(a) Express $\sin x+2 \cos x$ in the form $R \sin (x+\alpha)$ where $R$ and $\alpha$ are constants, $R>0$ and $0<\alpha<\frac{\pi}{2}$ Give the exact value of $R$ and give the value of $\alpha$ in radians to 3 decimal places. (3) The temperature, $\theta^{\circ} \mathrm{C}$, inside a room on a given day is modelled by the equation \[ \theta=5+\sin \left(\frac{\pi t}{12}-3\right)+2 \cos \left(\frac{\pi t}{12}-3\right) \quad 0 \leqslant t<24 \] where $t$ is the number of hours after midnight. Using the equation of the model and your answer to part (a), (b) deduce the maximum temperature of the room during this day, (c) find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

Public Answer

HILHUY The First Answerer