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.
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.
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