Law of cosines

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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, c be the sides of the triangle and A, B, C the angles opposite those sides. Then

This formula is useful to compute the third side of a triangle when two sides and the enclosed angle are known, and to compute the angles of a triangle if all three sides are known.

The law of cosines also shows that

iff cos C = 0 (since a, b > 0), which is equivalent to C being a right angle. (In other words, this is the Pythagorean Theorem and its converse.)

Table of contents

1 Derivation (for acute angles)
2 Law of cosines using vectors
3 See also
4 External link

Derivation (for acute angles)

Let a, b, c be the sides of the triangle and A, B, C the angles opposite those sides. Draw a line from angle B that makes a right angle with the opposite side, b. The length of this line is a sin C, and 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, b - a cos C and hypotenuse c. Therefore, according to the Pythagorean Theorem:

Law of cosines using vectors

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

Since

Using the dot product, we simplify this into:

External link

Several derivations of the Cosine Law, including Euclid's