Mathematics 9709 · AS & A Level · Circular measure
Circular measure — practice question
The diagram shows $OAB$ as a sector of a circle centred at $O$ with radius $r$. Point $C$ lies on $OB$ so that the angle $ACO$ is a right angle. The angle $AOB$ is $\alpha$ radians, and $AC$ splits the sector into two parts of equal area.
(i)[4]
Show that, after simplification, $\sin \alpha \cos \alpha = \frac{1}{2}\alpha$.
(ii)[5]
The solution to the equation in part (i) is given as $\alpha = 0.9477$, correct to $4$ decimal places. Determine the ratio perimeter of region $OAC$ : perimeter of region $ACB$, and present it in the form $k : 1$, with $k$ correct to $1$ decimal place.
(iii)[1]
Find the angle $AOB$ in degrees.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “The correct lengths are $OC=r\cos\alpha$ and $AC=r\sin\alpha$.” …