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 concepts: increasing and decreasing- and an extreme value for a function.

Definition ID Increasing/Decreasing
Definition EV Extreme Value for a Function

You can use the next examples to investigate visually the effects of linear composition prior to applying the core trigonometric functions on the increasing and decreasing shape of a more general trigonometric function by considering a mapping diagram of $f$ and the graph of $f$ where $f(x) = trig(Bx+C)$ where $trig(x) = \sin(x);\ trig(x) =\cos(x);$ or $trig(x)=\tan(x)$.
Example TRIG.ID.1 : $B =\frac {\pi}2, C = 0, f(x) = \sin(\frac {\pi}2 x)$
Example TRIG.ID.2 : $B = 1, C =\frac {\pi}2, f(x) = \cos(x+\frac {\pi}2)$

You can use the next example to investigate visually the effects of other linear compositions prior to applying the core trigonometric functions on the increasing and decreasing shape of a more general trigonometric function by considering a mapping diagram  of $f$ and the graph of $f$.

Example TRIG.DID.0 Dynamic Visualization of Increasing and Decreasing for Trigonometric Functions: Graphs, and Mapping Diagrams