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  "body": "\n## Statement\n\nLet $ABC$ be a [[hyperbolic triangle|GCZ8EEDCHS]] in the hyperbolic plane of constant curvature $-1$.\nLet $a$, $b$, $c$ be the lengths of the sides opposite $A$, $B$, $C$, and let $\\gamma$ be the angle at $C$.\nThen\n$$\\cosh c = \\cosh a \\cosh b - \\sinh a \\sinh b \\cos \\gamma,$$\nwhere $\\cosh$ and $\\sinh$ are the [[hyperbolic cosine|036696EMG2]] and [[hyperbolic sine|0H1JFKE7H4]].\n\n## Remarks\n\nProof idea: work in the [[hyperboloid model|809288G50Y]] with the form $B(x, y) = x_0 y_0 - x_1 y_1 - x_2 y_2$, where $\\cosh d(x, y) = B(x, y)$.\nAfter an isometry, $C = (1, 0, 0)$, $A = (\\cosh b, \\sinh b, 0)$ and $B = (\\cosh a, \\sinh a \\cos \\gamma, \\sinh a \\sin \\gamma)$.\nThen $\\cosh c = B(A, B) = \\cosh a \\cosh b - \\sinh a \\sinh b \\cos \\gamma$.\nFor $\\gamma = \\pi/2$ it gives the hyperbolic Pythagorean theorem $\\cosh c = \\cosh a \\cosh b$.\nFor small sides, $\\cosh x \\approx 1 + x^2/2$ and $\\sinh x \\approx x$, and the formula reduces to the Euclidean [[law of cosines|060ZVZBV3E]] $c^2 = a^2 + b^2 - 2ab \\cos \\gamma$.\nIn curvature $-k^2$ with $k \u003e 0$, the same formula holds with $a, b, c$ replaced by $ka, kb, kc$.\nThere 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.\n\n## Explanations\n\n### L1 human\n\nIf you know two sides of a hyperbolic triangle and the angle between them, this formula gives the third side.\nIt plays the role of the school law of cosines, with hyperbolic functions in place of squares.\nFor a right triangle it says $\\cosh c = \\cosh a \\cosh b$, the hyperbolic form of Pythagoras.\n\n### L2 human\n\nThe 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.\nSince 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.\nIt 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.\n",
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