In trigonometry, the law of cosines is a statement about arbitrary triangles which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Then,

This formula is useful for computing the third side of a triangle when two sides and their enclosed angles are known, and in computing the angles of a triangle if all three sides are known.

The law of cosines also shows that

   if and only if   

The statement cos C = 0 implies that C is a right angle, since a and b are positive. In other words, this is the Pythagorean theorem and its converse. Although the law of cosines is a broader statement of the Pythagorean theorem, it isn't a proof of the Pythagorean theorem, because the law of cosines derivation given below depends on the Pythagorean theorem.

Derivation (for acute angles)

Triangle

Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Draw a line from angle B that makes a right angle with the opposite side b. If the length of that line is x, then sin C = x/a, which implies x = a sin C.

That is, the length of this line is a sin C. Similarly, the length of the part of b that connects the foot point of the new line and angle C is a cos C. The remaining length of b is b − a cos C. This makes two right triangles, one with legs a sin C and b − a cos C and hypotenuse c. Therefore, according to the Pythagorean theorem:

because

Law of cosines using vectors

Using vectors and vector dot products, we can easily prove the law of cosines. If we have a triangle with vertices A, B, and C whose sides are the vectors a, b, and c, we know that:

since

Using the dot product, we simplify this into

See also

• triangulation
• law of sines
• Lambert's cosine law

External link

• Several derivations of the Cosine Law, including Euclid's