Challenging +1.2 This is a standard Further Maths question requiring systematic application of de Moivre's theorem to express sin^6θ in terms of multiple angles, followed by a straightforward integration with substitution. While it involves multiple steps and algebraic manipulation, the technique is well-practiced in FP1 and follows a predictable pattern without requiring novel insight.
7 Use de Moivre's theorem to express \(\sin ^ { 6 } \theta\) in the form
$$a + b \cos 2 \theta + c \cos 4 \theta + d \cos 6 \theta$$
where \(a , b , c , d\) are constants to be found.
Hence evaluate
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } 2 x d x$$
leaving your answer in terms of \(\pi\).
7 Use de Moivre's theorem to express $\sin ^ { 6 } \theta$ in the form
$$a + b \cos 2 \theta + c \cos 4 \theta + d \cos 6 \theta$$
where $a , b , c , d$ are constants to be found.
Hence evaluate
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } 2 x d x$$
leaving your answer in terms of $\pi$.
\hfill \mbox{\textit{CAIE FP1 2009 Q7 [9]}}