Solving Oblique Triangles: The Law of Cosines

The Law of Cosines states the following:

a² = b² + c² -2bc cos(A)

Alternative versions look like this:

b² = a² + c² -2ac cos(B)

c² = a² + b² -2ab cos(C)

In the last two formulas, the parts are simply interchanged to make the law easier under our convention of using a, b, c, A, B, and C to label triangles. The Law of Cosines is only one formula, not three.

This law is used primarily in two situations: when two sides and their included angle are given, and when three sides are given.

If two sides and their included angle are given, the next thing to calculate is the third side. The Law of Cosines, as shown above, is perfect for the situation. After the third side is calculated, the Law of Sines can be used to calculate either of the other two angles.

If three sides are given, the Law of Cosines must be manipulated a bit: For this situation, the Law of Cosines is most useful in this form: cos(A) = . Once one of the angles is known, the next can be calculated using the Law of Sines, and the third using subtraction, knowing that the angles of a triangle sum to 180 degrees.