How can I find the range of my function

Your computation effectively yields the following:

(3 Interval[{-1, 1}] - 3)/(2 Interval[{-1, 1}] + 10)
(* Out: Interval[{-3/4, 0}] *)

The WolframAlpha code effectively computes the minimum and maximum of the function; it's more like so:

Minimize[(3 Sin[x] - 3)/(2 Cos[x] + 10), x] // Simplify
(* Out: {-5/8, {x -> 2*(Pi - ArcTan[3/2])}} *)

I found Reduce is useful,

Reduce[y == (3 Sin[x] - 3)/(2 Cos[x] + 10), y, {x}, Reals]
% // Simplify

(* -(5/8) <= y <= 0 *)

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