The maximum value of the cosine function is 1. It has this value at $\displaystyle{x}={0}\usepackage{cmbright}\Huge\$ and then again at the end of one cycle. The value of x at that point is the length of one cycle, or one period of $\displaystyle{y}={\cos{{\left(\frac{{1}}{{2}}{x}\right)}}}\usepackage{cmbright}\Huge\$ . The period of $\displaystyle{y}={a}{\cos{{b}}}{x}\usepackage{cmbright}\Huge\$ is $\displaystyle\frac{{{2}\pi}}{{\left|{b}\right|}}\usepackage{cmbright}\Huge\$ , so the number we are looking for is:

$\displaystyle\frac{{{2}\pi}}{{\left|\frac{{1}}{{2}}\right|}}=\frac{{{2}\pi}}{{\frac{{1}}{{2}}}}=\frac{{{2}\pi}}{{1}}\cdot\frac{{2}}{{1}}={4}\pi\usepackage{cmbright}\Huge\$