| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Solve equations using trigonometric identities |
| Difficulty | Standard +0.8 Part (a) requires applying de Moivre's theorem to derive a multiple angle formula, involving binomial expansion of (cos θ + i sin θ)^5 and separating real/imaginary parts—a standard FP2 technique but algebraically intensive. Part (b) requires substituting both formulas to create a quintic equation in sin θ, factoring it, and solving multiple trigonometric equations in a given interval—this combines algebraic manipulation with systematic case analysis, going beyond routine exercises. |
| Spec | 1.05o Trigonometric equations: solve in given intervals4.02q De Moivre's theorem: multiple angle formulae |
\begin{enumerate}[label=(\alph*)]
\item Use de Moivre's theorem to show that
$$\sin 5\theta = 16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta$$ [5]
\end{enumerate}
Hence, given also that $\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find all the solutions of
$$\sin 5\theta = 5 \sin 3\theta$$
in the interval $0 \leq \theta < 2\pi$. Give your answers to 3 decimal places. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q7 [11]}}