7. \(\left[ \arccos x \right.\) and \(\arctan x\) are alternative notation for \(\cos ^ { - 1 } x\) and \(\tan ^ { - 1 } x\) respectively \(]\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-5_387_935_322_566}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi\) .
The curve has turning points at \(( 0 , \cos 1 ) , P , Q\) and \(R\) as shown in Figure 2.
- Find the coordinates of the points \(P , Q\) and \(R\) .
The curve \(C _ { 2 }\) has equation \(y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi\) .The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(S\) and \(T\) .
- Copy Figure 2 and on this diagram sketch \(C _ { 2 }\) stating the coordinates of the minimum point on \(C _ { 2 }\) and the points where \(C _ { 2 }\) meets or crosses the coordinate axes.
The coordinates of \(S\) are \(( \alpha , d )\) where \(0 < \alpha < \pi\) .
- Show that \(\alpha = \arccos \left( \frac { \pi } { 4 } \right)\) .
- Find the value of \(d\) in surd form and write down the coordinates of \(T\) .
The tangent to \(C _ { 1 }\) at the point \(S\) has gradient \(\tan \beta\) .
- Show that \(\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)\) .
- Find,in terms of \(\beta\) ,the obtuse angle between the tangent to \(C _ { 1 }\) at \(S\) and the tangent to \(C _ { 2 }\) at \(S\) .