Convergence of cosine in the calculator

Question

I noticed when playing with the calculator that if I repeatedly press "cos", starting with any number, it converges to $0.99984742$. Why does it happens??

Answer 1

The equation $$\text{cos}(x)=x$$ can be solved via the iteration $x_{n+1}=\text{cos}(x_n)$ for any start value $x_0$ because after at most $2$ iterations, we have an absolute value less than $1$ and in the interval $[-1,1]$ , the derivate of $\cos(x)$, namely $-\text{sin}(x)$ has absolute value less than $1$.

You get the fixpoint of $\text{cos}(x)$ , which is the (unique) real solution of $\text{cos}(x)=x$.

You calculated in degrees and solved $x=\cos(x°)$. In radians, the equation would be $$cos\left(\frac{\pi x}{180}\right)=x$$ , which has a unique solution as well. The convergence is very fast in this case because the derivate of $\text{cos}(\frac{\pi x}{180})$ is very small.

In radians, the solution is $0.73908513\cdots $

Answer 2

If you had used the iterated solution to obtain this way $ \cos x = x $ in radians you get:

$$ x= 0.73908513 $$

Your calculator setting in degrees produces this solution:

$$ x= 0.9998474...$$

This is a way to evaluate fixed points convergence. A sketch would be helpful to see how the small squares shrink to the fixed point in x-y plot.