Evenness, Oddness & Periodicity of Trigonometric Functions
Big Idea: Trigonometric functions have powerful symmetry and repetition properties that simplify calculations and deepen our understanding of their behavior.
Part 1: Even and Odd Functions — Review
Before diving into trig functions, let's recall the general definitions:
Even Function
A function $f$ is even if for every $x$ in its domain:
$$f(-x) = f(x)$$
Graphically: Symmetric about the y-axis.
Example: $f(x) = x^2$
Odd Function
A function $f$ is odd if for every $x$ in its domain:
$$f(-x) = -f(x)$$
Graphically: Symmetric about the origin (180° rotational symmetry).
Example: $f(x) = x^3$