# cosine rule or theorem

The cosine rule or theorem may refer to any of three theorems:

## The cosine theorem of plane trigonometry

This states that if *a*, *b*, and *c* are the sides,
and *α*, *β*, and *γ* the angles of
a triangle, then:

*a*^{2} = *b*^{2} + *c*^{ 2} - 2*bc* cos *α*

*b*^{2} = *c*^{2} + *a*^{ 2} - 2*ca* cos *β*

*c*^{2} = *a*^{2} + *b*^{ 2} - 2*ab* cos *γ*

i.e., the square of any one side is equal to the sum of the squares of the other two sides less twice the product of those sides and the cosine of the angle they include. The cosine theorem of plane trigonometry can be used to solve a triangle if the three sides of the triangle are given or if two sides and the included angles are given.

## The angle cosine theorem of spherical trigonometry

This states that if *a*, *b*, and *c* are the sides,
and *α*, *β*, and *γ* the angles of
a spherical triangle, then:

cos *α* = - cos *β* cos *γ*^{2} + sin *β* sin *γ* cos *a*

cos *β* = - cos *γ* cos *α*^{2} + sin *γ* sin *α* cos *b*

cos *γ* = - cos *α* cos *β*^{2} + sin *α* sin *β* cos *c*

## The cosine theorem for the sides of a spherical triangle

This states that if *a*, *b*, and *c* are the sides,
and *α*, *β*, and *γ* the angles of
a spherical triangle, then:

cos *a* = - cos *b* cos *c*^{2} + sin *b* sin *c* cos *α*

cos *b* = - cos *c* cos *a*^{2} + sin *c* sin *a* cos *β*

cos *c* = - cos *a* cos *b*^{2} + sin *a* sin *b* cos *γ*

The angle cosine theorem can be used if three angles of a spherical triangle are given, or if one side and the two adjacent angles are given. The cosine theorem for the sides of a spherical triangle can be used if three sides or two sides and the included angle are given.