Well that's pretty straightforward. You sketch the graphs of those trigonometric functions in the required interval (no particular skill required). Then you decide whether the pattern formed by the graph (yes they will have a pattern) will go on forever.
For example, your first equation would give a graph "bouncing ball but never losing height". So it is periodic (why should it stop ?). And your second equation will be even (symmetric by the y-axis) and will give a characteristic graph, and it will obviously be periodic.
Anyway, most functions involving sine, cosine or tangent are periodic.