I came up with this function: $2\left(\frac{1}{1+e^{\textstyle\frac{-6\sin^{-1}(\cos(x))}{\pi/2}}}-\frac12\right)$ to mimic a 'cosine'-esque function with flat peaks and valleys. Here it is as plotted by Wolfram Alpha:
What I was wondering is, is there a more elegant way to achieve this effect? (The values the function outputs need not be the same as those of this function - it only needs to look cosine-esque and have flat peaks and valleys).