Express \(8 \sin x - 15 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
Give the exact value of \(R\), and give the value of \(\alpha\), in radians, to 4 significant figures.
$$\mathrm { f } ( x ) = \frac { 15 } { 41 + 16 \sin x - 30 \cos x } \quad x > 0$$
Find
the minimum value of \(\mathrm { f } ( x )\)
the smallest value of \(x\) at which this minimum value occurs.
State the \(y\) coordinate of the minimum points on the curve with equation
$$y = 2 \mathrm { f } ( x ) - 5 \quad x > 0$$
State the smallest value of \(x\) at which a maximum point occurs for the curve with equation
$$y = - \mathrm { f } ( 2 x ) \quad x > 0$$
\section*{8. In this question you must show all stages of your working.
In this question you must show all stages of your working.}