Explanation

From 8sin θ − cos θ = 4 we get cos θ = 8sin θ − 4. Using cos²θ + sin²θ = 1 gives 65sin²θ − 64sin θ + 15 = 0, whose roots are sin θ = 3/5 and sin θ = 5/13. The root 5/13 gives cos θ = −12/13 < 0, placing θ outside (π/6, π/3); the root 3/5 gives cos θ = 4/5 > 0 with θ ≈ 36.9° in range. Hence sin θ = 3/5, option A.