Law of cosines

Math

Definition

A formula that relates the lengths of the sides of any triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem to non-right triangles and is used when you know two sides and the included angle (SAS) or all three sides (SSS).

Examples

  • Finding the third side of a triangle when two sides are 7 and 10 with an included angle of 60°
  • Determining an angle when all three side lengths are known
  • Calculating the distance between two ships given their distances from a lighthouse and the angle between them
Key Fact

c² = a² + b² − 2ab·cos(C)

Related Terms

Law of sines Pythagorean theorem Solving right triangles Ambiguous case

Study This Concept

Practice law of cosines with free review games in these units:

Trigonometry Unit 7: Law of Sines and Cosines