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

• 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

• Spherical law of cosines
• Law of haversines

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.