Law of Cosines

The Law of Cosines is a formula used in geometry to calculate the measure of an angle of a triangle or the length of one of its sides. It is an extension of the Pythagorean Theorem, which only applies to right triangles, to any kind of triangle.

Statement of the Law

In any triangle, with sides of lengths a, b, and c, and the angle opposite side c being γ (gamma), the Law of Cosines states that:

c² = a² + b² - 2ab cos(γ)

By rearranging the sides and angles, the Law of Cosines can also be written to solve for the other sides or angles:

a² = b² + c² - 2bc cos(α)

b² = a² + c² - 2ac cos(β)

Where α (alpha) is the angle opposite side a, and β (beta) is the angle opposite side b.

Usage

The Law of Cosines is primarily used when:

• You know the measures of all three sides of the triangle (SSS), and you want to find one of the angles.
• You know the measure of two sides and the included angle (SAS), and you want to find the third side.

Proof

The proof of the Law of Cosines can be derived by drawing an altitude in the triangle and then using the Pythagorean Theorem in each of the resulting right triangles, leading to the Law of Cosines.

Applications

The Law of Cosines has a wide range of applications in various fields such as physics, engineering, computer graphics, and navigation. It's also used extensively in trigonometry, calculus, and analytic geometry.