- (a) Express \(7 \sin 2 \theta - 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
(b) Hence solve, for \(0 \leqslant \theta < 90 ^ { \circ }\), the equation
$$7 \sin 2 \theta - 2 \cos 2 \theta = 4$$
giving your answers in degrees to one decimal place.
(c) Express \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\) in the form \(a \sin 2 \theta + b \cos 2 \theta + c\), where \(a\), \(b\) and \(c\) are constants to be found.
(d) Use your answers to part (a) and part (c) to deduce the exact maximum value of \(28 \sin \theta \cos \theta + 8 \sin ^ { 2 } \theta\)