3.1.1. (12 points) Calculate $\sin (\alpha+\beta)$, if $\sin \alpha+\cos \beta=\frac{1}{4}$ and $\cos \alpha+\sin \beta=-\frac{8}{5}$. Answer: $\frac{249}{800} \approx 0.31$.
Solution. Squaring the given equalities and adding them, we get

$$
\sin ^{2} \alpha+2 \sin \alpha \cos \beta+\cos ^{2} \beta+\cos ^{2} \alpha+2 \cos \alpha \sin \beta+\sin ^{2} \beta=\frac{1}{16}+\frac{64}{25}=\frac{1049}{400}
$$

from which we find $2 \sin (\alpha+\beta)+2=\frac{1049}{400}$, i.e., $\sin (\alpha+\beta)=\frac{249}{800} \approx 0.31$.