Jump to content

Half-side formula

From Wikipedia, the free encyclopedia
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]

For a triangle on a sphere, the half-side formula is[2]

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 is half the sum of the angles. Two more formulas can be obtained for and by permuting the labels

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

where semiperimeter is half the sum of the sides. Again, two more formulas can be obtained by permuting the labels

Half-tangent variant

[edit]

The same relationships can be written as rational equations of half-tangents (tangents of half-angles). If and then the half-side formula is equivalent to:

and the half-angle formula is equivalent to:

See also

[edit]

References

[edit]
  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.