Hyperbolic law of cosines

Remarks

Proof idea: work in the hyperboloid model with the form $B(x, y) = x_0 y_0 - x_1 y_1 - x_2 y_2$, where $\cosh d(x, y) = B(x, y)$. After an isometry, $C = (1, 0, 0)$, $A = (\cosh b, \sinh b, 0)$ and $B = (\cosh a, \sinh a \cos \gamma, \sinh a \sin \gamma)$. Then $\cosh c = B(A, B) = \cosh a \cosh b - \sinh a \sinh b \cos \gamma$. For $\gamma = \pi/2$ it gives the hyperbolic Pythagorean theorem $\cosh c = \cosh a \cosh b$. For small sides, $\cosh x \approx 1 + x^2/2$ and $\sinh x \approx x$, and the formula reduces to the Euclidean law of cosines $c^2 = a^2 + b^2 - 2ab \cos \gamma$. In curvature $-k^2$ with $k > 0$, the same formula holds with $a, b, c$ replaced by $ka, kb, kc$. There is also a dual law, $\cos \gamma = -\cos \alpha \cos \beta + \sin \alpha \sin \beta \cosh c$, which shows that the angles determine the triangle.

Going deeper

For a student

The law of cosines is the basic tool of hyperbolic trigonometry: the law of sines, the formulas for right triangles and the angle of parallelism all follow from it. Since the right-hand side grows like $e^{a + b}/4 \cdot (1 - \cos \gamma)$ for long sides, it shows that two geodesics leaving a point at a fixed angle diverge exponentially, a key feature of negative curvature. It is the hyperbolic member of a family with the spherical law $\cos c = \cos a \cos b + \sin a \sin b \cos \gamma$ and the Euclidean law as the flat limit.