Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
A circle with centre $O$ and radius $r$ is shown. The tangents to the circle at $A$ and $B$ intersect at $T$, and angle $AOB$ is $2x$ radians. The shaded region lies between the tangents $AT$ and $BT$, and the minor arc $AB$. Its area is equal to the area of the circle.
(a)[3]
Show that $x$ is a solution of the equation $\tan x = \pi + x$.
(b)[2]
There is a single root of this equation in the interval $0 < x < \tfrac{1}{2}\pi$. Verify by calculation that it is between $1$ and $1.4$.
(c)[3]
Apply the iterative formula $x_{n+1} = \tan^{-1}(\pi + x_n)$ to find the root correct to $2$ decimal places. Show 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 imply that $AT=r\tan x$ or $BT=r\tan x$” …