Range of rational function with harmonic denominator

9. $$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$ a. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta - \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. Given that $$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$ b.find the range of \(g\).

9

1. Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\) and \(R\) is positive and given in exact form. The function \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = \frac { 1 } { ( k + \cos \theta + 2 \sin \theta ) } , 0 \leq \theta \leq 2 \pi , k\) is a constant.
2. The maximum value of \(\mathrm { f } ( \theta )\) is \(\frac { ( 3 + \sqrt { 5 } ) } { 4 }\). Find the value of \(k\).

12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.

a) Express \(9 \cos x + 40 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). b) Find the maximum possible value of \(\frac { 12 } { 9 \cos x + 40 \sin x + 47 }\).

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 \(f(\theta)\), 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 \(\frac{1}{7 + 3\cos\theta + 4\sin\theta}\). [6]