The exact value of $\cos 0^\circ$ is $1$.

In trigonometry, the cosine function establishes a relationship between the angle of a right-angled triangle and the ratio of the length of the adjacent side to the hypotenuse. When the angle is $0$ degrees, the lengths of the adjacent side and the hypotenuse are effectively the same, as this angle represents the minimum possible value. Consequently, the ratio of the adjacent side to the hypotenuse is $1$, leading to the conclusion that $\cos 0^\circ = 1$.

To gain a deeper understanding, consider the unit circle, which is defined as a circle with a radius of $1$, centered at the origin of a coordinate plane. In this context, the cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle. For an angle of $0$ degrees, this intersection occurs at the point $(1, 0)$, making the x-coordinate $1$. Thus, we reaffirm that $\cos 0^\circ = 1$.

This principle is fundamental in trigonometry and is typically one of the first values learned. It serves various practical applications, including solving trigonometric equations, analyzing wave functions in physics, and even in computer graphics. Retaining the fact that $\cos 0^\circ$ equals $1$ can significantly aid in quickly resolving problems and grasping more complex trigonometric concepts.