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geometry > spherical geometry > spherical trigonometry > law of cosines

geometry > Euclidean geometry > trigonometry > law of cosines

geometry > non-Euclidean geometry > hyperbolic geometry > hyperbolic triangle > law of cosines

Preferred term

law of cosines

Definition(s)

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides ${\\displaystyle a,}$ ${\\displaystyle b,}$ and ${\\displaystyle c,}$ opposite respective angles ${\\displaystyle \\alpha ,}$ ${\\displaystyle \eta ,}$ and ${\\displaystyle \\gamma }$ (see Fig. 1), the law of cosines states:

${\\displaystyle {\egin{aligned}c^{2}&=a^{2}+b^{2}-2ab\\cos \\gamma ,\\\\a^{2}&=b^{2}+c^{2}-2bc\\cos \\alpha ,\\\^{2}&=a^{2}+c^{2}-2ac\\cos \eta .\\end{aligned}}}$

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if ${\\displaystyle \\gamma }$ is a right angle then ${\\displaystyle \\cos \\gamma =0,}$ and the law of cosines reduces to ${\\displaystyle c^{2}=a^{2}+b^{2}.}$
The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Law_of_cosines)

Broader concept(s)

Related concept(s)

triangle

Synonym(s)

cosine formula
cosine rule

In other languages

loi des cosinus

French

URI

http://data.loterre.fr/ark:/67375/PSR-X5T7H86P-S

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