4 Part of the track of a roller-coaster is modelled by a curve with the parametric equations
$$x = 2 \theta - \sin \theta , \quad y = 4 \cos \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi$$
This is shown in Fig. 8. B is a minimum point, and BC is vertical.
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\includegraphics[alt={},max width=\textwidth]{9ac55ae6-7a7f-47d0-a363-92da179be4ca-3_591_1437_433_391}
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\caption{Fig. 8}
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- Find the values of the parameter at A and B .
Hence show that the ratio of the lengths OA and AC is \(( \pi - 1 ) : ( \pi + 1 )\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). Find the gradient of the track at A .
- Show that, when the gradient of the track is \(1 , \theta\) satisfies the equation
$$\cos \theta - 4 \sin \theta = 2$$
- Express \(\cos \theta - 4 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\).
Hence solve the equation \(\cos \theta - 4 \sin \theta = 2\) for \(0 \leqslant \theta \leqslant 2 \pi\).