Math 2.0

Geometry Triangles and circles Theorem

Law of cosines

Builds on TriangleAngle at a pointEuclidean distance

Statement

Let ABCABC be a triangle in the Euclidean space ℝn\mathbb{R}^n. Let a=d(B,C)a = d(B, C), b=d(C,A)b = d(C, A) and c=d(A,B)c = d(A, B) be its side lengths, where dd is the Euclidean distance. Let γ=∠ACB\gamma = \angle ACB be its angle at CC. Then c2=a2+b2−2abcos⁡γ.c^2 = a^2 + b^2 - 2ab \cos \gamma.

Remarks

The proof in vector form is short. With u=A−Cu = A - C and v=B−Cv = B - C: c2=∥u−v∥2=∥u∥2+∥v∥2−2u⋅vc^2 = \|u - v\|^2 = \|u\|^2 + \|v\|^2 - 2 u \cdot v, and u⋅v=abcos⁡γu \cdot v = ab \cos \gamma. For γ=π/2\gamma = \pi / 2 the formula is the Pythagorean theorem. The formula lets you find the third side from two sides and the angle between them. It also gives each angle from the three sides. Mathlib states the formula for all points, with its convention for the angle at a repeated point.

History

Book II of Euclid's Elements contains the statement in the language of areas of rectangles. Propositions 12 and 13 of Book II treat the obtuse and the acute case.

Reading path

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Angle at a point, concept, no evidence levelAngle at a pointconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptTriangle, concept, no evidence levelTriangleconceptLaw of cosines, statement, no evidence levelLaw of cosinesstatement
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  1. Angle at a point concept
  2. Euclidean distance concept
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  4. Law of cosines statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_cosines, lead section
About this page
Tag060ZVZBV3E
URIm2://060ZVZBV3E
Aliasescosine-rule
Kindstatement, theorem
WritingW1 stated
MSC51M04
StatusThis copy of the site has no build data. Run m2 status 060ZVZBV3E.
UsesTriangle, Angle at a point, Euclidean distance
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