Standard +0.3 Part (a) is a standard harmonic form question requiring routine application of R cos(x-α) = R cos α cos x + R sin α sin x, solving for R and α. Part (b) extends this by finding the maximum of a rational function, requiring recognition that the denominator is minimized when the harmonic expression is minimized. This is a straightforward two-part question with clear methodology, slightly easier than average due to its predictable structure and standard techniques.
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 \(\frac { 12 } { 9 \cos x + 40 \sin x + 47 }\).
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 $\frac { 12 } { 9 \cos x + 40 \sin x + 47 }$.
\hfill \mbox{\textit{WJEC Unit 3 2022 Q11}}