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I was wondering if there was some easy way to evaluate a repeating pattern of $$\sin(\cos(\sin(\cos x)))$$ for an arbitrary number of $\sin(\cos(\cdots $'s. I typed it into desmos and notice if I typed in enough it hovered around a value near $.7$ - Does anyone know the actual value of this number or a finite representation of an arbitrary length string of sines and cosines?

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  • $\begingroup$ With arbitrary, do you mean an arbitrary finite composition or infinitely many, i.e. a limit? $\endgroup$ – klirk Sep 7 '17 at 2:27
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Let $$f(x) = \sin \,\left(\cos \, x \right)$$

You want to find a fixed point of $f$, and you suspect from experimentation that it is near $x\approx 0.7$.

This can also be written as the solution to:

$$\arcsin x = \cos x$$

You can see a graph of $\arcsin$ and $\cos$ and see their intersection here:

enter image description here


According to Wolfram | Alpha, you are correct:

$$x \,\approx \,\boxed{ 0.69482\,}$$

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  • $\begingroup$ So I figured that out this morning, but I was hoping for either: 1) a finite representation of the answer, or 2) a proof it is transcendental. $\endgroup$ – William Grannis Sep 7 '17 at 17:11
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    $\begingroup$ @skwarerüt As for (1) I put in the approximate decimal solution into the Inverse Symbolic Calculator here but did not find any meaningful results as I increased the number of decimal places that had to match. For (2), do you know whether or not it is transcendental? In general, it is quite difficult to prove whether numbers are transcendental ... I probably would not be able to prove or disprove whether this solution is transcendental. $\endgroup$ – Zubin Mukerjee Sep 7 '17 at 17:18
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If $y = \sin{(\cos{(\ldots \sin{(\cos{x})})})}$, then we should expect that $$\sin(\cos(y)) = y,$$

so you can find this value as the zero of $y-\sin(\cos(y)) = 0$.

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