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

To grasp why $\cos 60^\circ$ equals $\frac{1}{2}$, we can explore the properties of special triangles, notably the 30-60-90 triangle. This particular triangle is a right-angled triangle with angles measuring $30^\circ$, $60^\circ$, and $90^\circ$. The sides of a 30-60-90 triangle have a specific ratio: the side opposite the $30^\circ$ angle is the shortest and is half the length of the hypotenuse. The side opposite the $60^\circ$ angle measures $\sqrt{3}$ times the shortest side, while the hypotenuse is the longest side.

Let us consider a 30-60-90 triangle where the shortest side (opposite the $30^\circ$ angle) is set to $1$ unit. Consequently, the hypotenuse will be $2$ units, as it is twice the length of the shortest side. The side opposite the $60^\circ$ angle will then measure $\sqrt{3}$ units.

The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. For the $60^\circ$ angle in our 30-60-90 triangle, the adjacent side corresponds to the shortest side, which is $1$ unit, and the hypotenuse is $2$ units. Therefore, we can express this as:

This relationship is consistent across all 30-60-90 triangles, as the ratios between their sides remain unchanged. Thus, we can confidently state that the exact value of $\cos 60^\circ$ is always $\frac{1}{2}$.