What is the proof of the law of cosines?

What is the proof of the law of cosines?

The law of cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle. To prove it, we can use the coordinate system.

Can you explain how to set up the triangle in the coordinate system?

Sure! Place one vertex of the triangle at the origin (0,0), the second vertex at (c, 0), and the third vertex at (a, b). This way, you can easily apply the distance formula to find the lengths of the sides.

What are the lengths of the sides in terms of a, b, and c?

The lengths of the sides can be expressed as follows: the side opposite the angle C is 'a', opposite angle A is 'b', and opposite angle B is 'c'. Using the distance formula, you can derive the relationships.

Can you show me the full derivation?

Absolutely! Starting with the coordinates, you can calculate the lengths of the sides using the distance formula, then substitute these into the law of cosines formula to derive the proof step by step.