Solve sin(alpha+beta-gamma)=1/2,cos(beta+gamma-alpha)=1/2text{and}tan(gamma+alpha-beta)=1

$$\sin ( \alpha + \beta - \gamma ) = \frac { 1 } { 2 } , \cos ( \beta + \gamma - \alpha ) = \frac { 1 } { 2 } \text { and } \tan ( \gamma + \alpha - \beta ) = 1$$

Solve for γ

$$\gamma \in \mathrm{R}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\left(\gamma =2\pi n_{2}+\alpha +\beta -\frac{\pi }{6}\text{ or }\gamma =2\pi n_{2}+\alpha +\beta -\frac{5\pi }{6}\right)\text{ and }\exists n_{3}\in \mathrm{Z}\text{ : }\left(\gamma =2\pi n_{3}+\alpha -\beta +\frac{\pi }{3}\text{ or }\gamma =2\pi n_{3}+\alpha -\beta +\frac{5\pi }{3}\right)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\gamma =\pi n_{4}+\beta -\alpha +\frac{\pi }{4}\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\gamma =\pi n_{1}+\beta -\alpha +\frac{\pi }{2}$$

Solve for β

$$\beta \in \mathrm{R}$$

$$\exists n_{2}\in \mathrm{Z}\text{ : }\beta =\pi n_{2}+\alpha +\gamma -\frac{\pi }{4}$$

$$\exists n_{3}\in \mathrm{Z}\text{ : }\left(\gamma =\frac{23}{24}\pi +\left(-\frac{1}{2}\right)\pi n_{2}+\pi n_{3}\text{ or }\gamma =\frac{31}{24}\pi +\left(-\frac{1}{2}\right)\pi n_{2}+\pi n_{3}\right)\text{ and }\exists n_{4}\in \mathrm{Z}\text{ : }\left(\alpha =\frac{13}{24}\pi +\pi n_{4}+\left(-\frac{1}{2}\right)\pi n_{2}\text{ or }\alpha =\frac{29}{24}\pi +\pi n_{4}+\left(-\frac{1}{2}\right)\pi n_{2}\right)\text{ and }\nexists n_{1}\in \mathrm{Z}\text{ : }\beta =\pi n_{1}+\alpha +\gamma -\frac{\pi }{2}$$