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Half-side formula

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Title: Half-side formula  
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Subject: Law of sines, Spherical geometry, Law of tangents, Tangent half-angle formula, Spherical law of cosines, Law of cosines
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Half-side formula


In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles.

Formulas

\begin{align} \tan\left(\frac{a}{2}\right) & = R \cos (S- \alpha) \\[8pt] \tan \left(\frac{b}{2}\right) & = R \cos (S- \beta) \\[8pt] \tan \left(\frac{c}{2}\right) & = R \cos (S - \gamma) \end{align}

where

  • a, b, c are the lengths of the sides respectively opposite α, β, γ,
  • S = \frac{1}{2}(\alpha +\beta + \gamma) is half the sum of the angles, and
  • R=\sqrt{\frac {-\cos S}{\cos (S-\alpha) \cos (S-\beta) \cos (S-\gamma)}}.

The three formulas are really the same formula, with the names of the variables permuted.

See also

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