| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Standard +0.8 This question requires systematic application of compound angle formulae, algebraic manipulation to reach a tan form, and then solving a related equation using double angle substitution. While the techniques are standard P3 content, the multi-step algebraic manipulation and the need to recognize the connection between parts (a) and (b) elevates it above routine exercises. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Expands using compound angle identities: \(\sqrt{2}(\sin x \cos 45° + \cos x \sin 45°) = \cos x \cos 60° + \sin x \sin 60°\) | M1 | Attempts to use compound angle identities to produce equation in \(\sin x\) and \(\cos x\) |
| \(\sqrt{2}(\sin x \cos 45° + \cos x \sin 45°) = \cos x \cos 60° + \sin x \sin 60°\) o.e. | A1 | Correct equation in \(\sin x\) and \(\cos x\); may be implied by further work |
| Collects like terms: \(\cos x = (\sqrt{3}-2)\sin x\) | M1 | Use of correct exact trig values for \(\sin 45°, \cos 45°, \cos 60°, \sin 60°\); collection of like terms in \(\sin x\) and \(\cos x\); use of \(\tan x = \frac{\sin x}{\cos x}\) — attempt at two of three |
| \(\tan x \left(= \frac{1}{\sqrt{3}-2} = \frac{\sqrt{3}+2}{-1}\right) = -2-\sqrt{3}\) * | A1* | Proceeds to given answer with all previous marks scored; no errors in manipulation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or uses \(x + 45° = 2\theta\) o.e. | B1 | Implied by sight of \(\tan(2\theta - 45°) = -2-\sqrt{3}\) |
| Proceeds from \(\tan(2\theta \pm \alpha°) = -2-\sqrt{3} \Rightarrow 2\theta \pm \alpha° = 105°\) or \(285°\) | M1 | Must achieve angle of \(105°\) or \(285°\) or equivalent; allow radians (1.83, 4.97) |
| Correct order of operations to solve their \(2\theta \pm \alpha° = \ldots\) | dM1 | Dependent on previous M1; \(\tan(2\theta-45°)=-2-\sqrt{3} \Rightarrow 75°\) alone does not imply this |
| \(\theta = 75°, 165°\) | A1 | Both angles with no others given within the range |
# Question 7:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Expands using compound angle identities: $\sqrt{2}(\sin x \cos 45° + \cos x \sin 45°) = \cos x \cos 60° + \sin x \sin 60°$ | M1 | Attempts to use compound angle identities to produce equation in $\sin x$ and $\cos x$ |
| $\sqrt{2}(\sin x \cos 45° + \cos x \sin 45°) = \cos x \cos 60° + \sin x \sin 60°$ o.e. | A1 | Correct equation in $\sin x$ and $\cos x$; may be implied by further work |
| Collects like terms: $\cos x = (\sqrt{3}-2)\sin x$ | M1 | Use of correct exact trig values for $\sin 45°, \cos 45°, \cos 60°, \sin 60°$; collection of like terms in $\sin x$ and $\cos x$; use of $\tan x = \frac{\sin x}{\cos x}$ — attempt at two of three |
| $\tan x \left(= \frac{1}{\sqrt{3}-2} = \frac{\sqrt{3}+2}{-1}\right) = -2-\sqrt{3}$ * | A1* | Proceeds to given answer with all previous marks scored; no errors in manipulation |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses $x + 45° = 2\theta$ o.e. | B1 | Implied by sight of $\tan(2\theta - 45°) = -2-\sqrt{3}$ |
| Proceeds from $\tan(2\theta \pm \alpha°) = -2-\sqrt{3} \Rightarrow 2\theta \pm \alpha° = 105°$ or $285°$ | M1 | Must achieve angle of $105°$ or $285°$ or equivalent; allow radians (1.83, 4.97) |
| Correct order of operations to solve their $2\theta \pm \alpha° = \ldots$ | dM1 | Dependent on previous M1; $\tan(2\theta-45°)=-2-\sqrt{3} \Rightarrow 75°$ alone does not imply this |
| $\theta = 75°, 165°$ | A1 | Both angles with no others given within the range |
---\begin{enumerate}
\item In this question you must show all stages of your working.
\end{enumerate}
Solutions relying entirely on calculator technology are not acceptable.\\
(a) Given that
$$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$
show that
$$\tan x = - 2 - \sqrt { 3 }$$
(b) Hence or otherwise, solve, for $0 \leqslant \theta < 180 ^ { \circ }$
$$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$
\hfill \mbox{\textit{Edexcel P3 2024 Q7 [8]}}