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.