| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Standard +0.3 Part (i) is a standard triple angle proof using compound angle formulas (cos(2θ+θ)), requiring 3-4 routine steps. Part (ii) applies the proven identity to integrate cos³θ, which is a direct substitution followed by standard integration—methodical but not demanding novel insight. This is slightly above average difficulty due to the two-part structure and the need to recognize how to apply part (i) to part (ii), but both components are well-practiced A-level techniques. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct \(\cos(A + B)\) formula to express \(\cos 3\theta\) in terms of trig functions of \(2\theta\) and \(\theta\) | M1 | — |
| Use correct trig formulae and Pythagoras to express \(\cos 3\theta\) in terms of \(\cos\theta\) | M1 | — |
| Obtain a correct expression in terms of \(\cos\theta\) in any form | A1 | — |
| Obtain the given identity correctly | A1 | [4] |
| [SR: Give M1 for using correct formulae to express RHS in terms of \(\cos\theta\) and \(\cos 2\theta\), then M1A1 for expressing in terms of either only \(\cos 3\theta\) and \(\cos\theta\), or only \(\cos 2\theta\), \(\sin 2\theta\), \(\cos\theta\), and \(\sin\theta\), and A1 for obtaining the given identity correctly.] | — | — |
| (ii) Use identity and integrate, obtaining terms \(\frac{1}{4}(\frac{1}{3}\sin 3\theta) + \frac{1}{4}(3\sin\theta)\), or equivalent | B1 + B1 | — |
| Use limits correctly in an integral of the form \(k\sin 3\theta + l\sin\theta\) | M1 | — |
| Obtain answer \(\frac{2}{3} - \frac{3}{8}\sqrt{3}\), or any exact equivalent | A1 | [4] |
**(i)** Use correct $\cos(A + B)$ formula to express $\cos 3\theta$ in terms of trig functions of $2\theta$ and $\theta$ | M1 | —
Use correct trig formulae and Pythagoras to express $\cos 3\theta$ in terms of $\cos\theta$ | M1 | —
Obtain a correct expression in terms of $\cos\theta$ in any form | A1 | —
Obtain the given identity correctly | A1 | [4]
| [SR: Give M1 for using correct formulae to express RHS in terms of $\cos\theta$ and $\cos 2\theta$, then M1A1 for expressing in terms of either only $\cos 3\theta$ and $\cos\theta$, or only $\cos 2\theta$, $\sin 2\theta$, $\cos\theta$, and $\sin\theta$, and A1 for obtaining the given identity correctly.] | — | —
**(ii)** Use identity and integrate, obtaining terms $\frac{1}{4}(\frac{1}{3}\sin 3\theta) + \frac{1}{4}(3\sin\theta)$, or equivalent | B1 + B1 | —
Use limits correctly in an integral of the form $k\sin 3\theta + l\sin\theta$ | M1 | —
Obtain answer $\frac{2}{3} - \frac{3}{8}\sqrt{3}$, or any exact equivalent | A1 | [4]
---\begin{enumerate}[label=(\roman*)]
\item Prove the identity $\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta$. [4]
\item Using this result, find the exact value of
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \cos^3 \theta \, d\theta.$$ [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2010 Q7 [8]}}