- (a) Express \(3 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), 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 \(0 \leqslant \theta < 360 ^ { \circ }\), the equation
$$3 \cos \theta + 5 \sin \theta = 2$$
Give your answers to one decimal place.
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which
$$3 \cos \theta - 5 \sin \theta = 2$$
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