Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
Hence solve the equation
$$5 \cos \theta - 3 \sin \theta = 4$$
giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
Write down the least value of \(15 \cos \theta - 9 \sin \theta\) as \(\theta\) varies.