| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | May |
| Marks | 10 |
| Topic | Complex numbers 2 |
| Type | Integration using De Moivre identities |
| Difficulty | Challenging +1.2 This is a standard Further Maths question following a well-established template: prove a de Moivre identity, use binomial expansion to express a power of sin/cos in terms of multiple angles, then integrate. Part (a) is routine bookwork, part (b) requires careful algebraic manipulation but follows a known method, and part (c) is straightforward integration once part (b) is complete. While requiring multiple techniques and careful algebra, this represents a typical FM pure question without novel insight. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.02q De Moivre's theorem: multiple angle formulae |
4.
\begin{enumerate}[label=(\alph*)]
\item If $z = \cos \theta + \mathrm { i } \sin \theta$, use de Moivre's theorem to prove that
$$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
[3 marks]
\item Express $\sin ^ { 5 } \theta$ in terms of $\sin 5 \theta , \sin 3 \theta$ and $\sin \theta$\\[0pt]
[4 marks]
\item Hence show that
$$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }$$
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q4 [10]}}