law of cosines

Q: What is the law of cosines?

The law of cosines, also known as the cosine formula or cosine rule, is a generalization of the Pythagorean theorem . It relates the lengths of the sides of any triangle: c 2 = a 2 + b 2 − 2 a b c o s C , where C is the angle opposite side c . When two sides and the angle between them (SAS) or all three sides (SSS) are known, the law of cosines can be used to find an unknown side or angle, respectively, in these cases.

Q: What is the spherical law of cosines?

In spherical trigonometry , the spherical law of cosines relates angular distances on a sphere: cos c = cos a cos b + sin a sin b cos C . This identity is used in navigation and geodesy for calculating great-circle distances.

Q: Who contributed to the development of the law of cosines?

Greek mathematician Euclid and Persian mathematician Jamshīd al-Kāshī contributed to the development of the law of cosines. Euclid’s work laid the geometric basis, while al-Kāshī provided a statement of the law in the 15th century.

mathematics

Also known as: cosines, law of

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Anoushka Pant

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Last updated

Oct. 9, 2026 •History

What is the law of cosines?

What is the spherical law of cosines?

Who contributed to the development of the law of cosines?

law of cosines, generalization of the Pythagorean theorem relating the lengths of the sides of any triangle. If a, b, and c are the lengths of the sides and C is the angle opposite side c, then $c^{2} = a^{2} + b^{2} - 2 a b c o s C$ .When two sides and the angle between them (side–angle–side, or SAS) or all three sides (side–side–side, or SSS) are known, the law of cosines can be used to find an unknown side or angle, respectively, in these cases.

History

The geometric basis of the law of cosines can be found in the work of Greek mathematician Euclid. His work Elements, written c. 300 bce, presents propositions that are equivalent to the law, addressing cases for both obtuse and acute triangles. Though the concept of the cosine function had not yet been developed, these relationships were expressed using geometric constructions rather than trigonometric notation. In the 15th century, Persian mathematician Jamshīd al-Kāshī provided a statement of the law and demonstrated its use in triangulation. His version gained recognition and in France, the law is referred to as Le théorème d’Al-Kashi (French: “Al-Kashi’s Theorem”).

Variant: Spherical law of cosines

In spherical trigonometry, the law of cosines relates the angular distances between points on the surface of a sphere: $c o s c = c o s a c o s b + s i n a s i n b c o s C$ .Here, a, b, and c are the spherical side lengths of the triangle, measured by the angles they subtend at the center, and C is the angle between sides a and b. This identity is used in navigation and geodesy to account for the curvature of a spherical surface when calculating the great-circle distance—the shortest path between two points on a spherical surface, such as Earth.