Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram shows $A$ lying on the circumference of a circle whose centre is $O$ and whose radius is $r$. An arc of a circle centred at $A$ intersects the circumference at $B$ and $C$. The angle $OAB$ is $\theta$ radians. The shaded region is enclosed by the circle’s circumference and the arc centred at $A$ that joins $B$ and $C$. Its area is the same as half the area of the circle.
(i)[5]
Show that the following result is true: $\cos 2\theta = \frac{2 \sin 2\theta - \pi}{4\theta}$.
(ii)[3]
Using the iterative formula $\theta_{n+1} = \frac{1}{2} \cos^{-1}\left( \frac{2 \sin 2\theta_n - \pi}{4\theta_n} \right)$, together with the starting value $\theta_1 = 1$, find $\theta$ correct to 2 decimal places, recording each iteration to 4 decimal places.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or deduce $AB=2r\cos\theta$ or $AB^2=2r^2-2r^2\cos(\pi-2\theta)$” …