In radians, sin(11) is very nearly -1.
(It happens to be -0.9999902…)
Finally, I countered with cos(103993), which is 0.9999999998…
So what’s going on here?
All of this comes from the continued fraction for ,
It is well known that the convergents of a continued fraction (i.e., the fraction you obtain by truncating a continued fraction at some point) are the best rational approximations for the number. The first ten convergents of are
, , , , , , , , , .
Notice that , so . Thus . This was John D. Cook’s observation.
Similarly, , so , and . The other two approximations come from the next two convergents.
After the conversation on Twitter, I started playing a little more.
First, I noticed that
This comes from the third convergent. Since implies that , .
Actually, we can do better than that:
The approximation, implies that , so .
Next, I noticed that
This comes from , which implies that . Thus
Finally, I noticed that
In this case, gives , and . We obtain the result by substituting the previous approximation for .
Of course, we also have