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I was experimenting with the software geogebra, and playing with a couple of unusual trigonometric functions and I encountered a quite strange phenomena when I entered this input - $$f(\theta)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(....(\theta)))))))))$$ Let's assume the number of cosines approach infinity, though my input was 10-20 cosines.

I asked the program to graph this function and surprisingly the graph was straight-lined, as in for any value of $\theta$, the function is a constant value.

I then created a program in GBD Online to recheck if this was actually correct or just a software error. The program iterated $f(\theta)$ for all values of $\theta$ from $1$ to $999$ . And again, surprisingly, the values approached the constant of $0.739085$

The specifics of the program is here.

The graph in question is shown in this image.

My questions for this post are -

  1. Why does the function approach a constant?
  2. Do all trigonometric functions approach a constant with infinite iterations?
  3. If not, why only cosine?
Blue
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Abhimaan_09
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1 Answers1

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enter image description here

What is happening is that we can see in the repeated series a converging spiraling iterative solution for

$$ y= \cos(x). $$

Narasimham
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