8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model.
(d) Find the value of \(t\) when this minimum temperature occurs.

**(a)**

$R^2 = 3^2 + 4^2$ | M1

$R = 5$ | A1

$\tan \alpha = \frac{4}{3}$ | M1

$\alpha = 53...°$ | awrt 53° | A1 | (4)

**(b)**

Maximum value is 5 | B1 ft their R

At the maximum, $\cos(\theta - \alpha) = 1$ or $\theta - \alpha = 0$ | M1

$\theta = \alpha = 53...°$ | ft their α | A1 ft | (3)

**(c)**

$f(t) = 10 + 5\cos(15t - \alpha)°$ | Given

Minimum occurs when $\cos(15t - \alpha)° = -1$ | M1

The minimum temperature is $(10 - 5)° = 5°$ | A1 ft | (2)

**(d)**

$15t - \alpha = 180$ | M1

$t = 15.5$ | awrt 15.5 | M1 A1 | (3) [12]

8. (a) Express $3 \cos \theta + 4 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < 90 ^ { \circ }$.\\
(b) Hence find the maximum value of $3 \cos \theta + 4 \sin \theta$ and the smallest positive value of $\theta$ for which this maximum occurs.

The temperature, $\mathrm { f } ( t )$, of a warehouse is modelled using the equation

$$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$

where $t$ is the time in hours from midday and $0 \leqslant t < 24$.\\
(c) Calculate the minimum temperature of the warehouse as given by this model.\\
(d) Find the value of $t$ when this minimum temperature occurs.\\

\hfill \mbox{\textit{Edexcel C3 2009 Q8 [12]}}