The cosine rule, also known as the law of cosines, is a fundamental principle in trigonometry that connects the lengths of the sides of a triangle to the cosine of one of its angles. This formula is especially useful for solving triangles that are not right-angled. The general formula can be expressed as:

This formula can be rearranged in three different forms, depending on which angle and sides you are examining:

1. $c^2 = a^2 + b^2 - 2ab\cos(C)$
2. $a^2 = b^2 + c^2 - 2bc\cos(A)$
3. $b^2 = a^2 + c^2 - 2ac\cos(B)$

In these equations, $a$, $b$, and $c$ denote the lengths of the sides of the triangle, while $A$, $B$, and $C$ correspond to the angles opposite those sides. For instance, in the first formula, $c$ represents the side opposite angle $C$, while $a$ and $b$ are the other two sides.

The cosine rule is particularly advantageous when you know two sides of a triangle and the included angle (the angle formed between the two known sides) and need to calculate the length of the third side. Additionally, it can be employed to determine an angle if the lengths of all three sides are known. This versatility makes the cosine rule an essential tool for solving various triangle-related problems, especially in non-right-angled triangles where the Pythagorean theorem is not applicable.

Mastering the cosine rule can significantly improve your problem-solving abilities in geometry and trigonometry, simplifying your approach to a broad spectrum of questions, particularly in your GCSE Maths examinations.