1. It is known that $\sin x=\frac{3}{2} \sin y-\frac{2}{3} \cos y, \cos x=\frac{3}{2} \cos y-\frac{2}{3} \sin y$. Find $\sin 2 y$.
Answer. $\sin 2 y=\frac{61}{72}$.

Solution. Squaring both equalities and adding them term by term, we get $1=\frac{9}{4}-4 \sin y \cos y+\frac{4}{9}$, from which $2 \sin 2 y=\frac{61}{36}, \sin 2 y=\frac{61}{72}$.