$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$
Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),
- find the value of \(R\) and the value of \(\alpha\).
- Hence solve the equation
$$7 \cos 2 x - 24 \sin 2 x = 12.5$$
for \(0 \leqslant x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
- Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\), and \(c\) are constants to be found.
- Hence, using your answers to parts (a) and (c), deduce the maximum value of
$$14 \cos ^ { 2 } x - 48 \sin x \cos x$$