Math Law of Cosines

The law of cosines is a common formula used in trigonometry, the mathematics of triangles. Cosines, along with sines and tangents, can be used to determine lengths and angles in any kind of triangle.

History

Though not always called cosines, the concept of cosines dates back to ancient times, including an appearance in Euclid of Alexandria's "Elements," written in the third century B.C. The law of cosine's current form came along with algebraic notation developed in the early 1800s.

Usage

The law of cosines can solve any triangle problem in which the length of the three sides and any angle are involved. The law can determine any angle if the length of all three sides are known, or the length of a side if the other two sides and the angle opposite the unknown side are known.

Formula

The law of cosines, using the illustrated triangle as a guide, states that c^2 = a^2 + b^2 - 2ab(cos C). The same formula applies to any side of the triangle, using the cosine of the angle opposite of the side as the cosine.

Finding the Cosine

The cosine of 0 degrees is 1, the cosine of 90 degrees is 0 and the cosine of 180 degrees is -1. Cosines of other numbers are long decimal figures between -1 and 1, so determining a cosine of an angle requires either a cosine table or use of a cosine function that is standard on scientific calculators.

Solving for an Angle

The formula needs to be rearranged slightly to solve for an angle: (cos C) = (a^2 + b^2 - c^2)/2ab. Use a cosine table or the inverse cosine button, usually labeled cos^-1, on a scientific calculator to determine the angle from the solution.

References

• Oswego City School District

Resources

• Table of Cosines