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.

Formulas

On a unit sphere, the half-side formulas are.
:
\begin
\tan\left(\frac\right) & = R \cos (S- A) \\ pt\tan \left(\frac\right) & = R \cos (S- B) \\ pt\tan \left(\frac\right) & = R \cos (S - C)
\end

where
* ''a'', ''b'', ''c'' are the lengths of the sides respectively opposite angles ''A'', ''B'', ''C'',
* $S = \frac(A+B+ C)$ is half the sum of the angles, and
* $R=\sqrt.$

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.

See also

* Spherical law of cosines
* Law of haversines

References

{{reflist

Spherical trigonometry