| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Standard +0.3 This is a straightforward application of compound angle formulae requiring expansion of sin(θ-30°), algebraic manipulation to isolate tan θ, and solving a related equation. Part (a) is routine algebraic manipulation after applying standard formulae, while part (b) requires recognizing the structural similarity and applying a substitution. The multi-step nature and need for compound angle formula recall place it slightly above average, but it follows a standard template without requiring novel insight. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(2\sin(\theta-30°) = 5\cos\theta \Rightarrow 2\sin\theta\cos30° - 2\cos\theta\sin30° = 5\cos\theta\) | M1 | Attempts to use \(\sin(\theta-30°) = \sin\theta\cos(\pm30°) \pm \cos\theta\sin(\pm30°)\); condone omission of 2 on second term and slip on 5 |
| \(\div\cos\theta \Rightarrow 2\tan\theta\cos30° - 2\sin30° = 5\) | dM1 | Divides by \(\cos\theta\); may collect \(\sin\theta\) and \(\cos\theta\) terms first |
| \(\Rightarrow 2\tan\theta \times \frac{\sqrt{3}}{2} - 2\times\frac{1}{2} = 5\) | A1 | Fully correct equation in \(\tan\theta\) with \(\cos30°\) and \(\sin30°\) processed |
| \(\Rightarrow \sqrt{3}\tan\theta = 6 \Rightarrow \tan\theta = 2\sqrt{3}\) * | A1* | Correctly proceeds to given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts \(\arctan 2\sqrt{3}\) and subtracts \(20°\) | M1 | Allow \(\arctan 2\sqrt{3}\) followed by adding or subtracting \(20°\); answers with no working score 0 |
| \(x =\) awrt \(53.9°, 233.9°\) | A1, A1 | First A1: one correct value (allow awrt \(54°\) or \(234°\), or radians awrt 0.94 or 4.08); second A1: both values, no others in range; must be in degrees |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $2\sin(\theta-30°) = 5\cos\theta \Rightarrow 2\sin\theta\cos30° - 2\cos\theta\sin30° = 5\cos\theta$ | M1 | Attempts to use $\sin(\theta-30°) = \sin\theta\cos(\pm30°) \pm \cos\theta\sin(\pm30°)$; condone omission of 2 on second term and slip on 5 |
| $\div\cos\theta \Rightarrow 2\tan\theta\cos30° - 2\sin30° = 5$ | dM1 | Divides by $\cos\theta$; may collect $\sin\theta$ and $\cos\theta$ terms first |
| $\Rightarrow 2\tan\theta \times \frac{\sqrt{3}}{2} - 2\times\frac{1}{2} = 5$ | A1 | Fully correct equation in $\tan\theta$ with $\cos30°$ and $\sin30°$ processed |
| $\Rightarrow \sqrt{3}\tan\theta = 6 \Rightarrow \tan\theta = 2\sqrt{3}$ * | A1* | Correctly proceeds to given answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts $\arctan 2\sqrt{3}$ and subtracts $20°$ | M1 | Allow $\arctan 2\sqrt{3}$ followed by adding or subtracting $20°$; answers with no working score 0 |
| $x =$ awrt $53.9°, 233.9°$ | A1, A1 | First A1: one correct value (allow awrt $54°$ or $234°$, or radians awrt 0.94 or 4.08); second A1: both values, no others in range; must be in degrees |
---4. In this question you should show detailed reasoning.
\section*{Solutions relying entirely on calculator technology are not acceptable.}
\begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$
can be written in the form
$$\tan \theta = 2 \sqrt { 3 }$$
\item Hence, or otherwise, solve for $0 \leqslant x \leqslant 360 ^ { \circ }$
$$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$
giving your answers to one decimal place.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2021 Q4 [7]}}