I asked this question a while ago. I exchanged comments with a member(mixedmath) about the rigorous proofs that $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ and the addition formula for $\sin x$. He referred to the wikipedia article. However, I'm not sure if the proofs using pictures are rigorous enough. The proofs take it for granted that what an angle(measured by radian) is. IMO, a straightforward and yet rigorous definition of an angle is that as an arc length of the unit circle. This definition involves the limit or the sup of suitable sums of lengths of line segments. I don't see how this definition incorporates into the proofs. Simply put, are the proofs of the wikipedia article rigorous?
Rigorous proof of the Taylor expansions of sin $x$ and cos $x$ revisited
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calculus
algebra-precalculus
trigonometry