18 May 2011, 12:54 UTC

From time to time, I've written little programs to find almost-integers in certain forms.

### almost-integers from tan()

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.

Example:

>>> 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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