- (a) Write \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 \leqslant \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
(b) Show that the equation
$$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$
can be rewritten in the form
$$5 \cos 2 x - 2 \sin 2 x = c$$
where \(c\) is a positive constant to be determined.
(c) Hence or otherwise, solve, for \(0 \leqslant x < \pi\),
$$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$
giving your answers to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)