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cosine rule or theorem





There are three such theorems:
  • A theorem of plane trigonometry.
  • The angle cosine theorem of spherical trigonometry.
  • The cosine theorem for sides, also a theorem of spherical trigonometry.

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:
a2 = b2 + c 2 - 2bc cos α
b2 = c2 + a 2 - 2ca cos β
c2 = a2 + b 2 - 2ab 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 γ2 + sin β sin γ cos a
cos β = - cos γ cos α2 + sin γ sin α cos b
cos γ = - cos α cos β2 + 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 c2 + sin b sin c cos α
cos b = - cos c cos a2 + sin c sin a cos β
cos c = - cos a cos b2 + 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.


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   • GEOMETRY