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I can show that $\cos(\sin(x))$ is a contraction on $\mathbb{R}$ and hence by the Contraction Mapping Theorem it will have a unique fixed point. But what is the process for finding this fixed point? This is in the context of metric spaces, I know in numerical analysis it can be done trivially with fixed point iteration. Is there a method of finding it analytically?

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    Probably not. ${{{{{}}}}}$2012-11-28
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    The number $0.76816915673679597746208623955865641813208731218273718569186715\ldots$ does not ring a bell, at least, and it seems that we need a new Plouffe's inverter.2012-11-28
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    The Inverse Symbolic Calculator at http://isc.carma.newcastle.edu.au/advancedCalc returns a blue square with a white question mark inside it. I don't know what that means.2012-11-29
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    @Gerry, I believe it means you are pregnant. Or possibly someone you know.2012-12-21

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