jmc

Let $C$ be the point where line segment $\overline{AT}$ intersects the circle. The measure of $\angle RTB$ half the difference of the two arcs it cuts off: $$m\angle RTB=\frac{m\widehat{RB}-m\widehat{SC}}{2}.$$ Since $m\widehat{RS}=74^{\circ }$, $m\widehat{SC}=180^{\circ }-74^{\circ }-m\widehat{RB}$. Substituting this expression for $m\widehat{SC}$ as well as $28^{\circ }$ for $m\angle RTB$, we get $$28^{\circ }=\frac{m\widehat{RB}-(180^{\circ }-74^{\circ }-m\widehat{RB})}{2}.$$ Solve to find $m\widehat{RB}=\boxed{81}$ degrees.