If cot²θ − 3√3 cot θ + 6 = 0, where π/6 ≤ θ < π/2, then what is a value of sin θ + cos 2θ?

Let u = cot θ. Then u² − 3√3 u + 6 = 0. Using quadratic formula: u = [3√3 ± √(27 − 24)]/2 = [3√3 ± √3]/2. So u = 2√3 or u = √3. If cot θ = √3, then θ = π/6, sin θ = 1/2, cos 2θ = cos(π/3) = 1/2, sum = 1. If cot θ = 2√3, then tan θ = 1/(2√3), and sin θ + cos 2θ would be a different value. The value 1 is achieved at θ = π/6.