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]

Spherical triangle

Formulas

For a triangle on a sphere with radius $r$ , the half-side formulas are[2]

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

where

$a$ , $b$ , and $c$ are the lengths of the sides respectively opposite angles $A$ , $B$ , and $C$ ;
$S={\frac {1}{2}}(A+B+C)$ 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.

The sides $a$ , $b$ , and $c$ are normalized by the $1/ r$ factor so that they represent arc lengths on a unit sphere.

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.

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[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.

Half-side formula

Formulas

See also

References