IAL 2022 June Q8 | 國際數學站

(a) Use de Moivre’s theorem to show that

\begin{align*} \sin 5\theta \equiv\,& 16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta\\[2mm] \end{align*}

(5)

(b) Hence determine the five distinct solutions of the equation

\begin{align*} 16x^5 - 20x^3 + 5x + \frac{1}{5} =\,& 0\\[2mm] \end{align*}

giving your answers to 3 decimal places.

(5)

\begin{align*} \int_{0}^{\frac{\pi}{4}} (4 \sin^5 \theta - 5 \sin^3 \theta - 6 \sin \theta) \mathrm{d}\theta =\,& a\sqrt{2} + b\\[2mm] \end{align*}

where $a$ and $b$ are rational numbers to be determined.

(4)