| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Topic | Standard trigonometric equations |
| Type | Equation with non-equation preliminary part (sketch/proof/identity) |
| Difficulty | Moderate -0.3 Part (i) is a routine trigonometric equation requiring angle transformation and finding solutions in a given range—standard A-level fare worth 5 marks. Part (ii)(a) tests error analysis of a common algebraic mistake (dividing by sin x), which is conceptually important but straightforward to identify. Part (ii)(b) applies the corrected method to a slightly modified equation. Overall, this is a multi-part question testing standard techniques with no novel insight required, making it slightly easier than average due to its procedural nature and the scaffolding provided by the student's worked example. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
\begin{enumerate}[label=(\roman*)]
\item Solve, for $-90° \leqslant \theta < 270°$, the equation,
$$\sin(2\theta + 10°) = -0.6$$
giving your answers to one decimal place. [5]
\item \begin{enumerate}[label=(\alph*)]
\item A student's attempt at the question
"Solve, for $-90° < x < 90°$, the equation $7\tan x = 8\sin x$"
is set out below.
\begin{align}
7\tan x &= 8\sin x\\
7 \times \frac{\sin x}{\cos x} &= 8\sin x\\
7\sin x &= 8\sin x \cos x\\
7 &= 8\cos x\\
\cos x &= \frac{7}{8}\\
x &= 29.0° \text{ (to 3 sf)}
\end{align}
Identify two mistakes made by this student, giving a brief explanation of each mistake. [2]
\item Find the smallest positive solution to the equation
$$7\tan(4\alpha + 199°) = 8\sin(4\alpha + 199°)$$ [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM 2020 Q7 [9]}}