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Half-side formula

Relation between the side lengths and angles of a spherical triangle

Half-side formula
Summary

Relation between the side lengths and angles of a spherical triangle

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.

For a triangle \triangle ABC on a sphere, the half-side formula is \begin{align} \tan \tfrac12 a &= \sqrt{\frac{-\cos(S), \cos(S - A)} {\cos(S - B), \cos(S - C)} } \end{align}

where a, b, c are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles A, B, C respectively, and S = \tfrac12 (A+B+ C) is half the sum of the angles. Two more formulas can be obtained for b and c by permuting the labels A, B, C.

The polar dual relationship for a spherical triangle is the half-angle formula,

\begin{align} \tan \tfrac12 A &= \sqrt{\frac{\sin(s - b), \sin(s - c)} {\sin(s), \sin(s - a)} } \end{align}

where semiperimeter s = \tfrac12 (a + b + c) is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels A, B, C.

Half-tangent variant

The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If t_a = \tan \tfrac12 a, t_b = \tan \tfrac12 b, t_c = \tan \tfrac12 c,t_A = \tan \tfrac12 A, t_B = \tan \tfrac12 B, and t_C = \tan \tfrac12 C, then the half-side formula is equivalent to:

\begin{align} t_a^2 &= \frac{\bigl(t_Bt_C + t_Ct_A + t_At_B - 1\bigr)\bigl({-t_Bt_C + t_Ct_A + t_At_B + 1}\bigr)} {\bigl(t_Bt_C - t_Ct_A + t_At_B + 1\bigr)\bigl(t_Bt_C + t_Ct_A - t_At_B + 1\bigr)}. \end{align}

and the half-angle formula is equivalent to:

\begin{align} t_A^2 &= \frac{\bigl(t_a - t_b + t_c + t_at_bt_c\bigr)\bigl(t_a + t_b - t_c + t_at_bt_c\bigr)} {\bigl(t_a + t_b + t_c - t_at_bt_c\bigr)\bigl({-t_a + t_b + t_c + t_at_bt_c}\bigr)}. \end{align}

References

References

  1. (2007). "Handbook of Mathematics". Springer.
  2. Nelson, David. (2008). "The Penguin Dictionary of Mathematics". Penguin UK.
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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