Solve {l}{sinA=frac{sqrt{5}}{5}}{sinB=frac{sqrt{10}}{10}}{sinC=sin(A+B)}

$$\left. \begin{array} { l } { \sin A = \frac { \sqrt { 5 } } { 5 } } \\ { \sin B = \frac { \sqrt { 10 } } { 10 } } \\ { \sin C = \sin ( A + B ) } \end{array} \right.$$

Solve for A, B, C

$$A=2\pi n_{1}+\arcsin(\frac{\sqrt{5}}{5})\text{, }n_{1}\in \mathrm{Z}\text{, }B=2\pi n_{3}+\arcsin(\frac{\sqrt{10}}{10})\text{, }n_{3}\in \mathrm{Z}\text{, }C=2\pi n_{5}+\frac{\pi }{4}\text{, }n_{5}\in \mathrm{Z}$$

$$A=2\pi n_{1}+\arcsin(\frac{\sqrt{5}}{5})\text{, }n_{1}\in \mathrm{Z}\text{, }B=2\pi n_{3}+\arcsin(\frac{\sqrt{10}}{10})\text{, }n_{3}\in \mathrm{Z}\text{, }C=2\pi n_{6}+\frac{3\pi }{4}\text{, }n_{6}\in \mathrm{Z}$$

$$A=2\pi n_{1}+\arcsin(\frac{\sqrt{5}}{5})\text{, }n_{1}\in \mathrm{Z}\text{, }B=2\pi n_{4}-\arcsin(\frac{\sqrt{10}}{10})+\pi \text{, }n_{4}\in \mathrm{Z}\text{, }C=2\pi n_{7}-\arcsin(\frac{\sqrt{2}}{10})+2\pi \text{, }n_{7}\in \mathrm{Z}$$

$$A=2\pi n_{1}+\arcsin(\frac{\sqrt{5}}{5})\text{, }n_{1}\in \mathrm{Z}\text{, }B=2\pi n_{4}-\arcsin(\frac{\sqrt{10}}{10})+\pi \text{, }n_{4}\in \mathrm{Z}\text{, }C=2\pi n_{8}+\arcsin(\frac{\sqrt{2}}{10})+\pi \text{, }n_{8}\in \mathrm{Z}$$

$$A=2\pi n_{2}-\arcsin(\frac{\sqrt{5}}{5})+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }B=2\pi n_{3}+\arcsin(\frac{\sqrt{10}}{10})\text{, }n_{3}\in \mathrm{Z}\text{, }C=2\pi n_{9}+\arcsin(\frac{\sqrt{2}}{10})\text{, }n_{9}\in \mathrm{Z}$$

$$A=2\pi n_{2}-\arcsin(\frac{\sqrt{5}}{5})+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }B=2\pi n_{3}+\arcsin(\frac{\sqrt{10}}{10})\text{, }n_{3}\in \mathrm{Z}\text{, }C=2\pi n_{10}-\arcsin(\frac{\sqrt{2}}{10})+\pi \text{, }n_{10}\in \mathrm{Z}$$

$$A=2\pi n_{2}-\arcsin(\frac{\sqrt{5}}{5})+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }B=2\pi n_{4}-\arcsin(\frac{\sqrt{10}}{10})+\pi \text{, }n_{4}\in \mathrm{Z}\text{, }C=2\pi n_{11}+\frac{7\pi }{4}\text{, }n_{11}\in \mathrm{Z}$$

$$A=2\pi n_{2}-\arcsin(\frac{\sqrt{5}}{5})+\pi \text{, }n_{2}\in \mathrm{Z}\text{, }B=2\pi n_{4}-\arcsin(\frac{\sqrt{10}}{10})+\pi \text{, }n_{4}\in \mathrm{Z}\text{, }C=2\pi n_{12}+\frac{5\pi }{4}\text{, }n_{12}\in \mathrm{Z}$$