[cdancette]

for all functions, f(x) = f(0) + f'(0) * x for x very close to 0

[btilly]

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).

[Smaug123]

This is a specific instance of what LolWolf just said in the grandparent.

[dekhn]

yeah I didn't know that part, just the more specific one.

[jjaredsimpson]

It's the first term of the Maclaurin series

sin(x) = sin(0) + cos(1)x = 0 + x