Math 2.0

Geometry Triangles and circles Theorem

Law of sines

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 α=∠BAC\alpha = \angle BAC, β=∠CBA\beta = \angle CBA and γ=∠ACB\gamma = \angle ACB be its angles. Then asin⁡α=bsin⁡β=csin⁡γ.\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}.

Remarks

The common value of the three quotients is 2R2R, where RR is the radius of the circumcircle of the triangle. The angles of a triangle are in (0,π)(0, \pi), so their sines are positive and the quotients exist. From two angles and one side, the law of sines gives the other sides. From two sides and an angle that is not between them, it can give two different triangles. This is the ambiguous case.

History

Ptolemy knew an equivalent form, with chords in place of sines, in the 2nd century.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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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 sines, statement, no evidence levelLaw of sinesstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Angle at a point concept
  2. Euclidean distance concept
  3. Triangle concept
  4. Law of sines statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_sines, lead section
About this page
Tag0PASQ34GHG
URIm2://0PASQ34GHG
Aliasessine-rule
Kindstatement, theorem
WritingW1 stated
MSC51M04
StatusThis copy of the site has no build data. Run m2 status 0PASQ34GHG.
UsesTriangle, Angle at a point, Euclidean distance
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