Standard +0.3 This is a standard Further Maths question following a well-established procedure: apply de Moivre's theorem to express cos^4θ as a sum of multiple angle terms, then integrate. The method is algorithmic (binomial expansion of (e^{iθ} + e^{-iθ})^4, collect terms) and the integration is straightforward once the expression is obtained. While it requires knowledge of complex numbers and de Moivre's theorem (making it harder than typical A-level), it's a routine textbook exercise for FP1 students with no novel problem-solving required.
5 Use de Moivre's theorem to express \(\cos ^ { 4 } \theta\) in the form
$$a \cos 4 \theta + b \cos 2 \theta + c$$
where \(a , b , c\) are constants to be found.
Hence evaluate
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta d \theta$$
leaving your answer in terms of \(\pi\).
5 Use de Moivre's theorem to express $\cos ^ { 4 } \theta$ in the form
$$a \cos 4 \theta + b \cos 2 \theta + c$$
where $a , b , c$ are constants to be found.
Hence evaluate
$$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta d \theta$$
leaving your answer in terms of $\pi$.
\hfill \mbox{\textit{CAIE FP1 2011 Q5 [7]}}