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Basically, while i was watching a calculus of variations video, the guy said that the length with infinitesmal $\theta$ equals to $R\sin(d\theta)$ instead i find it as $2R(\sin(d\theta/2))$? What is wrong with that? I guess since it is so small it basically converges to that value but how do we prove that?

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Nothing is wrong, just $$\lim_{\theta\to 0}\frac{\ \sin(\theta)\ }{\ \theta\ } = 1$$ so $$\lim_{\theta\to 0}\frac{\ \sin(2\theta)\ }{\ 2\sin(\theta)\ } = 1$$ hence $$\sin(2\theta)\ \approx\ 2\sin(\theta)$$ for the theta angle small enough. And 'infinitesimal theta' means precisely that — the angle is small.

Compare, for example, Wikipedia Infinitesimal.

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  • $\begingroup$ so we can also say Rsinθ equals Rθ as well which is the arc length $\endgroup$
    – mark
    Jan 25 at 11:52

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