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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

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)

triangle

cosine formula
cosine rule

loi des cosinus

French

URI

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

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