How can the graph of $\cos(\sin x)$ be plotted? Can anyone please discuss the procedure for drawing and analyzing graphs like $\cos(\sin x)$ and $\sin(\cos x)$?
 A: Consider for example $f(x) = \cos(\sin x))$. Since $\sin x$ is $2\pi$-periodic, so is $f$ and so you need to analyze $f$ only on the interval $[-\pi,\pi]$. Since $\sin$ is odd and $\cos$ is even, the function $f$ is even and so and it is enough to analyze $f$ on $[0,\pi]$.
When $x \approx 0$, we have $\sin x \approx x$ so the graph of $f$ looks like the graph of $\cos$ near $x = 0$. Next, note that $\sin x$ is monotone increasing on $[0,\frac{\pi}{2}]$ mapping this interval to $[0,1]$ and $\cos$ is monotone decreasing on $[0,\pi]$ and in particular on $[0,1]$ and so $f$ is monotone decreasing on $[0,\frac{\pi}{2}]$ until it reaches $f(\frac{\pi}{2}) = \cos(1)$. Next, $\sin x$ is monotone decreasing on $[\frac{\pi}{2},\pi]$ mapping this interval to $[0,1]$ on which $\cos$ is also monotone decreasing and so the composition $f$ is monotone increasing on $[\frac{\pi}{2}, \pi]$ until it reaches $f(\pi) = 1$. Thus, we obtain a graph that looks like this (courtesy of FooPlot):

Of course, we could have performed the analysis above using calculus but since the functions involved are so simple, there is no need to go there.
