I used to do the same in school lessons. I can still remember the fixed point: 0.739085133, to the precision of the calculator I was using at the time. (Provided you're using radians. Which you should be.)
Fun fact about fixed points #1: Put a sheet of paper on a desk. Notice the precise area of the desk's surface that it covers. Now ball up the sheet of paper however you wish, without tearing it. Put it down again in such a way that every part of that sheet is somewhere over the original area that the paper occupied. Then there is at least one point on the paper that is precisely above its original location.
Fun fact about fixed points #2: Some functions, even simple ones suitable for a desktop calculator, do not reliably approach a fixed point when applied repeatedly to their own output. You can see this in action at Wikipedia "Cobweb plot" (http://en.wikipedia.org/wiki/Cobweb_plot) for the function known as the "logistic map", y = rx(1-x). Varying the initial value of x (really, x-subscript-zero) and the parameter r can cause the map to converge to a fixed point, to enter a repeating sequence of any length, or to remain in a finite range without ever repeating. This last behavior is known as "chaos".
