Trigonometry/Law of Cosines

Law of CosinesEdit


The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:[1]


where   is the angle between sides   and   .

Does the formula make sense?Edit

This formula had better agree with the Pythagorean Theorem when   .

So try it...

When   ,  

The   and the formula reduces to the usual Pythagorean theorem.


For any triangle with angles   and corresponding opposite side lengths   , the Law of Cosines states that




Dropping a perpendicular   from vertex   to intersect   (or   extended) at   splits this triangle into two right-angled triangles   and   , with altitude   from side   .

First we will find the lengths of the other two sides of triangle   in terms of known quantities, using triangle   .


Side   is split into two segments, with total length   .

  has length  
  has length  

Now we can use the Pythagorean Theorem to find   , since   .


The corresponding expressions for   and   can be proved similarly.

The formula can be rearranged:


and similarly for   and   .


This formula can be used to find the third side of a triangle if the other two sides and the angle between them are known. The rearranged formula can be used to find the angles of a triangle if all three sides are known. See Solving Triangles Given SAS.


  1. Lawrence S. Leff (2005-05-01). cited work. Barron's Educational Series. p. 326. ISBN 0764128922.