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# Half-side formula

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 Title: Half-side formula Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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,
• $S = \frac\left\{1\right\}\left\{2\right\}\left(\alpha +\beta + \gamma\right)$ is half the sum of the angles, and
• $R=\sqrt\left\{\frac \left\{-\cos S\right\}\left\{\cos \left(S-\alpha\right) \cos \left(S-\beta\right) \cos \left(S-\gamma\right)\right\}\right\}.$

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