Folkcor $^{2}$

Find the maximum value of the expression $x+y$, if $(2 \sin x-1)(2 \cos y-\sqrt{3})=0, x \in[0,3 \pi / 2], y \in$ $[\pi, 2 \pi]$.
Equality in the condition means that $\sin x=1 / 2$ or $\cos y=\frac{\sqrt{3}}{2}$. In each of these cases, the sum $x+y$ is maximal when each of the terms is maximal: in the first case, this is $x=5 \pi / 6, y=2 \pi$, and in the second $-x=$ $3 \pi / 2, y=11 \pi / 6$. In the second case, the result is greater: $5 \pi / 6+2 \pi=17 \pi / 6<3 \pi / 2+11 \pi / 6=10 \pi / 3$.

## Answer

$10 \pi / 3$.