Consider the core trigonometric functions: sine and cosine.

The values of these functions vary in predictable ways depending on the interval being considered.

This is apparent by reviewing the mapping diagrams along with the graphs in our initial examples for this section.

But first we review the key concept for periodic behavior for a function.

Definition PerF: Periodic Function

Period behavior of the core trigonometric functions.
In the following theorem, graphs and mapping diagrams visualize the facts that the sine and cosine  functions have period 2π while the tangent function has period π. These results are justified in all texts that cover trigonometric functions and are easily understood by considering the unit circle definitions of these functions.
Theorem TRIG.PERIOD The period for the sine, cosine, and tangent functions

You can use this next dynamic example to investigate visually the period effects of composition of the sine and cosine functions with the core linear functions $f_m(x)=mx$ on a mapping diagram of these functions and their connection to the wrapping function.

See also TRIG.APP Trigonometric Function Applications on the amplitude, period, and phase shift of wave functions.

Example TRIG.DPB.0 Dynamic Visualization of Periodic Behavior for Trigonometric Functions: Graphs, and Mapping Diagrams