The Law of Cosines

The Law of Cosines is a fundamental result in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It serves as a generalization of the Pythagorean Theorem, enabling calculations in non-right triangles where the simple relationships of right-angled geometry no longer apply. The formula is especially useful in scenarios where direct measurement of one side or angle is not feasible, such as in surveying, navigation, and engineering applications.

For any triangle with sides a, b, and c, and corresponding opposite angles A, B, and C, the Law of Cosines is given by:

Similar expressions can be written for the other sides:

This law is applicable in two primary situations: when two sides and the included angle are known, or when all three sides are known and the included angle is to be determined.

In addition to land surveying, the Law of Cosines is applied in astronomy to calculate distances between celestial bodies forming triangles from Earth's perspective. It is also used in robotics to determine joint angles in arm mechanisms and in physics to resolve forces acting at angles.

For example, consider a triangle where the sides a=7 units, b=10 units, and the included angle C=60^o. Using the Law of Cosines:

This shows how the Law of Cosines enables precise distance calculations even in complex geometries.