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

To understand why this is true, we can explore some concepts from trigonometry. In a right-angled 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 angle of $30^\circ$, we can utilize a special type of right-angled triangle known as the 30-60-90 triangle. This triangle features angles of $30^\circ$, $60^\circ$, and $90^\circ$, and its sides are in a specific ratio of $1 : \sqrt{3} : 2$.

In a 30-60-90 triangle, the side opposite the $30^\circ$ angle is half the length of the hypotenuse. If we assume the hypotenuse measures $2$ units, then the side opposite the $30^\circ$ angle (the shortest side) will measure $1$ unit, while the side adjacent to the $30^\circ$ angle (the longer leg) will measure $\sqrt{3}$ units.

Using the definition of cosine, we can express this relationship mathematically:

This value is exact and is derived from the geometric properties of the 30-60-90 triangle. Remembering this special triangle and its side ratios will enable you to quickly determine the cosine of $30^\circ$ and other related angles.