It is known that the function f (x) = sin (ω x + ϕ) (ω > 0, 0 ≤ ϕ≤ π) is even function, and the distance between two adjacent highest points on the image is 2 π. (I) find the analytic expression of F (x); (II) if α∈ (− π 3, π 2), f (α + π 3) = 13, find the value of sin (2 α + 5 π 3)

It is known that the function f (x) = sin (ω x + ϕ) (ω > 0, 0 ≤ ϕ≤ π) is even function, and the distance between two adjacent highest points on the image is 2 π. (I) find the analytic expression of F (x); (II) if α∈ (− π 3, π 2), f (α + π 3) = 13, find the value of sin (2 α + 5 π 3)

(I) ∵ if the distance between two adjacent highest points on an image is 2 π, ∵ t = 2 π, then ω = 2 π t = 1. ∵ f (x) = sin (x + ϕ) (2 points) ∵ f (x) is a even function, ∵ ϕ = k π + π 2 (K ∈ z), and 0 ≤ ϕ ≤ π, ϕ = π 2 F (x) = cosx. (5 points) (II) from the known cos (α + π 3) = 13, ∵ α ∈ (− π 3, π 2), ∵ α + π 3 ∈ (0, 5 π 6). Then sin (α + π 3) = 223. (8 points) ∵ sin (2 α + 5 π 3) = − sin (2 α + 2 π 3) = − 2Sin (α + π 3) cos (α + π 3) = − 429. (12 points)