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
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.)
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:




See also
Outside Links
Several derivations of the Cosine Law, including Euclid's