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}](../I/m/d27b59a3606eb05389629fd64fe20acb.png)
where
- a, b, c are the lengths of the sides respectively opposite α, β, γ,
-
is half the sum of the angles, and -
The three formulas are really the same formula, with the names of the variables permuted.
See also
References
- ↑ Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222.
- ↑ Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870.