**Symmetry and Composition of Trigonometric Functions**

Theorem: SCT: Symmetry of The Core Trigonometric Functions

Statement and justification for cosine as even and sine and Tangent as
odd symmetric functions. Visualize with mapping diagrams using unit
circle. Visualize with graphs and MD's.

Consider the trigonometric functions $f_s(x) = A\sin(Bx +C) $ and $f_c(x)= A\cos(Bx +C) $.

We explore how these forms are similar to the Vertex Form for quadratic functions. In particular they can
be used to understand the symmetry of the functions as a composition
of core functions.

In these examples we use SAGE and GeoGebra to visualize the composition for one linear core function with a core trigonometric function.

Compositions are a key to understanding the symmetry of trigonometric functions. The following key result on trigonometric functions as compositions is very useful.

This result says certain **trigonometric functions are built by composing
three core linear functions with a core trigonometric function:**
$trig(x)$ which can be any core trigonometric function with linear functions, $f_{+C}, f_B$, and $f_{*A}$.

Here is an example from TRIG.FORM that visualizes the theorem.

** FIX Example** TRIG.FORM.4
Suppose $f$
is a trigonometric function with leading coefficient $ =A=2$
and extreme value $f(1)=3$. Find the vertex form of the
trigonometric
function. Visualize $f$ with a mapping diagram that illustrates
the function as the composition of the four core functions
$f_{+3}∘f_{∗2}∘q∘f_{−1}$.

Understanding $f$ with compositions and mapping diagrams explains why trigonometric cosine functions of this form always have even symmetry with respect to an axis ($x = -C/B$).

You can use this next dynamic example with GeoGebra to investigate further the symmetry of a trigonometric functions in a mapping diagram of $f$.