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.
Other symmetries of the trigonometric functions are related to
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.
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