How can the Law of Cosines be expressed in relation to triangle sides?

The Law of Cosines is a fundamental relationship in triangle geometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, it can be expressed as ( c^2 = a^2 + b^2 - 2ab \cos(C) ), where ( c ) is the length of one side of the triangle, ( a ) and ( b ) are the lengths of the other two sides, and ( C ) is the angle opposite side ( c ).

This formula is particularly useful in various situations, such as determining a side length when the lengths of the other two sides and the included angle are known, or finding an angle when all three side lengths are known. The inclusion of the term ( -2ab \cos(C) ) adjusts for the angle ( C ), accounting for how the sides ( a ) and ( b ) interact based on their relative orientation—essentially capturing the impact of angle ( C ) on the relationship between the sides.

In contrast, the other options do not accurately describe the relationship defined by the Law of Cosines. The first option reflects the Pythagorean theorem, which only applies to right triangles. The