How do you find the exact value of cos 60°?

The exact value of cos 60° is 1/2.

To understand why cos 60° equals 1/2, we can use the properties of special triangles, specifically the 30-60-90 triangle. A 30-60-90 triangle is a right-angled triangle where the angles are 30°, 60°, and 90°. The sides of this triangle have a specific ratio: the side opposite the 30° angle is the shortest and is half the length of the hypotenuse, the side opposite the 60° angle is √3 times the shortest side, and the hypotenuse is the longest side.

Let's consider a 30-60-90 triangle where the shortest side (opposite the 30° angle) is 1 unit. This makes the hypotenuse 2 units (since it is twice the shortest side), and the side opposite the 60° angle is √3 units.

The cosine of an angle in a right-angled triangle is defined as the length of the adjacent side divided by the length of the hypotenuse. For the 60° angle in our 30-60-90 triangle, the adjacent side is the shortest side (1 unit), and the hypotenuse is 2 units. Therefore, cos 60° = adjacent/hypotenuse = 1/2.

This relationship holds true regardless of the size of the 30-60-90 triangle, as the ratios between the sides remain constant. Thus, the exact value of cos 60° is always 1/2.