Linear Composition with Core Trigonometric Functions

Simple modifications of sine, cosine or tangent functions often involve compositions with the core linear functions, giving functions in a form similar in some ways to the vertex form for quadratic polynomials:
[$A \ne 0, B\ne 0$]
$f_{trig}(x) = A \cdot trig (Bx+C) + D$

where $trig$ is  $\sin$, $\cos$ or $\tan$.

A general trigonometric function is a composition of core linear functions with a core trigonometric function.

Many features of the core trigonometric functions are retained and transformed when composed with linear functions as long as $A\ne 0$ and $B\ne 0$.
These features include the period [Subsection TRIG.PB] , extreme values [Subsection TRIG.ID] and the related amplitude, as well as phase shift.

To simplify slightly we consider the trigonometric functions with $D = 0$:
 $f_s(x) = A\sin(Bx +C) $, $f_c(x)= A\cos(Bx +C) $, and $f_t(x)= A\tan(Bx +C) $.

For $\sin$ and $\cos$ the extremes are $A$ and $-A$, the period is $2\pi/|B|$, the amplitude is $A$, and the phase shift is $-C/B$.

We explore how mapping diagrams for linear compositions with core trigonometric functions help us visualize and understand further these functions.
Four simple examples.

In these examples we use GeoGebra  to visualize the composition $f=h \circ g$ for one linear core function with a core trigonometric function.
Example TRIG.COMP.1: $ h(x) = \cos(x); g(x) = x-2; f(x) = \cos(x-2)$

Example TRIG.COMP.2: $ h(x) = x - 2; g(x) = \cos(x); f(x) = \cos(x)-2$

Example TRIG.COMP.3: $ h(x) = 2x ; g(x) = \cos(x); f(x) = 2\cos(x)$

Example TRIG.COMP.4: $ h(x) = \cos(x) ; g(x) = 2x; f(x) = \cos(2x)$

Four more examples: General trigonometric functions with combined compositions.
Example  TRIG.LCOMP.1: $f(x) =2\cdot trig(x-\pi/2)$ where $trig$ is  $\sin, \cos$ or $\tan$.

Example TRIG.LCOMP.2:  $f(x) =2\cdot trig(\frac x2 + \frac {\pi}2)$ where $trig$ is  $\sin, \cos$ or $\tan$.

Example TRIG.LCOMP.3:  $f(x) =2\cdot trig(2x + \frac {\pi}2)$ where $trig$ is  $\sin, \cos$ or $\tan$.

Example TRIG.LCOMP.4:  Suppose $f$ is a trigonometric function of the form  $f_s(x) = 2\sin(Bx +C) $ with period $\pi$, and  maximum value $f(1) =2$.  Find $B$ and one (of the many) possible value(s) for $C$. Visualize $f$ with a mapping diagram that illustrates the function as the composition of the three core functions  $f_{∗2}∘\sin∘f_{+C}∘f_{\times B}$

Understanding general trigonometric functions with compositions and mapping diagrams helps make sense for  many of the features of these functions.

You can use this next dynamic example with GeoGebra to investigate further the effects of linear compositions on a core trigonometric function with mapping diagrams.
In the example, visualizing the role of composition with the unit circle mapping diagrams that define these functions also helps make sense of the important connection to the core trigonometric functions.

Example TRIG.DCOMP.0: Dynamic Visualization of Linear Composition for Core Trigonometric Functions: Graphs, and Mapping Diagrams