Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The sketch displays the curve $y = \cos x$, for $0 \le x \le \frac{1}{2}\pi$. A rectangle $OABC$ has been drawn, with $B$ located on the curve at the point whose $x$-coordinate is $\theta$, and $A$ and $C$ lying on the axes, as shown. The shaded region $R$ is enclosed by the curve and by the lines $x = \theta$ and $y = 0$.
(i)[2]
Find an expression for the area of $R$ in terms of $\theta$.
(ii)[1]
The area of the rectangle $OABC$ is the same as 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 $\theta$ correct to $2$ decimal places. Record 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: “An attempt is made to integrate, with the limits $\theta$ and $\pi$ used.” …