| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Moderate -0.3 Part (a) requires straightforward application of double angle formulae (sin 2θ = 2sinθcosθ, cos 2θ = cos²θ - sin²θ) and rearrangement. Part (b) involves dividing by cos²θ to get a quadratic in tan θ, then solving—a standard technique. This is a routine multi-step question testing formula recall and algebraic manipulation with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use correct double angle formulae | M1 | e.g. \(2\sin\theta\cos\theta + \cos^2\theta - \sin^2\theta = 2\sin^2\theta\) |
| Obtain \(\cos^2\theta + 2\sin\theta\cos\theta - 3\sin^2\theta = 0\) from full and correct working | A1 | AG. Check conclusion is complete and matches the working |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Factorise to obtain \((\cos\theta - \sin\theta)(\cos\theta + 3\sin\theta) = 0\) | B1 | OE |
| Solve a quadratic in \(\sin\theta\) and \(\cos\theta\) to obtain a value for \(\theta\) | M1 | \(\tan\theta = 1\) or \(\tan\theta = -\frac{1}{3}\) |
| Obtain one correct value e.g. \(45°\) | A1 | |
| Obtain a second correct value e.g. \(161.6°\) and no others in the interval | A1 | Mark answers in radians \((0.785\) and \(2.82)\) as a misread. Accept awrt \(161.6\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain \(3\tan^2\theta - 2\tan\theta - 1 = 0\) | B1 | |
| Solve a 3 term quadratic in \(\tan\theta\) to obtain a value for \(\theta\) | M1 | \(\tan\theta = 1\) or \(\tan\theta = -\frac{1}{3}\) |
| Obtain one correct value e.g. \(45°\) | A1 | |
| Obtain a second correct value e.g. \(161.6°\) and no others in the interval | A1 | Mark answers in radians \((0.785\) and \(2.82)\) as a misread |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain \((\cos\theta + \sin\theta)^2 = (2\sin\theta)^2\) | B1 | |
| Solve to obtain a value for \(\theta\) | M1 | \(\tan\theta = 1\) or \(\tan\theta = -\frac{1}{3}\) |
| Obtain one correct value e.g. \(45°\) | A1 | |
| Obtain a second correct value e.g. \(161.6°\) and no others in the interval | A1 | Mark answers in radians \((0.785\) and \(2.82)\) as a misread |
| 4 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use correct double angle formulae | M1 | e.g. $2\sin\theta\cos\theta + \cos^2\theta - \sin^2\theta = 2\sin^2\theta$ |
| Obtain $\cos^2\theta + 2\sin\theta\cos\theta - 3\sin^2\theta = 0$ from **full and correct** working | A1 | AG. Check conclusion is complete and matches the working |
| | **2** | |
---
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Factorise to obtain $(\cos\theta - \sin\theta)(\cos\theta + 3\sin\theta) = 0$ | B1 | OE |
| Solve a quadratic in $\sin\theta$ and $\cos\theta$ to obtain a value for $\theta$ | M1 | $\tan\theta = 1$ or $\tan\theta = -\frac{1}{3}$ |
| Obtain one correct value e.g. $45°$ | A1 | |
| Obtain a second correct value e.g. $161.6°$ and no others in the interval | A1 | Mark answers in radians $(0.785$ and $2.82)$ as a misread. Accept awrt $161.6$ |
**Alternative Method 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain $3\tan^2\theta - 2\tan\theta - 1 = 0$ | B1 | |
| Solve a 3 term quadratic in $\tan\theta$ to obtain a value for $\theta$ | M1 | $\tan\theta = 1$ or $\tan\theta = -\frac{1}{3}$ |
| Obtain one correct value e.g. $45°$ | A1 | |
| Obtain a second correct value e.g. $161.6°$ and no others in the interval | A1 | Mark answers in radians $(0.785$ and $2.82)$ as a misread |
**Alternative Method 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain $(\cos\theta + \sin\theta)^2 = (2\sin\theta)^2$ | B1 | |
| Solve to obtain a value for $\theta$ | M1 | $\tan\theta = 1$ or $\tan\theta = -\frac{1}{3}$ |
| Obtain one correct value e.g. $45°$ | A1 | |
| Obtain a second correct value e.g. $161.6°$ and no others in the interval | A1 | Mark answers in radians $(0.785$ and $2.82)$ as a misread |
| | **4** | |
---4
\begin{enumerate}[label=(\alph*)]
\item Show that the equation $\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta$ can be expressed in the form
$$\cos ^ { 2 } \theta + 2 \sin \theta \cos \theta - 3 \sin ^ { 2 } \theta = 0$$
\item Hence solve the equation $\sin 2 \theta + \cos 2 \theta = 2 \sin ^ { 2 } \theta$ for $0 ^ { \circ } < \theta < 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q4 [6]}}