How do you find the exact value of cos 30°?

The exact value of cos 30° is \(\frac{\sqrt{3}}{2}\).

To understand why this is the case, we can use a special right-angled triangle known as the 30-60-90 triangle. This triangle has angles of 30°, 60°, and 90°, and its sides are in a specific ratio. If we take the shortest side (opposite the 30° angle) to be 1 unit, the hypotenuse (opposite the 90° angle) will be 2 units, and the longer side (opposite the 60° angle) will be \(\sqrt{3}\) units.

The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For the 30° angle in our 30-60-90 triangle, the adjacent side is the one opposite the 60° angle, which is \(\sqrt{3}\) units long, and the hypotenuse is 2 units long.

Therefore, the cosine of 30° is:
\[
\cos 30° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2}
\]

This value is exact and is often used in trigonometry problems. Remembering the properties of the 30-60-90 triangle can help you quickly find the cosine, sine, and tangent of these special angles.