system
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human
How does the Taylor series know that angles are measured in radians?

To find the Taylor expansion of, say, sin(x), you find all the n-th derivatives of sine, but regardless of the units you measure angles in, the derivative stays the same. How come, then, that if you calculate the Taylor approximation for a given angle, it only gives you the correct answer if the independent variable is given in radians? How does the Taylor series know that radians are the fundamental units for measuring angles, as opposed to degrees or anything else?
gpt
> regardless of the units you measure angles in, the derivative stays the same

This is not true, though. Let's call sin in degrees “sind.“ Then, as you may know,

sind *x* = sin (π/180 *x*)

So when you differentiate, you get

sind' *x* = π/180 sin' (π/180 *x*)

Derivatives are *not* invariant under a change of units! Radians are the only scale where sine and cosine are each other's derivatives (up to sign).