3 Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{54a69773-651f-4e2f-9a3c-06ea7c07098b-4_606_624_236_754}
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\caption{Fig. 8}
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In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations
$$x = 10 \cos \theta + 5 \cos 2 \theta , \quad y = 10 \sin \theta + 5 \sin 2 \theta , \quad ( 0 \leqslant \theta < 2 \pi )$$
where \(x\) and \(y\) are in metres.
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { \cos \theta + \cos 2 \theta } { \sin \theta + \sin 2 \theta }\).
Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(\theta = \frac { 1 } { 3 } \pi\). Hence find the exact coordinates of the highest point A on the path of C .
- Express \(x ^ { 2 } + y ^ { 2 }\) in terms of \(\theta\). Hence show that
$$x ^ { 2 } + y ^ { 2 } = 125 + 100 \cos \theta$$
- Using this result, or otherwise, find the greatest and least distances of C from O .
You are given that, at the point B on the path vertically above O ,
$$2 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$
- Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures.