| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Standard +0.8 Part (a) requires knowledge of the factor formula for sin A - sin B (or triple angle expansion), which is standard Further Maths content but not trivial. Part (b) requires recognizing that the expression from (a) can be used, then integrating a product of trig functions over non-standard limits and computing a numerical mean—this involves multiple steps including product-to-sum identities or substitution, and careful arithmetic. The combination of formula manipulation, integration technique, and numerical computation makes this moderately challenging but still within standard Further Maths scope. |
| Spec | 1.05l Double angle formulae: and compound angle formulae4.08e Mean value of function: using integral |
| Answer | Marks |
|---|---|
| \(\sin\theta - \sin 3\theta = 2\cos\left(\frac{\theta + 3\theta}{2}\right)\sin\left(\frac{\theta - 3\theta}{2}\right)\) | M1 |
| \(= 2\cos 2\theta \sin(-\theta)\) | A1 |
| \(= -2\cos 2\theta \sin(\theta)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 2\cos 2\theta \sin\theta + 7 = -\sin\theta + \sin 3\theta + 7\) | B1 | FT (a) |
| Mean value \(= \frac{1}{\pi}\int_{\pi}^{3} (-\sin\theta + \sin 3\theta + 7)d\theta\) | M1 | |
| \(= \frac{1}{\pi}\left[\cos\theta - \frac{1}{3}\cos 3\theta + 7\theta\right]_{\pi}^{3}\) | A1 | |
| \(= \frac{1}{\pi}\left(\cos 3 - \frac{1}{3}\cos 9 + 7 \times 3\right) - \frac{1}{\pi}\left(\cos 1 - \frac{1}{3}\cos 3 + 7\right)\) | m1 A1 | Use of limits |
| \(= 6.22\) | If M0, award SC1 for 6.22 unsupported |
## Part a)
$\sin\theta - \sin 3\theta = 2\cos\left(\frac{\theta + 3\theta}{2}\right)\sin\left(\frac{\theta - 3\theta}{2}\right)$ | M1 |
$= 2\cos 2\theta \sin(-\theta)$ | A1 |
$= -2\cos 2\theta \sin(\theta)$ | A1 |
## Part b)
$y = 2\cos 2\theta \sin\theta + 7 = -\sin\theta + \sin 3\theta + 7$ | B1 | FT (a)
Mean value $= \frac{1}{\pi}\int_{\pi}^{3} (-\sin\theta + \sin 3\theta + 7)d\theta$ | M1 |
$= \frac{1}{\pi}\left[\cos\theta - \frac{1}{3}\cos 3\theta + 7\theta\right]_{\pi}^{3}$ | A1 |
$= \frac{1}{\pi}\left(\cos 3 - \frac{1}{3}\cos 9 + 7 \times 3\right) - \frac{1}{\pi}\left(\cos 1 - \frac{1}{3}\cos 3 + 7\right)$ | m1 A1 | Use of limits
$= 6.22$ | | If M0, award SC1 for 6.22 unsupported\begin{enumerate}[label=(\alph*)]
\item Show that $\sin \theta - \sin 3\theta$ can be expressed in the form $a\cos b\theta \sin \theta$, where $a$, $b$ are integers whose values are to be determined. [3]
\item Find the mean value of $y = 2\cos 2\theta \sin \theta + 7$ between $\theta = 1$ and $\theta = 3$, giving your answer correct to two decimal places. [5]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2019 Q5 [8]}}