Fixed points of a first order equations

Question

I am trying to find the fixed points of the following system,

$$x'=x-\cos x$$

So to find fixed point you must set $$x'=0$$

So,

$$ \begin{split} x-\cos x &=0\\ x &=\cos x\\ x^* &\approx 0.739 \end{split} $$

But why is it $0.739$?

Answer 1

Because $$\cos(0.739085)= 0.739085$$ (angles measured in radians.) You can find that value by iteration: start with any value of $x_0$ and compute the iterative sequence $x_{n+1}=\cos(x_n)$. This can be done hitting the $\color{red}{\cos}$ button in a scientific calculator. I also recommend to draw he graph of $x$ and $\cos x$.