7. (a) Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 4 decimal places. The height of sea water, \(H\) metres, on a harbour wall is modelled by the equation $$H = 6 + 2.5 \cos \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \sin \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Calculate the times at which the model predicts that the height of sea water on the harbour wall will be 4.6 metres. Give your answers to the nearest minute.
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