To find the exact value of cos 36°, we can use the half-angle formula. The half-angle formula states that:
cos(θ/2) = √((1 + cos(θ)) / 2)
In this case, we take θ = 72°, since 36° is half of 72°. So, we need to find cos(72°) first.
We know that cos(72°) can be derived from the special angles in a pentagon or using the identity cos(72°) = sin(18°). To find cos(72°), we can also use:
cos(72°) = sin(90° – 72°) = sin(18°)
Now, using the value of sin(18°), which is known to be (√5 – 1)/4, we find:
cos(72°) = (√5 – 1)/4
Now, substituting this back into the half-angle formula:
cos(36°) = √((1 + cos(72°)) / 2)
Substituting the value of cos(72°):
cos(36°) = √((1 + (√5 – 1)/4) / 2)
Now simplifying this step by step:
cos(36°) = √((4/4 + (√5 – 1)/4) / 2)
= √(((4 + √5 – 1)/4) / 2)
= √(((3 + √5)/4) / 2)
= √(((3 + √5)/8))
Combining all of this, we find the exact value of cos(36°) as:
cos(36°) = √(5 + 1) / 4 = (√5 + 1) / 4
So, the exact value of cos 36° is (√5 + 1) / 4.