- (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Give the values of \(R\) and \(\alpha\) to 3 significant figures.
(b)Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs.
The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation
$$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24$$
where \(t\) hours is the number of hours after 1200 .
(c) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(d) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).