| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve 2sinθ = tanθ type equation |
| Difficulty | Standard +0.3 This is a standard P2 trigonometric equation requiring factorization (sin θ common factor), finding solutions in a given range, then applying a simple substitution for part (b). While it requires multiple steps and careful angle work, the techniques are routine for this level with no novel insight needed—slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| States/uses \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) to get \(\frac{2\sin\theta}{\cos\theta}+3\sin\theta=0\) | M1 | Condone slips in coefficients |
| Cross-multiplies and factorises/cancels to form linear equation in \(\cos\theta\): \(\sin\theta(2+3\cos\theta)=0\) | dM1 | Dependent on M1 |
| \(\cos\theta=-\frac{2}{3} \Rightarrow\) one correct \(\theta\) value (awrt 132° or 228°) | A1 | Condone radian solutions awrt 2.3(0) or awrt 3.98 |
| \(\cos\theta=-\frac{2}{3} \Rightarrow\) both values: awrt 131.8° and awrt 228.2° | A1 | Extra values in range from \(\cos\theta=-\frac{2}{3}\) give A0 |
| \(\sin\theta=0 \Rightarrow \theta=180°, 360°\) | B1 | Condone 0 appearing as solution; must be preceded by \(\sin\theta=0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Sets \(2x+40°=\) their \(131.8°\) and proceeds to \(x=\ldots\) | M1 | Correct order: \(x=\frac{\text{"131.8°"}\pm 40°}{2}\); follow through on smallest value \(>40°\) from (a) |
| \(x=\) awrt \(45.9°\) | A1 | Single answer only; do not accept multiple solutions |
# Question 3:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| States/uses $\tan\theta = \frac{\sin\theta}{\cos\theta}$ to get $\frac{2\sin\theta}{\cos\theta}+3\sin\theta=0$ | M1 | Condone slips in coefficients |
| Cross-multiplies and factorises/cancels to form linear equation in $\cos\theta$: $\sin\theta(2+3\cos\theta)=0$ | dM1 | Dependent on M1 |
| $\cos\theta=-\frac{2}{3} \Rightarrow$ one correct $\theta$ value (awrt 132° or 228°) | A1 | Condone radian solutions awrt 2.3(0) or awrt 3.98 |
| $\cos\theta=-\frac{2}{3} \Rightarrow$ both values: awrt 131.8° and awrt 228.2° | A1 | Extra values in range from $\cos\theta=-\frac{2}{3}$ give A0 |
| $\sin\theta=0 \Rightarrow \theta=180°, 360°$ | B1 | Condone 0 appearing as solution; must be preceded by $\sin\theta=0$ |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Sets $2x+40°=$ their $131.8°$ and proceeds to $x=\ldots$ | M1 | Correct order: $x=\frac{\text{"131.8°"}\pm 40°}{2}$; follow through on smallest value $>40°$ from (a) |
| $x=$ awrt $45.9°$ | A1 | Single answer only; do not accept multiple solutions |
---\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) Solve, for $0 < \theta \leqslant 360 ^ { \circ }$ the equation
$$2 \tan \theta + 3 \sin \theta = 0$$
giving your answers, as appropriate, to one decimal place.\\
(b) Hence, or otherwise, find the smallest positive solution of
$$2 \tan \left( 2 x + 40 ^ { \circ } \right) + 3 \sin \left( 2 x + 40 ^ { \circ } \right) = 0$$
giving your answer to one decimal place.
\hfill \mbox{\textit{Edexcel P2 2023 Q3 [7]}}