# Trigonometry/Law of Cosines

## Contents

## Law of Cosines[edit]

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 θ is the angle between sides *a* and *b*.

### 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.

## Permutations[edit]

For any triangle with angles A, B, and C and corresponding opposite side lengths *a*, *b*, and *c*, the Law of Cosines states that

### Proof[edit]

Dropping a perpendicular OC from vertex C to intersect AB (or AB extended) at O splits this triangle into two right-angled triangles AOC and BOC, with altitude *h* from side c.

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

*h*=*a*sin*B*

Side c is split into two segments, with total length *c*.

- OB has length
*a*cos*B* - AO has length
*c*-*a*cos*B*

Now we can use the Pythagorean Theorem to find *b*, since *b*^{2} = *AO*^{2} + *h*^{2}.

The corresponding expressions for *a* and *c* can be proved similarly.

The formula can be rearranged:

and similarly for cos(A) and cos(B).

## Applications[edit]

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.

## Notes[edit]

- ↑ Lawrence S. Leff (2005-05-01). cited work. Barron's Educational Series. p. 326. ISBN 0764128922. http://books.google.com/?id=y_7yrqrHTb4C&pg=PA326.