Answer
$$\frac{{dr}}{{d\theta }} = - \frac{r}{\theta }$$
Work Step by Step
$$\eqalign{
& \sin \left( {r\theta } \right) = \frac{1}{2} \cr
& {\text{differentiate both sides with respect to }}\theta \cr
& \frac{d}{{d\theta }}\left( {\sin \left( {r\theta } \right)} \right) = \frac{d}{{d\theta }}\left( {\frac{1}{2}} \right) \cr
& {\text{use the chain rule}} \cr
& \cos \left( {r\theta } \right)\frac{d}{{d\theta }}\left( {r\theta } \right) = \frac{d}{{d\theta }}\left( {\frac{1}{2}} \right) \cr
& {\text{use the product rule}} \cr
& \cos \left( {r\theta } \right)\left( {r\frac{d}{{d\theta }}\left( \theta \right) + \theta \frac{d}{{d\theta }}\left( r \right)} \right) = \frac{d}{{d\theta }}\left( {\frac{1}{2}} \right) \cr
& {\text{find derivatives}} \cr
& \cos \left( {r\theta } \right)\left( {r + \theta \frac{{dr}}{{d\theta }}} \right) = 0 \cr
& {\text{Solve for }}\frac{{dr}}{{d\theta }} \cr
& r\cos \left( {r\theta } \right) + \theta \cos \left( {r\theta } \right)\frac{{dr}}{{d\theta }} = 0 \cr
& \theta \cos \left( {r\theta } \right)\frac{{dr}}{{d\theta }} = r\cos \left( {r\theta } \right) \cr
& \frac{{dr}}{{d\theta }} = - \frac{{r\cos \left( {r\theta } \right)}}{{\theta \cos \left( {r\theta } \right)}} \cr
& \frac{{dr}}{{d\theta }} = - \frac{r}{\theta } \cr} $$
