> If you zoom in sufficiently at x = 0, f(x) = sin(x) looks indistinguishable from f(x) = x, whereas g(x) = cos(x) looks indistinguishable from g(x) = 1.

You can't use the plot as you only know the triangle definition yet. (And "looks indistinguidable" is rather handwavy).

> (also, sin(x) is negative approaching 0 from the left and positive approaching 0 from the right)

That only tells you that sin(0)=0