Repeatedly 'cosine'ing a number: convergence

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SUMMARY

The repeated application of the cosine function on a number in radians converges to approximately 0.73908513321516064165531208767387. This value is the solution to the equation cos(x) = x, indicating that the input must equal the output for convergence to occur. The discussion confirms that this phenomenon is mathematically established and can be demonstrated through iterative calculations.

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ChaoticLlama
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A curiosity I've had recently...

When repeatedly taking the cosine of a number in radians, it appears to converge to a value.

i.e. cos(cos(cos(...cos(x)...))) = 0.73908513321516064165531208767387...

Any thoughts/explanations/exact solution etc?

Thanks.
 
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Yes, this value does converge to the solution to the equation cos(x) = x, because the input, x, must be the same as the output, cos(x), if it converges.
 

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