The Law of Cosines

On this web page you can validate for yourself the Law of Cosines that you studied in Lesson 8.7 of Discovering Advanced Algebra: An Investigative Approach. You can think of the Law of Cosines as a modified Pythagorean relationship.


This sketch shows triangle ABC and the measurements of its angles and sides. Drag any of the vertices to change the triangle's size and shape, or press Make Right Triangle to change it to a right triangle. The values of c2, a2, b2, 2ab cos C, and a2 + b2 – 2ab cos C are calculated for you.

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  1. What do you observe about c2 and a2 + b2 – 2ab  cos C for any triangle? How does this validate the Law of Cosines (with respect to angle C)?
  2. What happens to the Law of Cosines when angle C is right?
  3. What can you say about 2ab cos C when angle C is acute? When angle C is obtuse?

If you'd like to algebraically derive the Law of Cosines, go to the Law of Cosines page for Discovering Geometry and look at the second sketch on the page.