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  "body": "\n## Definition\n\nLet $S^2 \\subset \\R^3$ be the unit [[sphere|T62369RFH6]] with the great circles as lines, as in [[spherical geometry|SQ9RQVANMG]].\nLet $A$, $B$, $C \\in S^2$ be linearly independent vectors of $\\R^3$.\nThe *spherical triangle* $ABC$ is the region of $S^2$ bounded by the three shorter great-circle arcs $BC$, $CA$, $AB$.\nIts sides are the arc lengths $a = \\arccos(B \\cdot C)$, $b = \\arccos(C \\cdot A)$, $c = \\arccos(A \\cdot B)$, each in $(0, \\pi)$.\nIts angle $\\gamma$ at $C$ is the angle between the planes of the arcs $CA$ and $CB$, measured inside the triangle.\nThe angles $\\alpha$ at $A$ and $\\beta$ at $B$ are defined in the same way.\n*Spherical trigonometry* is the study of the relations among these six quantities $a, b, c, \\alpha, \\beta, \\gamma$.\n\n## Remarks\n\nExample: for $A = (1, 0, 0)$, $B = (0, 1, 0)$, $C = (0, 0, 1)$, all sides and all angles equal $\\pi/2$.\nThe spherical law of cosines $\\cos c = \\cos a \\cos b + \\sin a \\sin b \\cos \\gamma$ checks: both sides are $0$.\nNon-example: the vectors $A = (1, 0, 0)$, $B = (-1, 0, 0)$ and $C = (0, 0, 1)$ are linearly dependent.\nThe points $A$ and $B$ are antipodal, there is no unique shorter arc $AB$, and $A, B, C$ do not form a spherical triangle.\nThe 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 $\\dfrac{\\sin a}{\\sin \\alpha} = \\dfrac{\\sin b}{\\sin \\beta} = \\dfrac{\\sin c}{\\sin \\gamma}$.\nOn a sphere of radius $R$, the sides are the arc lengths divided by $R$.\n",
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