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My friend asked me a trigonometric problem, but I got stuck when handling this: $$\sin{y}=(\sqrt{3}-1)\sin{\left(\frac{x}{2}\right)}\sin{\left(\frac{\pi}{3}+\frac{x}{2}-y\right)}$$ He asked, what are all the solution curves of the equation ? It seems that $x=3y$ is a solution, and we need to play with these trigonometry identity, and get $$\frac{\sin{y}}{\cos{y}}=\frac{1+\sqrt{3}\sin{x}-\cos{x}}{2+\sqrt{3}+\sin{x}+\sqrt{3}\cos{x}}$$ But now the solution curve is not obvious, hope to find some help, thanks!

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A graph of the relationship gives an interesting picture.

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  • $\begingroup$ Okay I see, and $x=3y$ is actually not a solution indeed. Thank you ! $\endgroup$ – Golbez Jan 22 '14 at 4:33
  • $\begingroup$ @Golbez, have you found what are the solutions? I hope there is some analytic solution as well. Anyway thank you for the non-trivial Question $\endgroup$ – lab bhattacharjee Jan 22 '14 at 4:39
  • $\begingroup$ I think it's just $y=k\pi+\arctan{\frac{1+\sqrt{3}\sin{x}-\cos{x}}{2+\sqrt{3}+\sin{x}+\sqrt{3}\cos{x}}}$ $\endgroup$ – Golbez Jan 22 '14 at 5:41
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Remember that $\sin x = \{ \exp(-i x) - \exp(ix) \} i$, where $i$ is the imaginary number. Then you can express the left hand side and the right hand side of your first displayed equation and work out a solution.

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Expanding the last sine in the first equation made you arriving to something looking as tan(y) = f(x), function f(x) being the ratio of two linear combinations of sin(x) and cos(x), as you arrived to. COngratulations for the lost of simplifications you did to arrive to something so simple.

So, just write y = ArcTan[f(x)] and you are done.

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  • $\begingroup$ Yes, you are right, and I just wonder whether this $\arctan$ can be simplified to some simpler forms, but it seems that this is impossible. $\endgroup$ – Golbez Jan 22 '14 at 6:00
  • $\begingroup$ @Golbez. No, it cannot be more simple. $\endgroup$ – Claude Leibovici Jan 22 '14 at 6:05

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