A function \(f\) has domain \(( - \infty , \infty )\) and is defined by \(f ( x ) = \cosh ^ { 3 } x - 3 \cosh x\).
Show that the graph of \(y = f ( x )\) has only one stationary point.
Find the nature of this stationary point.
State the largest possible range of \(f ( x )\).
When plotted on an Argand diagram, the four fourth roots of the complex number \(9 - 3 \sqrt { 3 } \mathrm { i }\) lie on a circle. Find the equation of this circle.
(a) By putting \(t = \tan \left( \frac { \theta } { 2 } \right)\), show that the equation
$$4 \sin \theta + 5 \cos \theta = 3$$
can be written in the form
$$4 t ^ { 2 } - 4 t - 1 = 0$$
Hence find the general solution of the equation
$$4 \sin \theta + 5 \cos \theta = 3$$