Half-side formula

Spherical triangle

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.[1]

Formulas

The half-side formulas are[2]


\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

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

See also

References

  1. Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222
  2. Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870.
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