Решить уравнение (с подробным объяснением) 2sin([latex] \frac{x}{2} [/latex]-[latex] \frac{p}{6} [/latex])=-1

[latex]2sin(\frac{x}{2}-\frac{\pi}{6})=-1\\sin(\frac{x}{2}-\frac{\pi}{6})=-\frac{1}{2}\\\frac{x}{2}-\frac{\pi}{6}=(-1)^{n+1}arcsin\frac{1}{2}+\pi n\\\frac{x}{2}-\frac{\pi}{6}=(-1)^{n+1}\frac{\pi}{6}+\pi n\\\frac{x}{2}=(-1)^{n+1}\frac{\pi}{6}+\frac{\pi}{6}+\pi n\\x=2*(-1)^{n+1}\frac{\pi}{6}+\frac{\pi}{3}+2\pi n, \; n\in Z;[/latex] [latex]\frac{x}{2}-\frac{\pi}{6}=arcsin\frac{1}{2}+2\pi n\\ \frac{x}{2}-\frac{\pi}{6}=\frac{\pi}{6}+2\pi n\\\frac{x}{2}=\frac{\pi}{3}+2\pi n\\x=\frac{2\pi}{3}+4\pi n, \; n\in Z;\\\\\frac{x}{2}-\frac{\pi}{6}=\pi -arcsin\frac{1}{2}+2\pi n\\\frac{x}{2}-\frac{\pi}{6}=\pi -\frac{\pi}{6}+2\pi n\\\frac{x}{2}-\frac{\pi}{6}=\frac{5\pi}{6}+2\pi n\\\frac{x}{2}=\pi +2\pi n\\x=2\pi n+4\pi n, \; n\in Z.[/latex]