Difference between revisions of "The Law of Cosines"

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== Licensing ==  
 
== Licensing ==  
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
* [* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Cosines Law of Cosines, Wikibooks] under a CC BY-SA license
+
* [https://en.wikibooks.org/wiki/Trigonometry/Law_of_Cosines Law of Cosines, Wikibooks] under a CC BY-SA license

Latest revision as of 17:00, 28 October 2021

Law of Cosines

Law-of-cosines1.svg

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

where is the angle between sides and .

Does the formula make sense?

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

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

Proof

Law-of-cosines2.svg

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

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.


Resources

Licensing

Content obtained and/or adapted from: