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I remember solving this before the crash. The ratio between this volume and the volume of the sphere is equal to the ratio between the area determined by the trihedron on the sphere and the area of the sphere. The region is a spherical triangle with angles $\alpha,\beta,\gamma$, and its area is thus $(\alpha+\beta+\gamma-\pi)\cdot R^2$. The volume we are looking for must then be $\frac{\alpha+\beta+\gamma-\pi}{4\pi}\cdot \frac{4\pi\cdot R^3}3=\frac{(\alpha+\beta+\gamma-\pi)\cdot R^3}3$.