6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). The coordinates of the points \(A , C\) and \(D\) are \(( - 1 , - 7,2 ) , ( 5,0,3 )\) and \(( - 1,3,3 )\) respectively, and the equation of the plane \(B C D\) is \(x + 2 y - 2 z = - 1\).
- Find, in either order, the coordinates of \(B\) and the length of \(A B\).
- Find the acute angle between the planes \(A C D\) and \(B C D\).
- (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
(b) By sketching the graphs of \(y = \sin 6 \theta\) and \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), or otherwise, find the other solution of the equation \(\sin 6 \theta = \sin 2 \theta\) in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\). - Use de Moivre's theorem to prove that
$$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right) .$$
- Hence show that one of the solutions obtained in part (i) satisfies \(\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )\), and justify which solution it is.