number of integers in the range of function

Question

3sinx + 2 (root 3) cosx+3

Answer 1

3^2 + (2√3)^2 = 9+12 = 21
so, if cosy = 3/√21, you have
3sinx + 2√3 cosx = √21 (3/√21 sinx + 2√3/√21 cosx)
= √21 sin(x+y)
so the range of
3sinx + 2√3 cosx + 3 is [3-√21 , 3+√21] ≈ [-1.58,7.58]
How many integers in that?

Answer 2

9 integers
@oobleck am i right or wrong

Answer 3

surely you don't need a confirmation of that result.
But yes, you are right: -1, 0, 1, 2, 3, 4, 5, 6, 7