Use de Moivre's theorem to show that \(\sin ^ { 6 } \theta = - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )\).
It is given that \(\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )\).
Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 6 } \left( \frac { 1 } { 4 } x \right) + \sin ^ { 6 } \left( \frac { 1 } { 4 } x \right) \right) \mathrm { d } x\).
Express each root of the equation \(16 c ^ { 6 } + 16 \left( 1 - c ^ { 2 } \right) ^ { 3 } - 13 = 0\) in the form \(\cos k \pi\), where \(k\) is a rational number.
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.