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How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ?

The iterative solution seems to be 1.417.

Can anybody suggest an algebraic solution ? Does it really exist ?

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    Why on earth do you want to solve that nasty looking thing? I'm pretty sure even $x = \cos x$ doesn't have a closed-form solution, so I wouldn't hold out much hope for a nice solution to your equation...2011-02-05
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    @Rahul. It is related to a contraction mapping problem.2011-02-05
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    I don't know if this helps you, but after I made the above comment I went looking for a proof that the solution to $x = \cos x$ cannot in fact be expressed in closed form. It turns out that, according to [Timothy Chow's 1999 article](http://www.jstor.org/pss/2589148), this is actually still open! It would be implied by [Schanuel's conjecture](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture), but that conjecture is not proven. Perhaps you can adapt Chow's argument to reduce your problem to Schanuel's conjecture too.2011-02-05
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    Shanuel would show it is transcendental, I guess, but that does *not* mean it cannot be expressed in closed form.2011-06-14
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    In fact, Schanuel implies much more, the algebraic independence of $\pi$ and $e$, which Timothy Chow was able to extend to show that no solution to $x = \cos x$ can be expressed in closed form (as precisely defined in Chow's paper). See Theorem 1 on p. 443 of Chow's paper. A URL for a freely available version is .2011-07-14

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You can easily get extremely accurate approximations using (for example) Wims function calculator, by searching the root $x^*$ of $$ f(x) = x - \bigg[\frac{1}{2}\cos \bigg(\frac{2}{3}\sin \bigg(\frac{3}{4}x\bigg)\bigg) + 1\bigg] $$ (say between $-5$ and $5$). Computing its value up to a mild number of digits gives $$ x^* = 1.417520004.... $$ While Inverse symbolic calculator does not recognize this, it does lead to the approximation $$ x^* \approx \frac{{10 \sin (1)}}{{e^{\exp (\gamma )} }} = 1.417520089... , $$ where $\gamma$ is Euler's constant ($=0.5772156649...$).