Challenging +1.2 This is a standard Further Maths question testing De Moivre's theorem and binomial expansion techniques. The first part is routine bookwork (showing the cos/sin identities). The main challenge is expanding (z^n - 1/z^n)^6 using binomial theorem and collecting terms, which requires careful algebra but follows a well-established method. The integration at the end is straightforward once the expression is obtained. While multi-step and requiring sustained accuracy, this is a typical FP1 examination question without novel insight required.
Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that
$$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2]
Hence express \(\sin^6 \theta\) in the form
$$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$
where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4]
Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]
Given that $z = e^{i\theta}$ and $n$ is a positive integer, show that
$$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2]
Hence express $\sin^6 \theta$ in the form
$$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$
where the constants $p$, $q$, $r$, $s$ are to be determined. [4]
Hence find the mean value of $\sin^6 \theta$ with respect to $\theta$ over the interval $0 \leq \theta \leq \frac{1}{4}\pi$. [5]
\hfill \mbox{\textit{CAIE FP1 2003 Q8 [11]}}