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 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...
The and the formula reduces to the usual Pythagorean theorem.
For any triangle with angles A, B, and C and corresponding opposite side lengths a, b, and c, the Law of Cosines states that
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 b2 = AO2 + h2.
The corresponding expressions for a and c can be proved similarly.
The formula can be rearranged:
and similarly for cos(A) and cos(B).
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.