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Maybe this is precalculus? Either way, here is a question that I am curious about. Take a circle of radius R and sweep out an arc length S_{AB}with endpoints 'A' and 'B' over angle theta. For a short enough arc length, I believe that we could approximate S_{AB}by the chord length AB.

I am trying quantify "when" the ratio S_{AB}/R is such that the approximation is a good one. I guess a good start is to establish some relationships. From the picture below, we see that the arc length is given by S_{AB}=R*theta and the chord length is given by AB = 2*R*sin(theta/2).

So I believe we should now ask when does R*theta ≈ 2*R*sin(theta/2).

I know from other problems we often employ the approximation that if an angle 'X' is "small enough", then sin(X)≈X. It looks like this would help here since if we let sin(theta/2) = theta/2, then the approximation above becomes an identity. I am just having trouble figuring out how to relate this all back to the ratio S_{AB}/R ? What if we said that we already know that for some critical value of the angle X we can approximate sin(X) = X. We will call that "known" value X_{cr}. So if theta/2 < X_{cr}then S_{AB}≈AB. So

[itex]\theta/2 < X_{cr}\Rightarrow \theta < 2*X_{cr}[/itex] and from the arc length relationship S_{AB}= r*theta we can assert that when [itex]S_{AB}/R < 2*X_{cr}[/itex], the approximation is good.

Can someone let me know if they think my logic is flawed? I have never done something like this from scratch before

Thanks!

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# Homework Help: When is arc length ≈ chord length

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