CAIE FP1 2017 June — Question 8

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2017
SessionJune
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
8
  1. Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\) and hence express \(16 \sin ^ { 5 } \theta\) in the form \(\sin 5 \theta + p \sin 3 \theta + q \sin \theta\), where \(p\) and \(q\) are integers to be determined.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta\).
8 (i) Let $z = \cos \theta + \mathrm { i } \sin \theta$. Show that $z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta$ and hence express $16 \sin ^ { 5 } \theta$ in the form $\sin 5 \theta + p \sin 3 \theta + q \sin \theta$, where $p$ and $q$ are integers to be determined.\\

(ii) Hence find the exact value of $\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } 16 \sin ^ { 5 } \theta \mathrm {~d} \theta$.\\

\hfill \mbox{\textit{CAIE FP1 2017 Q8}}
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