We will prove here that for every x, at least one of the numbers |sin(x)|, |sin(x + 1)|, and |sin(x + 2)| is greater than 1/2. We will use a simple geometric argument, in particular we will use the fact that π/6 is slightly more than 1/2, hence π/3 is slightly more than 1; the value π/6 is obviously crucial here since sine is equal to 1/2 there.

We will do the proof by exploring all possible situations. If |sin(x)| > 1/2 we are done. So assume that this is not true. Then x must be between nπ − π/6 and nπ + π/6 for some integer n (see picture below).

If x were in the right half of this interval, then x + 1 must be greater than nπ + π/6 (since π/6 < 1) but smaller than nπ + 5π/6, so |sin(x + 1)| > 1/2 and we are done.

Thus the last case is when x is in the left half, that is, between nπ − π/6 and nπ.

Then the number x + 2 must again be greater than nπ + π/6 (since π/3 < 2) but smaller than nπ + 5π/6, so |sin(x + 2)| > 1/2. Since this was the last case left, we covered all possibilities and always found one large value as needed.