How can I to solve $ \cos x = 2x$? I would like to get an approx. solution to the equation: $ \cos x = 2x$, I don't need an exact solution just some approx. And I need a solution using elementary mathematics (without derivatives etc).
Take a pocket calculator, start with $0$ and repeatedly type [cos], [$\div$], [2], [=]. This will more or less quickly converge to a value $x$ such that $\frac{\cos x}2=x$, just what you want.
if $\cos x=2x$ which means 
$$x=\frac {1}{2.22131587} \,\mathrm{rad},$$

$$x=28.65957881\,\mathrm{Grad},$$

$$x=25.79362093\,\mathrm{Degree}$$

point:
$$\cos x=1-\frac {x^2}{2!}+\frac {x^4}{4!}-\frac {x^6}{6!}+\cdots$$

$0 &lt; x &lt; +\frac {1}{2.5}\,\mathrm{rad}$ because $\cos x=2x$



This is also a mathematical method, i mean Estimate the amount through math without using a computer or calculator!.

$$\cos x \simeq 1-\frac {x^2}{2!}=2x,$$  

$$x^2+4x-2=0,$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

where the $a=1$ and $b=4$ and $c=-2$ then $x\simeq 0.45$