1. (a) Express \(2 \sin x - 4 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 significant figures. In a town in Norway, a student records the number of hours of daylight every day for a year. He models the number of hours of daylight, \(H\), by the continuous function given by the formula $$H = 12 + 4 \sin \left( \frac { 2 \pi t } { 365 } \right) - 8 \cos \left( \frac { 2 \pi t } { 365 } \right) , \quad 0 \leqslant t \leqslant 365$$ where \(t\) is the number of days since he began recording.
(b) Using your answer to part (a), or otherwise, find the maximum and minimum number of hours of daylight given by this formula. Give your answers to 3 significant figures.
(c) Use the formula to find the values of \(t\) when \(H = 17\), giving your answers to the nearest integer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)