| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Small angle approximation |
| Type | Simplify single fraction to numerical value |
| Difficulty | Standard +0.8 This requires knowing small angle approximations (sin θ ≈ θ, cos θ ≈ 1 - θ²/2, tan θ ≈ θ), applying them to multiples of θ (tan 3θ ≈ 3θ, cos 2θ ≈ 1 - 2θ²), then simplifying the resulting algebraic fraction. It's more than routine substitution as students must handle the denominator carefully and recognize the cancellation pattern, but it's a standard Further Maths small angle question without requiring deep insight. |
| Spec | 1.05e Small angle approximations: sin x ~ x, cos x ~ 1-x^2/2, tan x ~ x |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{\theta \tan 3\theta}{\cos 2\theta - 1} \approx \frac{\theta \times 3\theta}{1 - \frac{(2\theta)^2}{2} - 1}\) | M1 | Attempts to use the small angle approximations |
| \(\approx \frac{3\theta^2}{-2\theta^2}\) | M1 | Substitutes correctly and attempts to simplify |
| \(\approx -\frac{3}{2}\) | A1 | Uses both identities and simplifies to \(-\frac{3}{2}\) |
## Question 1:
| $\frac{\theta \tan 3\theta}{\cos 2\theta - 1} \approx \frac{\theta \times 3\theta}{1 - \frac{(2\theta)^2}{2} - 1}$ | M1 | Attempts to use the small angle approximations |
|---|---|---|
| $\approx \frac{3\theta^2}{-2\theta^2}$ | M1 | Substitutes correctly and attempts to simplify |
| $\approx -\frac{3}{2}$ | A1 | Uses both identities and simplifies to $-\frac{3}{2}$ |
---\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}
$$\frac { \theta \tan 3 \theta } { \cos 2 \theta - 1 }$$
\hfill \mbox{\textit{Edexcel PMT Mocks Q1 [3]}}