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.
Four more examples: General trigonometric functions with combined compositions.
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