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Half-side formula

Relation between the side lengths and angles of a spherical triangle From Wikipedia, the free encyclopedia

Half-side formula
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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]

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Spherical triangle

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

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Half-tangent variant

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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:

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References

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