Range of simple harmonic function

6. (a) Express \(\sqrt { 5 } \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 significant figures.
(b) Solve, for \(- \pi < \theta < \pi\), $$\sqrt { 5 } \cos \theta - 2 \sin \theta = 0.5$$ giving your answers to 3 significant figures. [Solutions based entirely on graphical or numerical methods are not acceptable.] $$\mathrm { f } ( x ) = A ( \sqrt { 5 } \cos \theta - 2 \sin \theta ) + B \quad \theta \in \mathbb { R }$$ where \(A\) and \(B\) are constants. Given that the range of f is $$- 15 \leqslant f ( x ) \leqslant 33$$ (c) find the value of \(B\) and the possible values of \(A\).

6

1. Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(0 < \alpha < \frac { \pi } { 2 }\).
2. Write down the range of the function $$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$

Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac{1}{2}\pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\). [6]