Half-side formula

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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[edit]

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

  • 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[edit]

References[edit]

  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 .