The exact value of $\cos 30^\circ$ is given by:

To comprehend why this is true, we can refer to a specific type of right triangle known as the 30-60-90 triangle. This triangle features angles measuring $30^\circ$, $60^\circ$, and $90^\circ$, and the lengths of its sides are in a distinct ratio. If we designate the shortest side (the one opposite the $30^\circ$ angle) as $1$ unit, then the hypotenuse (the side opposite the $90^\circ$ angle) measures $2$ units, and the longer side (the side opposite the $60^\circ$ angle) is $\sqrt{3}$ units long.

In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to that of the hypotenuse. For the $30^\circ$ angle in our 30-60-90 triangle, the adjacent side is the one opposite the $60^\circ$ angle, which has a length of $\sqrt{3}$ units, while the hypotenuse measures $2$ units.

Thus, we can express the cosine of $30^\circ$ as follows:

This value is exact and frequently utilized in various trigonometry problems. Familiarizing yourself with the properties of the 30-60-90 triangle can be extremely beneficial for quickly determining the cosine, sine, and tangent of these special angles.