From time to time, I've written little programs to find almost-integers in certain forms.
Most recently, I hit upon the form tan(i)/tan(j) where i and j are both integers. I was at first surprised to see that there were in fact a lot of the form tan(2x)/tan(x) ≅ 2, but a little thought revealed why: These were values of x that were just above multiples of π.
>>> x=355; tan(2*x)/tan(x); x/pi 2.000000001817364 113.00000959524569
Reduce x modulo π, and the formula becomes tan(2ϵ)/tan(ϵ) ≅ 2ϵ/ϵ ≅ 2. Mystery solved.
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