Trigonometry/Law of Cosines

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Law of Cosines[edit | edit source]


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 | edit source]

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 | edit source]

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

Proof[edit | edit source]


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 .

Applications[edit | edit source]

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 | edit source]

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