Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
(2 marks)
Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.