Combine a sinθ + b cosθ into one trigonometric function r sin(θ+φ). This is essential for finding maximum and minimum values and solving equations.
Combination Formula
a sinθ + b cosθ = r sin(θ + φ)
where r = √(a²+b²), cos φ = a/r, sin φ = b/r
Example ①
Combine sinθ + √3 cosθ into one trigonometric function.
Answer:r = √(1+3) = 2。cos φ = 1/2、sin φ = √3/2 → φ = π/3。
sinθ + √3 cosθ = 2sin(θ + π/3)
Combine sinθ + √3 cosθ into one trigonometric function.
Answer:r = √(1+3) = 2。cos φ = 1/2、sin φ = √3/2 → φ = π/3。
sinθ + √3 cosθ = 2sin(θ + π/3)
Example ②
Combine √3 sinθ − cosθ and find its maximum and minimum values.
Answer:r = 2。cos φ = √3/2、sin φ = −1/2 → φ = −π/6。
√3 sinθ − cosθ = 2sin(θ − π/6)。Maximum value 2, minimum value −2
Combine √3 sinθ − cosθ and find its maximum and minimum values.
Answer:r = 2。cos φ = √3/2、sin φ = −1/2 → φ = −π/6。
√3 sinθ − cosθ = 2sin(θ − π/6)。Maximum value 2, minimum value −2
Key Point
- r = √(a²+b²) is always positive
- Determine the quadrant of φ using both cos φ = a/r and sin φ = b/r
- After combination, the maximum is r and the minimum is −r
Practice Problems
- Combine sinθ − cosθ and find its maximum and minimum values.
- Find the maximum value of 2sinθ + 2√3 cosθ.
- Solve sinθ + cosθ ≥ 1（0 ≤ θ ≤ 2π）.
Answers and Explanations
- Answer:√2 sin(θ−π/4)。Maximum value √2, minimum value −√2
Explanation: r = √2、φ = −π/4（cos φ = 1/√2、sin φ = −1/√2）。 - Answer:4
Explanation: r = √(4+12) = 4。 - Answer:0 ≤ θ ≤ π/2
Explanation: √2 sin(θ+π/4) ≥ 1 → sin(θ+π/4) ≥ 1/√2。θ+π/4 ∈ [π/4, 9π/4] and the range where sin ≥ √2/2 is [π/4, 3π/4]。Therefore, θ ∈ [0, π/2]。