Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram shows point $A$ lying on the circumference of a circle with centre $O$ and radius $r$. A circular arc centred at $A$ cuts 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 $A$ joining $B$ to $C$. Its area is equal to half the area of the circle.
(i)[5]
Demonstrate that $\cos 2\theta = \frac{2\sin 2\theta - \pi}{4\theta}$.
(ii)[3]
Use the iterative formula $\theta_{n+1} = \frac{1}{2}\cos^{-1}\left(\frac{2\sin 2\theta_n - \pi}{4\theta_n}\right)$, starting from $\theta_1 = 1$, to determine $\theta$ correct to 2 decimal places, and show 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: “Indicate or infer $AB=2r\cos\theta$ or $AB^2=2r^2-2r^2\cos(\pi-2\theta)$” …