Find specific 4 curves touching $y=\cos10x+\cos21x$.

Question

The following is the graph of $y=\cos10x+\cos21x$.

You can see that there seems to be four curves that can touch this graph. I tried $y=\cos(x/2+\pi/2\pm\pi)+1$ and $y=-\cos(x/2\pm\pi/2)-1$:

But unfortunately, they cut the graph. What actually are that four curves touching the graph? Thanks.

Answer 1

Take the one that goes through $(0,2)$. The peaks are roughly at $x=4n\pi/21$, call them $x_n=4n\pi/21+\beta_n$. I will assume $\beta_n$ are small.
At these peaks, $$\frac{dy}{dx}=-10\sin10x-21\sin21x=0\\ 10\sin\left(\frac{40n\pi}{21}+10\beta_n\right)+21\sin(4n\pi+21\beta_n)=0\\ 10\left(\sin\frac{40n\pi}{21}+10\beta_n\cos\frac{40n\pi}{21}\right)+21^2\beta_n\approx0\\ \beta_n\approx\frac{10\sin(2n\pi/21)}{10^2\cos(2n\pi/21)+21^2}<\frac1{40}$$ The height of the peak is $\cos10x+\cos21x$, which is roughly $$y=\cos\left(\frac{40n\pi}{21}+10\beta_n\right)+\cos(21\beta_n)\\ \approx\cos\left(\frac{2n\pi}{21}-10\beta_n\right)+1-(21\beta_n)^2/2\\ \approx\cos\left(\frac x2-\frac{21}2\beta_n\right)+1-5\beta_n\sin(x/2)\\ \approx\cos\left(\frac x2-\frac{11}2\beta_n\right)+1\\ \approx1+\cos\left(\frac x2-\frac18\sin\frac x2\right)\\ \approx1+\cos(\frac x2)+\frac18\sin^2\frac x2$$

Answer 2

Hint: try using the sum to product identities: http://www.sosmath.com/trig/prodform/prodform.html for $\cos(a)+\cos(b)$. Notice that in your graph above the amplitude is a function of time.