What is so special about this number?

Question

I noticed a strange thing with my calculator.
When I start with any number like 1,2,3 or 1.2, 1.34 .... or even 0.
And repeatedly take the cosine function of this number.
I get the same following number. I don't thing this is a coincidence since it's happening with any number I try.

0.99984774153108811295981076866798

It's pretty astonishing the accuracy this number has. I wouldn't have asked this question if only only few 4 or 5 decimals of every number matched but it's it's 32 decimal places I get for every number I try.
You got to try it yourself to believe it.
I want to know if there's a reason behind this? And why don't other functions like sine or tangent show similar properties? Note that the calculator is set to degrees.

Answer 1

Your calculator must be operating in degrees. Since $0.9998\ldots$ degrees is very close to $0$ (being less than $1/90$ of the way from $0$ radians to $\pi/2$ radians), its cosine must be very close to $1$. What you are finding is the fixed point of the function $\cos \theta$, where $\theta$ is expressed in degrees—that is, the number of degrees $\theta$ where

$$ \theta = \cos \theta $$

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Here's a graphical depiction of $\theta$ and $\cos \theta$ (with $\theta$ expressed in degrees). The fixed point is the intersection of these two curves:

Since it's difficult to see this intersection at the above scale, here it is zoomed in, and you'll see that the intersection occurs very close to $(1, 1)$; in fact, it is $(0.9998\ldots, 0.9998\ldots)$, as you discovered:

Answer 2

First, the number is a (the) fixed point $x_0$ of the map $x \mapsto \cos x^{\circ}$; here, $\cdot^{\circ}$ denotes interpreting $x$ as an angle measure of $x$ degree. Alternatively, we can avoid mention of degrees by saying this number is (the) fixed point of the map $T : x \mapsto \cos \frac{\pi x}{180}$.

Applying the cosine function to any real number gives a number in the interval $I := [-1, 1]$, so we can just as well ask why repeatedly applying cosine to any number $x$ in this interval gives a sequence $x, Tx, T^2 x, \ldots$ converging to this particular number, and the standard tool for proving this is the Banach Fixed-Point Theorem:

On this interval, the derivative stays small: $|T'(x)| \leq \sin \frac{\pi}{180} < 1$ for all $x \in I$, so for any $x, y \in I$, we have $|\cos x - \cos y| \leq \left(\sin \frac{\pi}{180}\right) |x - y|$, and hence the map $\cos$ is a contraction on that interval. Thus, by the B.F.-P.T., the map $T : x \mapsto \cos \frac{\pi x}{180}$ has a unique fixed point, and for every starting point $x \in I$ (and hence, by our previous observation, for every $x \in \Bbb R$), the sequence $(x, Tx, T^2 x, \ldots)$ converges to $x_0$.

The unit of degree is somewhat arbitrary, so it's more common to consider the corresponding operator $x \mapsto \cos x$ (mechanically, this just amounts to putting your calculator in degree mode). In that case, the fixed point is the Dottie Number, $0.739085\!\ldots$.