| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Solve equations using trigonometric identities |
| Difficulty | Standard +0.8 This FP3 question requires using De Moivre's theorem and binomial expansion to derive a multiple angle identity, then solving a trigonometric equation using that result. Part (i) is a guided derivation requiring careful algebraic manipulation of complex exponentials (5 marks suggests multiple steps), while part (ii) requires recognizing that the equation simplifies to cos θ = -1/2 and finding solutions in the given range. The techniques are standard for FP3 but require precision and multiple steps, placing it moderately above average difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}[label=(\roman*)]
\item By expressing $\cos \theta$ in terms of $e^{i\theta}$ and $e^{-i\theta}$, show that
$$\cos^5 \theta \equiv \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta).$$ [5]
\item Hence solve the equation $\cos 5\theta + 5\cos 3\theta + 9\cos \theta = 0$ for $0 \leqslant \theta \leqslant \pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 Q4 [9]}}