When is arc length ≈ chord length

[Saladsamurai]

Homework Statement

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
$\theta/2 < X_{cr}\Rightarrow \theta < 2*X_{cr}$ and from the arc length relationship S_AB = r*theta we can assert that when $S_{AB}/R < 2*X_{cr}$, 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!

 

[Hurkyl]

Your argument looks like a typical reasonable heuristic one.


However, do note that you are paying attention to relative error. If x and sin(x) are "close", then relatively speaking, Rx and Rsin(x) are equally "close".

However, if absolute error matters, your condition on a good approximation will depend on both R and x.

 

[Saladsamurai]

However, if absolute error matters, your condition on a good approximation will depend on both R and x.

Hi Hurkyl I am wondering, isn't this the same as saying that that my condition of a good approximation depends on how close the value of the ratio S/R is to 2*X_cr ? Seeing as the angle is given by S/R.

Thanks!

 

[sagardip]

Generally, we assume sinx~x for x<10degrees. So maybe that would be of some help to you

 

[gomunkul51]

Yes you can ! :)

but it all depends on how accurate you want to be and/or if you don't have an easy alternative.

*basically its a good approximation, you don't need to guess just crunch the numbers and check by how much percent sin(x) and x differ for a range of values.

 

[Hurkyl]

Hi Hurkyl I am wondering, isn't this the same as saying that that my condition of a good approximation depends on how close the value of the ratio S/R is to 2*X_cr ?

Nope.

While the value S_AB/R tells you everything about the relative error when approximating S_AB with AB, it doesn't tell you anything about the absolute error.

 

[Saladsamurai]

Nope.

While the value S_AB/R tells you everything about the relative error when approximating S_AB with AB, it doesn't tell you anything about the absolute error.

OK. So perhaps you are saying that for some fixed R, the choice of X will fix my absolute error? That is, if given the choice between 2 different values of X1 and X2, then the one that makes sin(X) closest to X is the one with the lowest absolute error.

Sorry, I am just trying to get a feel for what you are saying.