| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | September |
| Marks | 10 |
| Topic | Trigonometric equations in context |
| Type | Solve tan·sin or tan·trig product |
| Difficulty | Standard +0.3 Part (i) requires converting sin x tan x = 5 to sin²x/cos x = 5, then solving a quadratic in cos x - standard technique but slightly non-routine. Part (ii) involves reading amplitude from a graph, finding coordinates using sine function properties, and solving a transformed trig equation - all straightforward applications of A-level techniques with no novel insight required. The 'show detailed reasoning' requirement and multiple parts add some length but not conceptual difficulty. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}
\item In this question you must show detailed reasoning.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.\\
(i) Solve, for $0 \leq x < 360 ^ { \circ }$, the equation
$$\sin x \tan x = 5$$
giving your answers to one decimal place.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-24_641_732_778_680}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation
$$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$
where $A$ is a constant and $\theta$ is measured in radians.\\
The points $P , Q$ and $R$ lie on the curve and are shown in Figure 1.\\
Given that the $y$ coordinate of $P$ is 7\\
(a) state the value of $A$,\\
(b) find the exact coordinates of $Q$,\\
(c) find the value of $\theta$ at $R$, giving your answer to 3 significant figures.\\
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q11 [10]}}