| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2021 |
| Session | October |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify expression to polynomial form |
| Difficulty | Standard +0.3 This is a straightforward application of small angle approximations (sin θ ≈ θ, cos θ ≈ 1 - θ²/2) with basic algebraic manipulation. Students substitute the approximations, expand (1 - θ²/8)² for the cosine term, and collect like terms. While it requires careful algebra and understanding of the approximations, it's a standard textbook exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| \(4\sin\frac{\theta}{2} \approx 4\left(\frac{\theta}{2}\right)\), \(3\cos^2\theta \approx 3\left(1-\frac{\theta^2}{2}\right)^2\) | M1 | Attempts to use at least one correct approximation within the given expression. Either \(\sin\frac{\theta}{2} \approx \frac{\theta}{2}\) or \(\cos\theta \approx 1 - \frac{\theta^2}{2}\) or e.g. \(\sin\theta \approx \theta\) if they write \(\cos^2\theta\) as \(1-\sin^2\theta\) |
| \(4\sin\frac{\theta}{2} + 3\cos^2\theta \approx 4\left(\frac{\theta}{2}\right) + 3\left(1 - \frac{\theta^2}{2}\right)^2\) | dM1 | Attempts to use correct approximations with the given expression to obtain expression in terms of \(\theta\) only. Depends on first M1. |
| \(= 2\theta + 3(1 - \theta^2 + \ldots) = 3 + 2\theta - 3\theta^2\) | A1 | Correct terms following correct work. Allow terms in any order; ignore extra terms if correct or incorrect. |
## Question 4:
$4\sin\frac{\theta}{2} \approx 4\left(\frac{\theta}{2}\right)$, $3\cos^2\theta \approx 3\left(1-\frac{\theta^2}{2}\right)^2$ | M1 | Attempts to use at least one correct approximation within the given expression. Either $\sin\frac{\theta}{2} \approx \frac{\theta}{2}$ or $\cos\theta \approx 1 - \frac{\theta^2}{2}$ or e.g. $\sin\theta \approx \theta$ if they write $\cos^2\theta$ as $1-\sin^2\theta$
$4\sin\frac{\theta}{2} + 3\cos^2\theta \approx 4\left(\frac{\theta}{2}\right) + 3\left(1 - \frac{\theta^2}{2}\right)^2$ | dM1 | Attempts to use correct approximations with the given expression to obtain expression in terms of $\theta$ only. Depends on first M1.
$= 2\theta + 3(1 - \theta^2 + \ldots) = 3 + 2\theta - 3\theta^2$ | A1 | Correct terms following correct work. Allow terms in any order; ignore extra terms if correct or incorrect.
---\begin{enumerate}
\item Given that $\theta$ is small and measured in radians, use the small angle approximations to show that
\end{enumerate}
$$4 \sin \frac { \theta } { 2 } + 3 \cos ^ { 2 } \theta \approx a + b \theta + c \theta ^ { 2 }$$
where $a , b$ and $c$ are integers to be found.
\hfill \mbox{\textit{Edexcel Paper 2 2021 Q4 [3]}}