| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify single fraction to numerical value |
| Difficulty | Moderate -0.3 This is a straightforward application of small angle approximations (cos θ ≈ 1 - θ²/2, sin θ ≈ θ) requiring substitution and algebraic simplification. It's slightly easier than average because it's a direct recall question with minimal steps, though students must remember to apply approximations to 4θ and 3θ correctly. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Attempts either \(\sin 3\theta \approx 3\theta\) or \(\cos 4\theta \approx 1 - \frac{(4\theta)^2}{2}\) in \(\frac{1-\cos 4\theta}{2\theta \sin 3\theta}\) | M1 | 1.1b |
| Attempts both \(\sin 3\theta \approx 3\theta\) and \(\cos 4\theta \approx 1 - \frac{(4\theta)^2}{2}\) giving \(\frac{1-\left(1-\frac{(4\theta)^2}{2}\right)}{2\theta \times 3\theta}\) and attempts to simplify | M1 | 2.1 |
| \(= \frac{4}{3}\) oe | A1 | 1.1b |
# Question 1:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts either $\sin 3\theta \approx 3\theta$ or $\cos 4\theta \approx 1 - \frac{(4\theta)^2}{2}$ in $\frac{1-\cos 4\theta}{2\theta \sin 3\theta}$ | M1 | 1.1b |
| Attempts both $\sin 3\theta \approx 3\theta$ and $\cos 4\theta \approx 1 - \frac{(4\theta)^2}{2}$ giving $\frac{1-\left(1-\frac{(4\theta)^2}{2}\right)}{2\theta \times 3\theta}$ and attempts to simplify | M1 | 2.1 |
| $= \frac{4}{3}$ oe | A1 | 1.1b |
**Notes:** Condone missing bracket on the $4\theta$ so $\cos 4\theta \approx 1 - \frac{4\theta^2}{2}$ would score the method. Expect simplification to a single term in $\theta$. Look for answer of $k$ but condone $k\theta$ following a slip. A1 requires both identities used, simplifies to $\frac{4}{3}$, no incorrect lines. Condone awrt 1.33.
**Alt:** $\frac{1-\cos 4\theta}{2\theta \sin 3\theta} = \frac{1-(1-2\sin^2 2\theta)}{2\theta \sin 3\theta} = \frac{2\sin^2 2\theta}{2\theta \sin 3\theta} = \frac{2\times(2\theta)^2}{2\theta \times 3\theta} = \frac{4}{3}$
---\begin{enumerate}
\item Given that $\theta$ is small and is measured in radians, use the small angle approximations to find an approximate value of
\end{enumerate}
$$\frac { 1 - \cos 4 \theta } { 2 \theta \sin 3 \theta }$$
(3)
\hfill \mbox{\textit{Edexcel Paper 1 2018 Q1 [3]}}