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.)

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## 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: