If $\alpha$ + $\beta$ + $\gamma$ = 2$\pi$, then the system of equations

x + (cos $\gamma$)y + (cos $\beta$)z = 0

(cos $\gamma$)x + y + (cos $\alpha$)z = 0

(cos $\beta$)x + (cos $\alpha$)y + z = 0

has :

<p>Given $\alpha$ + $\beta$ + $\gamma$ = 2$\pi$</p> <p>$$\Delta = \left| {\matrix{ 1 & {\cos \gamma } & {\cos \beta } \cr {\cos \gamma } & 1 & {\cos \alpha } \cr {\cos \beta } & {\cos \alpha } & 1 \cr } } \right|$$</p> <p>$$ = 1 - {\cos ^2}\alpha - \cos \gamma (\cos \gamma - \cos \alpha \cos \beta ) + \cos \beta (\cos \alpha \cos \gamma - \cos \beta )$$</p> <p>$$ = 1 - {\cos ^2}\alpha - {\cos ^2}\beta - {\cos ^2}\gamma + 2\cos \alpha \cos \beta \cos \gamma $$</p> <p>$$ = {\sin ^2}\alpha - {\cos ^2}\beta - \cos \gamma (\cos \gamma - 2\cos \alpha \cos \beta )$$</p> <p>$$ = - \cos (\alpha + \beta )\cos (\alpha - \beta ) - \cos \gamma (\cos (2\pi - (\alpha - \beta )) - 2\cos \alpha \cos \beta )$$</p> <p>$$ = - \cos (2\pi - \gamma )\cos (\alpha - \beta ) - \cos \gamma (\cos (\alpha + \beta ) - 2\cos \alpha \cos \beta )$$</p> <p>$$ = - \cos \gamma \cos (\alpha - \beta ) + \cos \gamma (\cos \alpha \cos \beta + \sin \alpha \sin \beta )$$</p> <p>$= - \cos \gamma \cos (\alpha - \beta ) + \cos \gamma \cos (\alpha - \beta )$</p> <p>$= 0$</p> <p>So, the system of equation has infinitely many solutions.</p>