| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Derive triple angle then solve equation |
| Difficulty | Standard +0.3 This is a structured multi-part question with clear guidance at each step. Part (i) is a standard derivation using addition formulae and double angle identities. Part (ii) involves algebraic substitution following explicit instructions. Part (iii) requires solving a trigonometric equation and back-substituting. While it requires multiple techniques and careful algebra, the question provides substantial scaffolding and uses well-practiced methods, making it slightly easier than the average A-level question. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(\sin(A + B)\) formula to express \(\sin 3\theta\) in terms of trig. functions of \(2\theta\) and \(\theta\) | M1 | |
| Use correct double angle formulae and Pythagoras to express \(\sin 3\theta\) in terms of \(\sin\theta\) | M1 | |
| Obtain a correct expression in terms of \(\sin\theta\) in any form | A1 | |
| Obtain the given identity | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Substitute for \(x\) and obtain the given answer | B1 | [1] |
| (iii) Carry out a correct method to find a value of \(x\) | M1 | |
| Obtain answers 0.322, 0.799, −1.12 | A1 + A1 + A1 | [4] |
**(i)** Use $\sin(A + B)$ formula to express $\sin 3\theta$ in terms of trig. functions of $2\theta$ and $\theta$ | M1 |
Use correct double angle formulae and Pythagoras to express $\sin 3\theta$ in terms of $\sin\theta$ | M1 |
Obtain a correct expression in terms of $\sin\theta$ in any form | A1 |
Obtain the given identity | A1 | [4]
**[SR: Give M1 for using correct formulae to express RHS in terms of $\sin\theta$ and $\cos 2\theta$, then M1A1 for expressing in terms of $\sin\theta$ and $\sin 3\theta$ only, or in terms of $\cos\theta, \sin\theta, \cos 2\theta$ and $\sin 2\theta$, then A1 for obtaining the given identity.]**
**(ii)** Substitute for $x$ and obtain the given answer | B1 | [1]
**(iii)** Carry out a correct method to find a value of $x$ | M1 |
Obtain answers 0.322, 0.799, −1.12 | A1 + A1 + A1 | [4]
**[Solutions with more than 3 answers can only earn a maximum of A1 + A1.]**8 (i) By first expanding $\sin ( 2 \theta + \theta )$, show that
$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
(ii) Show that, after making the substitution $x = \frac { 2 \sin \theta } { \sqrt { 3 } }$, the equation $x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0$ can be written in the form $\sin 3 \theta = \frac { 3 } { 4 }$.\\
(iii) Hence solve the equation
$$x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0$$
giving your answers correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P3 2014 Q8 [9]}}