Just for funsies (but may answer your question): every (differentiable) function is linear at a small enough time scale. This includes e^x or pretty much every function you listed.
Which is what differential geometry is based on - you take a surface (for example - the one dimensional functions you mentioned work too, but those are too trivial). Then you associate to each point a plane (re-centered at to the origin). The collection of all vectors that fit into the plane is a vector space called a tangent space, and the collection of all tangent spaces is a tangent bundle. And now you've set up differential geometry and can study it.
This approximation has vanishing error from a linear function (as the interval decreases) and (most importantly!) we only have finite noisy samples; so it’s essentially indistinguishable (in a statistical way) for a small enough interval given some variance.
The approximation is particularly good for sin(x) because the next term in the Taylor series, f''(0) * x^2 / 2, happens to be 0. So the error is O(x^3) rather than the more common O(x^2).