Use the Factor Theorem to show that \(4 x - 3\) is a factor of
$$16 x ^ { 3 } + 11 x - 15$$
Given that \(x = \cos \theta\), show that the equation
$$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$
can be written in the form
$$16 x ^ { 3 } + 11 x - 15 = 0$$
Hence show that the only solutions of the equation
$$27 \cos \theta \cos 2 \theta + 19 \sin \theta \sin 2 \theta - 15 = 0$$
are given by \(\cos \theta = \frac { 3 } { 4 }\).