The maximum value of 3cosθ + 5sin(θ - π/6) for any real value of θ is:

y = 3cosθ + 5sin(θ - π/6)
y = 3cosθ + 5(sinθcos(π/6) - cosθsin(π/6))
y = 3cosθ + 5(sinθ(√3/2) - cosθ(1/2))
y = 3cosθ + (5√3/2)sinθ - (5/2)cosθ
y = (1/2)cosθ + (5√3/2)sinθ
Let Rcos(θ - α) = (1/2)cosθ + (5√3/2)sinθ
Rcosα = 1/2
Rsinα = 5√3/2
tanα = 5√3
R² = (1/2)² + (5√3/2)² = 1/4 + 75/4 = 76/4 = 19
R = √19
Therefore, y = √19cos(θ - α)
The maximum value of y is √19.