Spherical trigonometry

Remarks

Example: for $A = (1, 0, 0)$, $B = (0, 1, 0)$, $C = (0, 0, 1)$, all sides and all angles equal $\pi/2$. The spherical law of cosines $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$ checks: both sides are $0$. Non-example: the vectors $A = (1, 0, 0)$, $B = (-1, 0, 0)$ and $C = (0, 0, 1)$ are linearly dependent. The points $A$ and $B$ are antipodal, there is no unique shorter arc $AB$, and $A, B, C$ do not form a spherical triangle. The standard relations are the law of cosines for sides above, the law of cosines for angles $\cos \gamma = -\cos \alpha \cos \beta + \sin \alpha \sin \beta \cos c$, and the law of sines $\dfrac{\sin a}{\sin \alpha} = \dfrac{\sin b}{\sin \beta} = \dfrac{\sin c}{\sin \gamma}$. On a sphere of radius $R$, the sides are the arc lengths divided by $R$.