To find the exact value of tan(15°), we can use the half-angle formula for tangent. The half-angle formula is given by:
tan(θ/2) = √(1 – cos(θ)) / (1 + cos(θ))
In this case, we can consider 15° as half of 30°. Hence, let’s set θ = 30°. We need to find cos(30°), which is a known value:
cos(30°) = √3 / 2
Now, substituting this into the half-angle formula:
tan(15°) = √(1 – cos(30°)) / (1 + cos(30°))
Substituting cos(30°):
tan(15°) = √(1 – √3 / 2) / (1 + √3 / 2)
Simplifying this:
1 – √3 / 2 = 2/2 – √3/2 = (2 – √3) / 2.
1 + √3 / 2 = 2/2 + √3/2 = (2 + √3) / 2.
Including these into the formula gives:
tan(15°) = √((2 – √3) / 2) / ((2 + √3) / 2)
The 2’s cancel out, leading us to:
tan(15°) = √(2 – √3) / (2 + √3)
This is the exact value of tan(15°) expressed using the half-angle formula.