What is the exact value of cos 30?

The exact value of cos(30°) is √(3) / 2.

We can find this using the cosine half-angle identity property which is as follows:

• If an angle, x/2, lies in the first or fourth quadrant, then cos(x/2) = √((1 + cos(x)) / 2).
• If an angle, x/2, lies in the second or third quadrant, then cos(x/2) = -√((1 + cos(x)) / 2).

There are a few facts that will help us to see that we can use the cosine half-angle identity to evaluate cos(30°) exactly.

1. Notice that 30° = (60/2)°, so cos(30°) = cos((60/2)°)
2. The angle 30° lies in the first quadrant on a graph.
3. cos(60°) = 1/2

Therefore, we can use the first instance of the cosine half-angle identity with x = 60 to evaluate cos(30°) exactly.

• cos(30°)
• = cos((60/2)°)
• = √((1 + cos(60°)) / 2
• = √((1 + 1/2)) / 2)
• = √((3/2) / 2))
• = √(3/4)
• = √(3) / √(4)
• = √(3) / 2

Using this identity gives that the exact value of cos(30°) is √(3) / 2.