Trigonometry/Law of Cosines< Trigonometry
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:
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...
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.