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...

spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous...

{b}{R}}} where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:...

the cosine of the angle θ between the observer's line of sight and the surface normal; I = I0 cos θ. The law is also known as the cosine emission law or...

the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from...

{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}}.} The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary...

being the law of cosines. The law of sines can be generalized to higher dimensions on surfaces with constant curvature. With the side of length a as the...

written using cosines (sometimes called the spherical law of cosines, not to be confused with the law of cosines for plane geometry) instead of haversines...

accessible enclosed distance. The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem: c 2 = a...

{\displaystyle \gamma } ⁠ the angles opposite those sides. Applying the law of cosines we get cos ⁡ γ = a 2 + b 2 − c 2 2 a b {\displaystyle \cos \gamma ={\frac...

+\beta )}}.} The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be...

\\\end{aligned}}} which is the law of cosines. There are two ternary operations involving dot product and cross product. The scalar triple product of three vectors is...

law of cosines is sometimes referred to as the théorème d'Al-Kashi. In discussing decimal fractions, Struik states that (p. 7): "The introduction of decimal...

)\sin(\angle \mathrm {BPC} )}}} This can be deduced from the spherical law of cosines, but can also be found by other means. In m-dimensional Euclidean space...

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 law of cosines gives four...

Book I of Euclid's Elements. The theorem is an immediate corollary of the law of cosines. For two triangles with sides { a , b , c } {\displaystyle \{a,b...

equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its...

consequence of – another existing theorem. The 'Porism' can be viewed on pages 36 and 37 of DROC (Harvard electronic copy) "Sine, Cosine, and Ptolemy's...

faces of the tetrahedron adjacent to the edge P i P j {\displaystyle P_{i}P_{j}} . The law of cosines for a tetrahedron, which relates the areas of the...

tetrahedra is 5-dimensional. See: Law of sines The law of cosines for the tetrahedron relates the areas of each face of the tetrahedron and the dihedral...

the polarization of an algebraic form. Inner product space – Vector space with generalized dot product Law of cosines – Generalization of Pythagorean theorem...

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 law of cosines in...

proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines. Let the triangle have sides a , b , c {\displaystyle...

the Earth, the cosine of the central angle is near 0.99999999). For modern 64-bit floating-point numbers, the spherical law of cosines formula, given...

circumradius. The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known. The law states, a 2 + b...

the law of cosines for sides reduces to the planar law of cosines and two of Napier's analogies reduce to Mollweide's formulas above. But the law of cosines...

specific trigonometric laws or formulas. For instance, propositions twelve and thirteen of book two of the Elements are the laws of cosines for obtuse and acute...

triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without...

)^{3}{\textrm {d}}\theta .} Here s(θ) can be found in terms of q and θ using the Law of cosines. The steps needed to evaluate the integral, together with...

angle γ (gamma), where the adjacent leg equals 0, the cosine of γ also equals 0. The law of cosines formulates that c 2 = a 2 + b 2 − 2 a b cos ⁡ θ {\displaystyle...