Solve {sinalpha+sinbeta=1

$$\{ \sin \alpha + \sin \beta = 1$$

Solve for α, β (complex solution)

$$\alpha =2\pi n_{1}+i\ln(2)+\left(-i\right)\ln(2i+\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }+\left(-1\right)\left(\left(\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }\right)\left(\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }+4i\right)\right)^{\frac{1}{2}})\text{, }n_{1}\in \mathrm{Z}\text{, }\beta \in \mathrm{C}$$

$$\alpha =2\pi n_{2}+i\ln(2)+\left(-i\right)\ln(2i+\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }+\left(\left(\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }\right)\left(\left(-1\right)e^{i\beta }+e^{\left(-i\right)\beta }+4i\right)\right)^{\frac{1}{2}})\text{, }n_{2}\in \mathrm{Z}\text{, }\beta \in \mathrm{C}$$

Solve for α, β

$$\alpha =arcSin(\left(-1\right)SinI(\beta )+1)+2\pi n_{1}\text{, }n_{1}\in \mathrm{Z}\text{, }\beta \in \begin{bmatrix}2n_{20}\pi ,2\pi n_{20}+\pi \end{bmatrix}\text{, }n_{20}\in \mathrm{Z}\text{; }\alpha =\pi +2\pi n_{2}+\left(-1\right)arcSin(\left(-1\right)SinI(\beta )+1)\text{, }n_{2}\in \mathrm{Z}\text{, }\beta \in \begin{bmatrix}2n_{20}\pi ,2\pi n_{20}+\pi \end{bmatrix}\text{, }n_{20}\in \mathrm{Z}$$