What is the expression for the Law of Cosines regarding side a?

The Law of Cosines is a fundamental relation in triangle geometry that connects the lengths of the sides of a triangle to the cosine of one of its angles. For any triangle with sides ( a ), ( b ), and ( c ) opposite to angles ( A ), ( B ), and ( C ) respectively, the equation specifically for side ( a ) is written as:

[ a^2 = b^2 + c^2 - 2bc \cos(A) ]

This formula reflects how the square of side ( a ) can be expressed in terms of the squares of the other two sides, ( b ) and ( c ), and includes a term that accounts for the angle opposite to side ( a ). The term ( -2bc \cos(A) ) serves to adjust for the angle via the cosine function, thus integrating both side lengths and the included angle into the relation.

This formulation is particularly useful in determining the length of a side when two sides and the included angle are known, allowing for many practical applications in various fields such as physics, engineering, and computer graphics where triangle properties are essential. The negative sign before the cosine term is crucial because it indicates that more acute angles