We have, f(θ) = (3sin(θ) – 4cos(θ) – 10)
(3sin(θ) + 4cos(θ) – 10) = (9sin2(θ) – 16cos2(θ)) – 10(3sinθ + 4cosθ) – 10(3sinθ – 4cosθ)
= (9sin2(θ) – 16cos2(θ))– 10(3sinθ + 4cosθ + 3sinθ – 4cosθ)
= (9sin2(θ) – 16cos2(θ)) – 60sin(θ)
= 25sin2θ – 60sin(θ) – 16
= (5 sinθ – 6)2 – 36 – 16
= (5 sinθ – 6)2 – 52
Hence, the minimum value of f(θ) = 121 – 52 = 69