What Is the Law of Sines and Cosines?

The law of sines and law of cosines are two different equations relating the measure of the angles of a triangle to the length of the sides. The laws apply to any triangle, not just right-angled triangles.

Draw a triangle. Label the angles A, B and C and the opposite sides a, b and c. The law of sines says that the sines of the angles are proportional to the lengths of the opposite sides. That means sin A/a = sinB/b = sinC/c.

The law of cosines says that for any triangle, the square of the length of any side equals the sum of the squares of the other two sides minus 2 times the product of those two sides times the cosine of the included angle. If seeking the length of side a, for example, use the formula a^2 = b^2 + c^2 – (2bc cosA), and then take the square root of both sides of the equation.