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Spherical Law of Cosines Using Sides

The spherical law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

The cosine of the angular length of the third side equals the product of the cosine of the angle opposite the third side and the sines of the angular lengths of the first and second sides plus the product of the cosines of the angular lengths of the first and second sides.

Formula

Cos[QuantityVariable["c", "Angle"]] == Cos[QuantityVariable["a", "Angle"]]*Cos[QuantityVariable["b", "Angle"]] + Cos[QuantityVariable["γ", "Angle"]]*Sin[QuantityVariable["a", "Angle"]]*Sin[QuantityVariable["b", "Angle"]]

symbol description physical quantity
c third side angular length "Angle"
a first side angular length "Angle"
b second side angular length "Angle"
γ angle opposite third side "Angle"

Forms

Examples

Get the resource:

In[1]:=
ResourceObject["Spherical Law of Cosines Using Sides"]
Out[1]=

Get the formula:

In[2]:=
FormulaData[ResourceObject["Spherical Law of Cosines Using Sides"]]
Out[2]=

Use some values:

In[3]:=
FormulaData[ ResourceObject[ "Spherical Law of Cosines Using Sides"], {QuantityVariable[ "c","Angle"] -> Quantity[\[Pi]/2, "Radians"], QuantityVariable["\[Gamma]","Angle"] -> Quantity[\[Pi]/2, "Radians"], QuantityVariable["a","Angle"] -> Quantity[\[Pi]/2, "Radians"]}]
Out[3]=

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