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.^[1]

Formulas

On a unit sphere, the half-side formulas are^[2]

{\begin{aligned}\tan \left({\frac {a}{2}}\right)&=R\cos(S-A)\\[8pt]\tan \left({\frac {b}{2}}\right)&=R\cos(S-B)\\[8pt]\tan \left({\frac {c}{2}}\right)&=R\cos(S-C)\end{aligned}}

where

a, b, c are the lengths of the sides respectively opposite angles A, B, C,
$S={\frac {1}{2}}(A+B+C)$ is half the sum of the angles, and
$R={\sqrt {\frac {-\cos S}{\cos(S-A)\cos(S-B)\cos(S-C)}}}.$

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

To generalize to a sphere of arbitrary radius r, the lengths a,b,c must be replaced with

$a\rightarrow a/r$
$b\rightarrow b/r$
$c\rightarrow c/r$

so that a,b,c all have length scales, instead of angular scales.

References

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

[1] Bronshtein, I. N.; Semendyayev, K. A.; Musiol, Gerhard; Mühlig, Heiner (2007), Handbook of Mathematics, Springer, p. 165, ISBN 9783540721222[1]

[2] Nelson, David (2008), The Penguin Dictionary of Mathematics (4th ed.), Penguin UK, p. 529, ISBN 9780141920870.

[1]

[2]

Formulas

See also

References