Final answer:
The tangent of 157.5° is found using the half-angle formula, which is tan(157.5°) = √(2 - √2). Since 157.5° is half of 315°, we use the cosine of 315° in our calculation to get the final result.
Step-by-step explanation:
To find tan 157.5° using the half-angle formula, we can start by identifying an angle whose tangent is known and can be doubled to get 157.5°. We know that 157.5° is half of 315°. The half-angle formula for tangent is:
tan(θ/2) = ± √((1 - cos(θ)) / (1 + cos(θ)))
Since we are working with 157.5°, we consider θ = 315°. The cosine of 315° is √2/2 and the tangent has to be positive because 157.5° lies in the second quadrant where sine is positive, and tangent is the ratio of sine to cosine:
tan(157.5°) = √((1 - √2/2) / (1 + √2/2))
We simplify under the square root:
tan(157.5°) = √((2 - √2) / (2 + √2)) √((2 - √2) / (2 - √2))
This rationalizes the denominator, leading to:
tan(157.5°) = √((4 - 2√2) / (2))
Finally, we divide by 2 under the square root to get:
tan(157.5°) = √(2 - √2)
Therefore, using the half-angle formula, tan 157.5° is √(2 - √2).