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 on a sphere, the half-side formula is.
where , , are the angular lengths (measure of central angle, arc lengths normalized to a sphere of unit radius) of the sides opposite angles , , 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
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 Spherical law of cosines Law of haversines
References