| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | General solution — find all solutions |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of the general solution formula for cos θ = cos α (namely θ = ±α + 360n°), requiring only substitution and simple algebra. Part (b) involves algebraic manipulation using the product-to-sum formula or factor form of cosines, but with the hint provided, it's essentially pattern matching and verification rather than deep problem-solving. This is slightly easier than average for Further Maths content. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
**(a) Find the general solution, in degrees, of the equation**
$$\cos(5x + 40°) = \cos 65°$$
**(5 marks)**
**(b) Given that**
$$\sin \frac{\pi}{12} = \frac{\sqrt{3} - 1}{2\sqrt{2}}$$
**express $\sin \frac{\pi}{12}$ in the form $\cos \frac{\pi}{4} \cos(ap) + \cos(bp)$, where $a$ and $b$ are rational. (3 marks)**3
\begin{enumerate}[label=(\alph*)]
\item Find the general solution, in degrees, of the equation
$$\cos \left( 5 x + 40 ^ { \circ } \right) = \cos 65 ^ { \circ }$$
\item Given that
$$\sin \frac { \pi } { 12 } = \frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$$
express $\sin \frac { \pi } { 12 }$ in the form $\left( \cos \frac { \pi } { 4 } \right) ( \cos ( a \pi ) + \cos ( b \pi ) )$, where $a$ and $b$ are rational.\\
(3 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2013 Q3 [8]}}