Law of cosines

The law of cosines relates a triangle’s side lengths to the cosine of one of its angles. It generalizes the Pythagorean theorem to non-right triangles.

Pythagorean theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. When the included angle is 90°, the law reduces exactly to this theorem.
Triangulation: A method for locating a point by measuring directions or distances from known positions. Known baselines and angles can determine unknown distances through the law.
Cosine: A trigonometric function that gives the ratio of a right triangle’s adjacent side to its hypotenuse. The law uses cosine to encode the effect of an included angle.
Law of sines: A triangle relation stating that each side divided by the sine of its opposite angle has the same value. It is often more direct when an opposite side-angle pair is known.
Dot product: An operation on vectors that equals the product of their magnitudes and the cosine of the angle between them. Expanding a squared vector difference with the dot product yields the law’s cosine term.
Oblique triangle: A triangle that contains no right angle. The law directly handles these triangles, where right-triangle formulas do not suffice.
Triangle: A polygon with three sides, three vertices, and three interior angles. The law applies to the side-angle relationships of any Euclidean triangle.
Right triangle trigonometry: The use of trigonometric ratios to relate angles and sides in right triangles. Its basic ratios require a right angle, while the cosine law does not.
Triangle inequality: A rule stating that the sum of any two sides of a triangle exceeds the third side. The law’s side lengths must still satisfy these bounds to form a triangle.
Side-angle-side: A triangle-congruence condition in which two sides and their included angle are specified. These measurements determine the opposite side using the law.

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