1.2: A sector and a shaded area

Figure 1 shows sector $\displaystyle ORS$ of a circle with centre $\displaystyle O$ and radius $\displaystyle 4\,\mathrm{cm}$. The size of angle $\displaystyle ROS$ is $\displaystyle \theta$ radians. The area of sector $\displaystyle ORS$ is $\displaystyle 2\pi\,\mathrm{cm}^2$.

(a) Find the exact value of $\displaystyle \theta$.

(b) Find the perimeter, in $\displaystyle \mathrm{cm}$ to 3 significant figures, of the sector $\displaystyle ORS$.

The point $\displaystyle T$ lies on $\displaystyle OR$ such that $\displaystyle OT:TR=1:3$. The shaded region is bounded by $\displaystyle TR$, $\displaystyle TS$ and the arc $\displaystyle RS$. The area of this region is $\displaystyle A\,\mathrm{cm}^2$.

(c) Find the exact value of $\displaystyle A$.