Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram shows triangle $ABC$, where angle $ABC$ is a right angle and $BC = a$. A circular arc, centred at $C$ and with radius $a$, connects $B$ to point $M$ on $AC$. The angle $ACB$ is $\theta$ radians. The area of sector $CMB$ is one third of the area of triangle $ABC$.
(i)[2]
Show that $\theta$ obeys the equation $\tan \theta = 3\theta$.
(ii)[3]
There is one root of this equation in the interval $0 < \theta < \frac{1}{2}\pi$. Apply the iterative formula $\theta_{n+1} = \tan^{-1}(3\theta_n)$ to find the root correct to $2$ decimal places. Record each iteration to $4$ decimal places.
Worked solution & mark scheme
This 5-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Using the formulae $\tfrac{1}{2} r^2 \theta$ and $\tfrac{1}{2} bh$, set up an equation in $a$ and $\theta$” …