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 plane triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines says

where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c.

Some schools also describe the notation as follows:

Where C represents the same as γ and the rest of the parameters are the same.

The formula above could also be represented in other form:

The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90° or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem:

The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known.

By changing which sides of the triangle play the roles of a, b, and c in the original formula, one discovers that the following two formulas also state the law of cosines:

Though the notion of the cosine was not yet developed in his time, Euclid's Elements, dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines. The case of obtuse triangle and acute triangle (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions 12 and 13 of Book 2. Trigonometric functions and algebra (in particular negative numbers) being absent in Euclid's time, the statement has a more geometric flavor:

Proposition 12
In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle. —Euclid's Elements, translation by Thomas L. Heath.

Using notation as in Fig. 2, Euclid's statement can be represented by the formula

This formula may be transformed into the law of cosines by noting that CH = (CB) cos(π − γ) = −(CB) cos γ. Proposition 13 contains an entirely analogous statement for acute triangles.

The theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form.