Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.