \dfrac{\pi}{30}, \dfrac{3\pi}{10}, \dfrac{\pi}{15}, \dfrac{2\pi}{15} \right), \left( \dfrac{\pi}{15}, \dfrac{7\pi}{15}, \dfrac{\pi}{10}, \dfrac{7\pi}{30} \right), \left( \dfrac{\pi}{10}, \dfrac{13\pi}{30}, \dfrac{2\pi}{15}, \dfrac{4\pi}{15} \right), \left( \dfrac{4\pi}{15}, \dfrac{7\pi}{15}, \dfrac{3\pi}{10}, \dfrac{11\pi}{30} \right), \left( \dfrac{\pi}{30}, \dfrac{11\pi}{30}, \dfrac{\pi}{10}, \dfrac{\pi}{10} \right), \left( \dfrac{7\pi}{30}, \dfrac{13\pi}{30}, \dfrac{3\pi}{10}, \dfrac{3\pi}{10} \right), \left( \dfrac{\pi}{15}, \dfrac{4\pi}{15}, \dfrac{\pi}{10}, \dfrac{\pi}{6} \right), \left( \dfrac{2\pi}{15}, \dfrac{8\pi}{15}, \dfrac{\pi}{6}, \dfrac{3\pi}{10} \right), \left( \dfrac{\pi}{12}, \dfrac{5\pi}{12}, \dfrac{\pi}{10}, \dfrac{3\pi}{10} \right), \left( \dfrac{\pi}{10}, \dfrac{3\pi}{10}, \dfrac{\pi}{6}, \dfrac{\pi}{6} \right) \right\}.$$

1. Is my interpretation correct? Do I miss any possibilities?
2. What is about the non-rational solutions? Is there any way to solve the equation over $$\mathbb{R}$$?
• The question does not make much sense over $\mathbb R$, as for most choices of $x_1,x_2,x_3$, there is a solution $(x_1,x_2,x_3,\pi^{-1}\arcsin(\sin(\pi x_1)\sin(\pi x_2)/\sin(\pi x_3)))$. There is nothing to classify here. – Emil Jeřábek Jul 22 '20 at 8:39
• What do you mean by "solving over R"? This equation defines a hypersurface in $R^4$ which evidently contains infinitely many points. What does it mean to solve? Parametrize this hypersurface? – Alexandre Eremenko Jul 22 '20 at 12:31
• I'm glad you found my paper. – Gerry Myerson Jul 22 '20 at 12:36