Symmetry  of  Trigonometric Functions

We begin with a statement and visual justification for cosine as an even symmetric function and sine and tangent as odd symmetric functions.
Theorem:TRIG.SYM. Symmetry of The Core Trigonometric Functions

Other symmetries of the trigonometric functions are related to
Theorem:TRIG.TID. Trigonometric  Identities.

Consider the trigonometric functions $f(x) = A trig(Bx +C) $ where $trig = \sin, \cos,$ or $\tan$.

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 these functions as a composition of core functions.

Three simple examples.
In these examples we use GeoGebra  to visualize symmetry with the composition for one linear core function with a core trigonometric function.
Example TRIG.SYM.1 : $ f(x) = \cos(x); g(x) = x-2$
Example TRIG.SYM.2 :$ f(x) = x - 2; g(x) = \cos(x)$
Example TRIG.SYM.3 :$ f(x) = 2x ; g(x) = \cos(x)$

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.SYM.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  $f_{+3}∘f_{∗2}∘q∘f_{−1}$.

Symmetry of Trigonometric Functions: Suppose $f(x) =A\cos(Bx +C) $.

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 $f$.

Example TRIG.DSYMM.0 Dynamic Visualization of Symmetry for  Trigonometric Functions: Graphs, and Mapping Diagrams