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Spherical Law of Tangents

The spherical law of tangents is a statement about the relationship between the tangents of two angles of a triangle on a sphere and the angular lengths of the opposing sides.

The cotangent of the average of the angular lengths of first and second sides times the tangent of half the difference between the angular lengths of the first and second sides equals the cotangent of the average of the angles opposite the first and second sides times the tangent of half the difference between the angles opposite the first and second sides.

Formula

Cot[(QuantityVariable["a", "Angle"] + QuantityVariable["b", "Angle"])/2]*Tan[(QuantityVariable["a", "Angle"] - QuantityVariable["b", "Angle"])/2] == Cot[(QuantityVariable["α", "Angle"] + QuantityVariable["β", "Angle"])/2]*Tan[(QuantityVariable["α", "Angle"] - QuantityVariable["β", "Angle"])/2]

symbol description physical quantity
a first side angular length "Angle"
b second side angular length "Angle"
α angle opposite first side "Angle"
β angle opposite second side "Angle"

Forms

Examples

Get the resource:

In[1]:=
ResourceObject["Spherical Law of Tangents"]
Out[1]=

Get the formula:

In[2]:=
FormulaData[ResourceObject["Spherical Law of Tangents"]]
Out[2]=

Use some values:

In[3]:=
FormulaData[ ResourceObject[ "Spherical Law of Tangents"], {QuantityVariable["b","Angle"] -> Quantity[\[Pi]/2, "Radians"], QuantityVariable["a","Angle"] -> Quantity[\[Pi]/2, "Radians"]}]
Out[3]=

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