## Task A-3.1.

If $2 \sin x - 3 \cos y = a$ and $2 \cos x + 3 \sin y = b$, what is $\sin (x - y)$?
## Solution.

By squaring and adding the given equalities in sequence, we get

$$
\begin{array}{cl}
4\left(\sin ^{2} x+\cos ^{2} x\right)-12(\sin x \cos y-\cos x \sin y)+9\left(\cos ^{2} y+\sin ^{2} y\right)=a^{2}+b^{2}, & 1 \text { point } \\
4-12(\sin x \cos y-\cos x \sin y)+9=a^{2}+b^{2}, & 2 \text { points } \\
13-12 \sin (x-y)=a^{2}+b^{2}, & 2 \text { points } \\
\sin (x-y)=\frac{13-a^{2}-b^{2}}{12} & 1 \text { point }
\end{array}
$$