2016 · Q78
If A = sin²θ + cos⁴θ where 0 ≤ θ ≤ π/2, then which is correct?
Elementary MathematicsTrigonometry
- A1 ≤ A ≤ 2
- B3/4 ≤ A ≤ 1
- C13/16 ≤ A ≤ 2
- D3/4≤A≤13/16
Explanation
Put u = sin²θ ∈ [0,1]: then A = u+(1−u)² = u²−u+1 = (u−1/2)²+3/4. The minimum is 3/4 at u = 1/2 and the maximum is 1 at the endpoints u = 0, 1. So 3/4 ≤ A ≤ 1 (option B).