Trigonometry/Law of Cosines

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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.

PermutationsEdit

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

 
 
 

ProofEdit

 

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   .

ApplicationsEdit

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.

NotesEdit