10 The diagram shows part of the graphs of \(y = \cos ^ { 2 } x\) and \(y = 5 \sin 2 x\) for small positive values of \(x\). The graphs meet at the point \(A\) with \(x\)-coordinate \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{022274c9-7ed2-4436-ae97-d410d7d566fc-14_652_561_402_735}
- Find the exact area contained between the two graphs (between \(x = 0\) and \(x = \alpha\) ) and the \(y\)-axis. Give your answer in terms of \(\alpha , \cos 2 \alpha\) and/or \(\sin 2 \alpha\).
- Using the fact that \(\alpha\) is a small positive solution to the equation \(\cos ^ { 2 } x = 5 \sin 2 x\), show that \(\alpha\) satisfies approximately the equation \(\alpha ^ { 2 } + 10 \alpha - 1 = 0\), if terms in \(\alpha ^ { 3 }\) and higher are ignored.
- Use the equation in part (b) to find an approximate value of \(\alpha\), correct to 3 significant figures.