| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | September |
| Marks | 5 |
| Topic | Small angle approximation |
| Type | Identify error in approximation usage |
| Difficulty | Moderate -0.8 This question tests small angle approximations with straightforward algebraic manipulation in part (a), then asks students to identify a common error (forgetting to convert degrees to radians) and verify the approximation numerically. While it requires understanding of when approximations apply, the mathematical content is routine for Further Maths students and involves mainly substitution and basic algebra rather than problem-solving or novel insight. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Using my calculator, \(1 + 4\cos(5°) + 3\cos^2(5°) = 7.962\), to 3 decimal places. |
| Using the approximation \(8 - 5\theta^2\) gives \(8 - 5(5)^2 = -117\) |
| Therefore, \(1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2\) is not true for \(\theta = 5°\) |
\begin{enumerate}[label=(\alph*)]
\item Given that $\theta$ is small, use the small angle approximation for $\cos \theta$ to show that
$$1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$$ [3]
\end{enumerate}
Adele uses $\theta = 5°$ to test the approximation in part (a).
Adele's working is shown below.
\begin{tabular}{|l|}
\hline
Using my calculator, $1 + 4\cos(5°) + 3\cos^2(5°) = 7.962$, to 3 decimal places. \\
\\
Using the approximation $8 - 5\theta^2$ gives $8 - 5(5)^2 = -117$ \\
\\
Therefore, $1 + 4\cos \theta + 3\cos^2 \theta \approx 8 - 5\theta^2$ is not true for $\theta = 5°$ \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\roman*)]
\item Identify the mistake made by Adele in her working.
\item Show that $8 - 5\theta^2$ can be used to give a good approximation to $1 + 4\cos \theta + 3\cos^2 \theta$ for an angle of size $5°$ [2]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q4 [5]}}