| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Standard +0.3 This is a multi-part trigonometric equation question requiring standard techniques: (a) finding a minimum point using phase shift knowledge, (b) solving a trig equation by expanding cos(x-30°) and using standard methods, and (c) deducing roots from the transformation x→2x. While it requires multiple steps and understanding of transformations, all techniques are routine for Further Maths students with no novel insight required. Slightly easier than average due to the structured guidance through parts. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-05_700_1281_884_488}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$y = 5 \cos ( x - 30 ) ^ { \circ } \quad x \geqslant 0$$
The point $P$ on the curve is the minimum point with the smallest positive $x$ coordinate.
\begin{enumerate}[label=(\alph*)]
\item State the coordinates of $P$.
\item Solve, for $0 \leqslant x < 360$, the equation
$$5 \cos ( x - 30 ) ^ { \circ } = 4 \sin x ^ { \circ }$$
giving your answers to one decimal place.\\
(4)
\item Deduce, giving reasons for your answer, the number of roots of the equation
$$5 \cos ( 2 x - 30 ) ^ { \circ } = 4 \sin 2 x ^ { \circ } \text { for } 0 \leqslant x < 3600$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q5 [8]}}