So I was messing around on my iPhone calculator trying to find the the precision of the calculator by finding at what point sin(x) was equal to x. I found myself repeating the sine function sin(sin(sin(....sin(x)...)))). Predictably the limit of this repeated operation of taking the sine was 0.
I then wondered what would happen if I did the same thing with cos(x). Theoretically it should approach 1 since it cos(x) <= 1 and the cosine of that would be even closer to 1. However, since only cos(0) would yield one I expected that the result would be close to 1, but not exactly 1. At this point I realized that it made a difference between radians and degrees, so I chose degrees and took the cosine repeatedly of an arbitrary number and I found that this yielded a result of roughly .999847741531088... each time. Stunningly, it also approached this limit relatively quickly, usually being close after only 4 cosine operations.
I found that using radians also produced a similar limit of around .73906... but it took much longer to approach this value. I messed around and found other limits and interesting behavior by taking other patterns like sin(cos(sin(cos(...(cos(x)...)))).
Why do these limits exist, and what is special about these particular numbers?
I know this is not rigorous mathematics, but I think it is interesting that such limits exist, especially the .9998477... limit for repeated cosine operations in degrees.
Turns out these values aren't arbitrary. Rather, they are the approximate solutions to $\cos x = x$ in radians and degrees.
(In the last line of your post you mean "...for repeated cosine operations in degrees".)