Challenging +1.8 This Further Maths question requires manipulating Maclaurin series for a product of three trigonometric functions (non-trivial algebraic work collecting terms to x³), then solving a transcendental equation using the approximation. It demands strong series manipulation skills and insight to use the cubic approximation effectively, going well beyond standard single-function expansions.
\begin{enumerate}[label=(\roman*)]
\item Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2x \sin 4x\) up to and including the term in \(x^3\). [4]
\item Hence find, in exact surd form, an approximation to the least positive root of the equation \(2\sin x \sin 2x \sin 4x = x\). [3]
\begin{enumerate}[label=(\roman*)]
\item Use the Maclaurin series for $\sin x$ to work out the series expansion of $\sin x \sin 2x \sin 4x$ up to and including the term in $x^3$. [4]
\item Hence find, in exact surd form, an approximation to the least positive root of the equation $2\sin x \sin 2x \sin 4x = x$. [3]
</end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 Q7 [7]}}