x=\frac\pi9+\frac{2\pi k}{3}} \atop {x=\frac{\pi n}{3}, \ n=2k+1==>\ x=\frac\pi3+\frac{2\pi k}{3}}} \right.
x\in[-2\pi;\pi]\\
k=-3:x=\frac{\pi}{9}-2\pi=-\frac{17\pi}{9};\ \ x=\frac{\pi}{3}-2\pi=-\frac{5\pi}{3};\\ " alt="\sin(3x-\frac{\pi}{6})=\frac{1}{2};\\
3x-\frac\pi6=(-1)^n\arcsin\frac12+\pi n. n\in Z\\
3x-\frac\pi6=(-1)^n\frac{\pi}{6}+\pi n, n\in Z\\
3x=\frac\pi6(1+(-1)^n)+\pi n, n\in Z\\
x=\frac{\pi}{18}(1+(-1)^n)+\frac{\pi n}{3}, n\in Z\\
\left[ {{x=\frac{\pi}{9}+\frac{\pi n}{3},\ n=2k==>x=\frac\pi9+\frac{2\pi k}{3}} \atop {x=\frac{\pi n}{3}, \ n=2k+1==>\ x=\frac\pi3+\frac{2\pi k}{3}}} \right.
x\in[-2\pi;\pi]\\
k=-3:x=\frac{\pi}{9}-2\pi=-\frac{17\pi}{9};\ \ x=\frac{\pi}{3}-2\pi=-\frac{5\pi}{3};\\ " align="absmiddle" class="latex-formula">
значит имеем такие решения
