Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts a circle whose centre is $O$ and whose radius is $r$. The tangents to the circle at $A$ and $B$ intersect at $T$, while the angle $AOB$ is $2x$ radians. The shaded part is enclosed by the tangents $AT$ and $BT$, together with the minor arc $AB$. Its perimeter is the same as the circle’s circumference.
(a)[3]
Show that $x$ obeys the equation $\tan x = \pi - x$.
(b)[2]
This equation has a single root in the interval $0 < x < \tfrac{1}{2}\pi$. Verify, by calculation, that this root is between $1$ and $1.3$.
(c)[3]
Use the iterative formula $x_{n+1} = \tan^{-1}(\pi - x_n)$ to find the root correct to 2 decimal places. Record 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: “State, or make clear, $AT=r\tan x$ or $BT=r\tan x$” …