7 A curve is defined by the parametric equations
$$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$
where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).
- Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
- Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
- Hence find the values of \(t\) at the points \(P\) and \(Q\).