$$\sin (A+B)= \sin A+ \sin B$$
$\left\{\begin{matrix}A=2\pi n_{4}\text{, }n_{4}\in \mathrm{Z}\text{; }A=2\pi n_{5}-B\text{, }n_{5}\in \mathrm{Z}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }B=2\pi n_{1}\\A=2\pi n_{2}+B+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }&\exists n_{3}\in \mathrm{Z}\text{ : }B=\pi n_{3}+\frac{\pi }{2}\\A\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }B=2\pi n_{1}\end{matrix}\right.$
$\left\{\begin{matrix}B=2\pi n_{4}\text{, }n_{4}\in \mathrm{Z}\text{; }B=2\pi n_{5}-A\text{, }n_{5}\in \mathrm{Z}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }A=2\pi n_{1}\\B=2\pi n_{2}+A+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }&\exists n_{3}\in \mathrm{Z}\text{ : }A=\pi n_{3}+\frac{\pi }{2}\\B\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }A=2\pi n_{1}\end{matrix}\right.$