1 Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence find the range of the function \(\mathbf { f } ( \theta )\), where $$f ( \theta ) = 7 + 3 \cos \theta + 4 \sin \theta \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi .$$ Write down the greatest possible value of \(\frac { 1 } { 7 + 3 \cos \theta + 4 \sin \theta }\).