Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts a semicircle with diameter $AB$, centre $O$ and radius $r$. The shaded part is the minor segment bounded by the chord $AC$, and its area is one third of the area of the semicircle. The angle $CAB$ is $\theta$ radians.
(a)[4]
Show that the relationship $\theta = \frac{1}{3}(\pi - 1.5\sin 2\theta)$ holds.
(b)[2]
Verify by calculation that $0.5 < \theta < 0.7$ is true.
(c)[3]
Use an iterative formula derived from the equation in part (a) to find $\theta$ correct to $3$ decimal places. Write the value from each iteration to $5$ decimal places.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or infer $\angle AOC=\pi-2\theta$” …