Answer
$u_r=\cos{\theta}\cos{(r\cos{\theta})}$, $u_\theta=-r\sin{\theta}\cos{(r\cos{\theta})}$.
Work Step by Step
$u(r,\theta)=\sin{(r\cos{\theta})}$
In order to find $u_r$ we treat $\theta$ as a constant and differentiate with respect to $r$.
$u_r=\cos{\theta}\cos{(r\cos{\theta})}$
In order to find $u_\theta$ we treat $r$ as a constant and differentiate with respect to $\theta$.
$u_\theta=-r\sin{\theta}\cos{(r\cos{\theta})}$
