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)+D $ where $trig = \sin, \cos,$ or $\tan$.
These trigonometric functions are built by composing four core linear functions with a core trigonometric function: $trig(x)$ which can be any core trigonometric function with linear functions, $f_{+D}, f_{+C}, f_{*B}$, and $f_{*A}$. Thus $f = f_{+D}∘ f_{*A}∘ trig∘ f_{+C} ∘ f_{*B}$.

We explore how the symmetry of these functions, similar to the quadratic functions, can be understood 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: $A=B=1; trig(x) = \cos(x); C = -2; D =0; f(x)=cos(x-2).$
Example TRIG.SYM.2:$ A=B=1; trig(x) = \cos(x); C =0; D = -2; f(x)=cos(x)-2$
Example TRIG.SYM.3:$A=2; B=1; trig(x) = \cos(x); C = D =0; f(x)=2cos(x).$

Symmetry of Cosine Functions: Suppose $f(x)=A\cos(Bx +C)+D $. Understanding $f$ with compositions and mapping diagrams explains why trigonometric cosine functions of this form always have even symmetry with respect to an axis ($x = -C/B$).

Example TRIG.SYM.4: Suppose $f$ is a cosine function with amplitude $ =A=2$ and  maximum value $f(1)=3$ and period $2 \pi$. Find the composition form of the trigonometric function. Visualize $f$ with a mapping diagram that illustrates the function as the composition of the four core functions  $f_{+1}∘f_{∗2}∘\cos∘f_{−1}$ with even symmetry with respect to the axis $x = 1$..

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