Infinitely Concatenated Sine and Cosine Using a graphing calculator, if one concatenates sine and cosine repeatedly, i.e.
$$y=\sin(\cos(\sin(\cos(x))))$$
the graph appears to approach a horizontal line, suggesting that at infinite concatenation, there is a single value of the function for all $x$.  Is this correct?  If so is this value known?
 A: This is an example of an attractive fixed point.
A fixed point of the function $x\mapsto \sin(\cos(x))$ is a number $x_0$ satisfying $x_0 = \sin(\cos(x_0))$, i.e. the number you put in is the same number that you get out.  An attractive fixed point is one for which, if $x$ is sufficiently close to $x_0$, then $\sin(\cos(x))$ is even closer to $x_0$, and may be made as close as desired by iterating the process a large enough number of times.  In this case, "sufficiently" close appears to include all numbers in the domain.
The Banach fixed-point theorem is applicable here.  The derivative of $x\mapsto\sin(\cos(x))$ is bounded in absolute value by a number less than $1$; from that it follows from the mean value theorem that this is a contraction mapping.  And the space is complete, i.e. closed under limits of Cauchy sequences.  Hence the theorem is applicable.  The theorem says that every function to which it is applicable has exactly one attractive fixed point.
In fact if you draw the graph of $y=\sin(\cos(x))$ and that of $y=x$, you see that they intersect exactly once, and the $x$-coordinate of the intersection point is then the one fixed point.
A: Taking infinite concatenation into account,and writing $y$ as $$y=\sin(\cos(y))$$
Wolfram Alpha gives the solution to be a constant namely $$0.694819690731$$
