This article is about the trigonometric identity. For the cosine law of optics, see Lambert's cosine law.
Fig. 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.
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 and opposite respective angles and (see Fig. 1), the law of cosines states:
The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then and the law of cosines reduces to
The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.
trigonometry, the lawofcosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosineof one of its angles...
spherical trigonometry, the lawofcosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous...
the "lawofcosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar lawofcosines from...
and angles in scalene triangles, with the other being the law of cosines. The lawof sines can be generalized to higher dimensions on surfaces with constant...
+\beta )}}.} The lawof tangents, although not as commonly known as the lawof sines or the lawofcosines, is equivalent to the lawof sines, and can be...
{\displaystyle x} and y {\displaystyle y} , sine and cosine can be expressed in terms of real sines, cosines, and hyperbolic functions as sin z = sin x...
accessible enclosed distance. The lawofcosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem: c 2 = a...
written using cosines (sometimes called the spherical lawofcosines, not to be confused with the lawofcosines for plane geometry) instead of haversines...
{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.} The lawofcosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary...
{b}{R}}} where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic lawofcosines that applies to all hyperbolic triangles:...
\\\end{aligned}}} which is the lawofcosines. There are two ternary operations involving dot product and cross product. The scalar triple product of three vectors is...
equal either 30° or 150°. Using the lawofcosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its...
law. In the parallelogram on the right, let AD = BC = a, AB = DC = b, ∠ B A D = α . {\displaystyle \angle BAD=\alpha .} By using the lawofcosines in...
two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on Spherical lawofcosines gives four...
polarization of an algebraic form. Inner product space – Generalization of the dot product; used to define Hilbert spaces Lawofcosines – Property of all triangles...
)\sin(\angle \mathrm {BPC} )}}} This can be deduced from Spherical lawofcosines Atropisomer "Angle Between Two Planes". TutorVista.com. Archived from...
tetrahedron adjacent to the edge PiPj. The lawofcosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral angles...
tetrahedra is 5-dimensional. See: Lawof sines The lawofcosines for the tetrahedron relates the areas of each face of the tetrahedron and the dihedral...
by the lawofcosines and lawof sines (also called the cosine rule and sine rule). The triangle inequality states that the sum of the lengths of any two...
+\cot \beta ={\frac {c}{h_{c}}}.} The lawofcosines can be expressed in terms of the cotangent instead of the cosine, which brings the triangle's area S...
function Laws for solution of triangles: Lawofcosines Spherical lawofcosinesLawof sines Lawof tangents Lawof cotangents Mollweide's formula List of integrals...
to the relation of Pythagorean triples to the Gaussian integers. Triangles with an angle of 60° are a special case of the LawofCosines: c 2 = a 2 − a...
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![{\displaystyle {\begin{aligned}c^{2}&=a^{2}+b^{2}-2ab\cos \gamma ,\\[3mu]a^{2}&=b^{2}+c^{2}-2bc\cos \alpha ,\\[3mu]b^{2}&=a^{2}+c^{2}-2ac\cos \beta .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd9a9823b7eec03c9da379cfba653a92126cb4ff)
