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
- c2 = a2 + b2 - 2ab cos C
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
- c2 = a2 + b2
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.)
Derivation
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:
- c2 = (a sin C)2 + (b - a cos C)2
- c2 = a2sin2C + b2 - 2ab cos C + a2cos2C
- c2 = a2(sin2C + cos2C) + b2 - 2ab cos C
- c2 = a2 + b2 - 2ab cos C
See also
