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O-Level Trigonometric functions, identities and equations: worked solution
2 marks. Full working, one step per line.
Question
Hence find the least value of 1/((3cosθ + 5sinθ)² + 1) for 0 ≤ θ ≤ π/2.
Worked answer
Write 3 cos theta + 5 sin theta = R cos(theta - alpha) with R = sqrt(3² + 5²) = sqrt(34). Its maximum value is sqrt(34), attained at alpha = arctan(5/3) approx 59 degrees, which lies in [0, pi/2]. The fraction 1/((3cos theta + 5 sin theta)² + 1) is least when the denominator is greatest, i.e. when (3 cos theta + 5 sin theta)² is greatest = 34. Then the denominator = 34 + 1 = 35, so the least value = 1/35.
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This question is part of O-Level Trigonometric functions, identities and equations, in O-Level Additional Maths (A-Maths).
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