IAL 2021 Jan Q8 | 國際數學站

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Given that $z = e^{i\theta}$

(a) show that $z^n + \dfrac{1}{z^n} = 2\cos n\theta$

where $n$ is a positive integer.

(2)

(b) Show that

\begin{align*} \cos^6\theta = \frac{1}{32}\left(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10\right) \end{align*}

(5)

\begin{align*} \cos 6\theta + 6\cos 4\theta + 15\cos 2\theta = 0 \qquad 0 \leqslant \theta \leqslant \pi \end{align*}

Give your answers to $3$ significant figures.

(4)

(d) Use calculus to determine the exact value of

\begin{align*} \int_{0}^{\frac{\pi}{3}} \left(32\cos^6\theta - 4\cos^2\theta\right)\mathrm{d}\theta \end{align*}

Solutions relying entirely on calculator technology are not acceptable.

(5)