The exact value of $\cos 60^\circ$ is $\frac{1}{2}$.

In trigonometry, the cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For the specific angle of $60^\circ$, this ratio simplifies to $\frac{1}{2}$. This fundamental value is commonly encountered in GCSE Mathematics.

To understand why $\cos 60^\circ$ is equal to $\frac{1}{2}$, consider an equilateral triangle where all angles measure $60^\circ$ and all sides are of equal length. If you draw a line from one vertex to the midpoint of the opposite side, you create two right-angled triangles. Each of these right-angled triangles will have angles measuring $30^\circ$, $60^\circ$, and $90^\circ$. The hypotenuse of each right-angled triangle corresponds to the side of the equilateral triangle, while the length of the adjacent side to the $60^\circ$ angle is half the length of the hypotenuse. Thus, the cosine of $60^\circ$ is determined by the ratio of the adjacent side (which is half the hypotenuse) to the hypotenuse itself, resulting in the value of $\frac{1}{2}$.

This value is part of a set of standard trigonometric values that you should memorize, as they frequently appear on exams and are essential for solving various problems involving right-angled triangles. Remembering that $\cos 60^\circ$ equals $\frac{1}{2}$ can facilitate quick solutions to trigonometric equations and enhance your understanding of triangle properties.