Looking over some old posts of @icecolbeveridge I came across this curiosity:

13 cos(ln(2)) ~ 10.

Why should cos(ln(2)) be very close to 10/13 ... any reason?

Or is it "Just because"?

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@ColinTheMathmo @icecolbeveridge \(10/13\) happens to be one of the continued fraction convergents of \(\cos\log 2\): \[\cos\log 2\approx\cfrac1{1+\cfrac1{3+\cfrac13}}=\frac{10}{13}\]

@tpfto Have you derived that CF by combining the individual functions, or did you simply take the floating point value and compute the CF from that?

If the former, that might constitute an explanation. If the latter, it's just saying "Yes, it's close to 10/13".

Does that make sense?

CC: @icecolbeveridge

@ColinTheMathmo @icecolbeveridge yes, that one just came from the numerical value. The partial denominators look like \(0, 1, 3, 2, 1, 726, 1, 14, \dots\), and the sequence of approximants goes like \(1, 3/4, 7/9, 10/13, 7267/9447, 7277/9460, 109145/141887, \dots\), where we get the jump from \(10/13\) to \(7267/9447\) because of the suddenly large \(726\). So "10/13 just happens to be a good approximation" is a good summary.

@tpfto @icecolbeveridge Yup, the suddenly large value shows that it's a good approximation ... agreed.

So we still don't have a good reason ... my feeling is that there might not be one.

Just a coincidence.

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