# Law of cosines

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

The law of cosines also shows that

*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 |

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:

## See also

## External link