Mathematics 9709 · AS & A Level · Numerical solution of equations
Numerical solution of equations — practice question
The diagram depicts triangle $ABC$, where $AB = AC = a$ and angle $BAC = \theta$ radians. Two semicircles are drawn externally on $AB$ and $AC$ as diameters. An arc of a circle centred at $A$ connects $B$ to $C$. The shaded segment has the same area as the two semicircles combined.
(i)[3]
Show that $\displaystyle \theta=\frac12\pi+\sin\theta$.
(ii)[2]
Verify by calculation that $\theta$ is between $2.2$ and $2.4$.
(iii)[3]
Use an iterative formula derived from the equation in part (i) to find $\theta$ 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: “Use the appropriate area formulae for a segment and a semicircle to build an equation in $\theta$” …