Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts a semicircle $ACB$ with centre $O$ and radius $r$. The angle $BOC$ is $x$ radians. The shaded segment has an area equal to one quarter of the semicircle’s area.
(i)[3]
Show that $x$ obeys $x = \frac{3}{4}\pi - \sin x$.
(ii)[2]
This equation has a single root. Check by calculation that the root lies between $1.3$ and $1.5$.
(iii)[3]
Apply the iterative formula $x_{n+1} = \frac{3}{4}\pi - \sin x_n$ to find the root correct to $2$ decimal places. Record the outcome of 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: “Using $\tfrac12r^2\theta$ and $\tfrac12r^2\sin\theta$, or an equivalent pair of formulae, set up an equation” …