Algebra and Trigonometry, 5th Edition

8.2 The Law of Cosines

• Use the law of cosines to solve triangles.

• Determine whether the law of sines or the law of cosines should be applied to solve a triangle.

The law of sines is used to solve triangles given a side and two angles (AAS and ASA) or given two sides and an angle opposite one of them (SSA). A second law, called the law of cosines, is needed to solve triangles given two sides and the included angle (SAS) or given three sides (SSS).

The Law of Cosines

To derive this property, we consider any $\mathrm{\Delta }ABC$ placed on a coordinate system. We position the origin at one of the vertices—say, C—and the positive half of the x-axis along one of the sides—say, side CB. Let $\left(x, y\right)$ be the coordinates of vertex A. Point B has coordinates $(a,$