Graphing gives the solution to be $0.739,$ but I want to know how to find that value mathematically. I imagine I'd have to use trigonometric identities, but how to go about it?
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The solution is the same as the solution for $x=\cos(x)$. Do you see why ? – Peter Nov 11 '21 at 18:57
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1... therefore, it suffices to use iteration $x_{n+1}=\cos(x_n)$ – Jean Marie Nov 11 '21 at 19:00
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related – Peter Nov 11 '21 at 19:00
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1If you love playing pocket calculator, you could try repeatingly using $\cos (.)$ started with any number. – Ng Chung Tak Nov 12 '21 at 01:11
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$$\cos x=\arccos x$$
$\arccos$ has domain $[-1,1]$ and is a decreasing function, $\arccos(\cos1)=1,$ and $\cos x\not>1.$
Thus, the required solution lies within $[\cos 1,1].$ On this domain,
$$\cos x=\arccos x\\\iff \cos(\cos x)=\cos(\arccos x)\\\iff x=\cos(\cos x);$$ there is no analytic solution.
Now, $\sin(\cos x)<1,$ so $$\Big|\frac {\mathrm d}{\mathrm dx}\cos(\cos x)\Big|=\lvert(\sin x)\sin(\cos x)\rvert\\<1.$$ Therefore, within $[\cos 1,1],$ the required solution is an attracting fixed point of $\cos(\cos x),$ and consequently of $\cos x.$
Thus, by solving $x_{n+1}=\cos x_n$ iteratively (using $x_0=1,$ say), the required solution is $0.739,$ the Dottie number.
ryang
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