| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Topic | Trigonometric equations in context |
| Type | Show then solve substituted equation |
| Difficulty | Standard +0.3 This is a standard A-level trig equation question with clear scaffolding. Part (a) requires routine algebraic manipulation (expanding, using tan = sin/cos, and the Pythagorean identity), while part (b) applies the given result with a straightforward substitution θ = 2x. The quadratic formula and range restriction are standard techniques. Slightly easier than average due to the helpful structure. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}
\item In this question you must show detailed reasoning.
\end{enumerate}
\section*{Solutions relying entirely on calculator technology are not acceptable.}
(a) Show that the equation
$$( 3 \cos \theta - \tan \theta ) \cos \theta = 2$$
can be written as
$$3 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
(b) Hence solve for $- \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }$
$$( 3 \cos 2 x - \tan 2 x ) \cos 2 x = 2$$
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q11 [7]}}