Three rays are going out from point $O$ in space, forming pairwise angles $\alpha, \beta$ and $\gamma$ with $0^o<\alpha \le \beta \le \gamma <180^o$. Prove that $\sin \frac{\alpha}{2}+ \sin \frac{\beta}{2} > \sin \frac{\gamma}{2}$.
Draw a sphere with radius $\frac{1}{2}$ and center $O$. Draw a triangle with the intersections of the 3 rays and the sphere. We get that the sidelengths of the triangle are $\sin\frac{\alpha}{2}$, etc... So this is just triangle inequality.