Standard +0.3 This is a standard C4 harmonic form question with routine application of R cos(x-α) method followed by finding extrema, plus a straightforward trigonometric identity proof and equation solving. All techniques are textbook exercises requiring no novel insight, though the multi-part structure and identity manipulation place it slightly above average difficulty.
Express \(3 \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
(3 marks)
Hence find the minimum value of \(3 \cos x + 2 \sin x\) and the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) where the minimum occurs. Give your value of \(x\) to the nearest \(0.1 ^ { \circ }\).
Show that \(\cot x - \sin 2 x = \cot x \cos 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
Hence, or otherwise, solve the equation
$$\cot x - \sin 2 x = 0$$
in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $3 \cos x + 2 \sin x$ in the form $R \cos ( x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving your value of $\alpha$ to the nearest $0.1 ^ { \circ }$.\\
(3 marks)
\item Hence find the minimum value of $3 \cos x + 2 \sin x$ and the value of $x$ in the interval $0 ^ { \circ } < x < 360 ^ { \circ }$ where the minimum occurs. Give your value of $x$ to the nearest $0.1 ^ { \circ }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that $\cot x - \sin 2 x = \cot x \cos 2 x$ for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\item Hence, or otherwise, solve the equation
$$\cot x - \sin 2 x = 0$$
in the interval $0 ^ { \circ } < x < 180 ^ { \circ }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2013 Q3 [12]}}