Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts the curve $y = \cos x$, for $0 \le x \le \frac{\pi}{2}$. A rectangle $OABC$ is shown, with $B$ lying on the curve and having $x$-coordinate $\theta$, while $A$ and $C$ lie on the axes, as illustrated. The shaded region $R$ is enclosed by the curve and by the lines $x = \theta$ and $y = 0$.
(i)[2]
Find the area of $R$ as a function of $\theta$.
(ii)[1]
The area of rectangle $OABC$ matches the area of $R$. Show that $\theta = \frac{1 - \sin \theta}{\cos \theta}$.
(iii)[3]
Use the iterative formula $\theta_{n+1} = \frac{1 - \sin \theta_n}{\cos \theta_n}$, starting from $\theta_1 = 0.5$, to find the value of $\theta$ correct to $2$ decimal places. Give each iterate to $4$ decimal places.
Worked solution & mark scheme
This 6-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Carry out the integration and apply the limits $\theta$ and $\pi$” …