A vector of magnitude '$a$'  is turned through angle $\theta$ . The magnitude of change in the vector is given by:

Let $\overrightarrow{a}$ is rotated through angle  $\theta$ to get  $\overrightarrow{b}$
$\therefore \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=a$ and angle between $\overrightarrow{a}$ and  $\overrightarrow{b}$ is  $\theta$
The magnitude of change in vector $\overrightarrow{a}$  is 
$|\overrightarrow{b}-\overrightarrow{a}|=\sqrt{{\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{a}\right|}^2+2\left|\overrightarrow{b}\right|\left|\overrightarrow{a}\right|}\cos (\pi -\theta )$
$=a\sqrt{2(1-\cos \theta )}=a\sqrt{(4{\sin }^2\theta /2)}$
$=|2a\sin \theta /2|$