Question

What happens if I use `cos` repeatedly?

like if I use cos(cos(cos(cos(cos(cos(x)))))) upto a hundred times?

Answer 1

It converges to the Dottie number.

Explanation:

The value will converge to the Dottie number `=0.739085...`

If we graph `y=cos(x)` and `y=x` the Dottie number is the unique value `x_0` where they intersect, that is, the solution to the equation `x = cos(x)`.

graph{(y-x)(y-cos(x))=0 [-10, 10, -5, 5]}

There is no known closed form for the Dottie number, but, as asked, it can be approximated by repeatedly applying the cosine function to any given value (although there are other sequences which converge to it more quickly). In other words, it is the limit as `n->oo` of the sequence `x_(n+1) = cos(x_n)` where `x_0` is any real number.

Another way of looking at it is to note that `cos(theta)` is the `x` coordinate of the point where the ray forming an angle `theta` with the `x`-axis intersects the unit circle. Taking this `x`-coordinate and using it as a new angle and repeating this process is equivalent to the sequence above.

If we did this many times, we would note that the rays would start to converge. The `x` coordinate of that point where the ray they are converging to intersect the unit circle is the Dottie number.