Check if a function is periodic and find its fundamental period
Select Function
f(x)=sin(x)
f(x)=cos(x)
f(x)=tan(x)
f(x)=x+
Result
Derivation
Periodicity Rules
sin(kx), cos(kx): period = 2pi/|k|
tan(kx): period = pi/|k|
Linear/poly: NOT periodic (except constant)
Sum of periodic functions: LCM of periods
A function is periodic if it repeats at regular intervals. Trigonometric functions are inherently periodic. Most other functions (linear, quadratic, exponential) are not periodic. Constant functions are trivially periodic.
⚠Constant functions are periodic with any period T. Non-constant polynomial functions are never periodic. Only trigonometric and related functions are periodic.
What Is Periodicity?
A periodic function repeats its values at regular intervals. The smallest positive interval is the fundamental period. Sine, cosine, and tangent are the most common periodic functions. Periodicity is a key property in Fourier analysis and signal processing.
Sine/Cosine
Period=2pi/k. sin(x+2pi)=sin(x). cos(x) has same period. Both bounded between -1 and 1.
Tangent
Period=pi/k (half that of sine). tan(x+pi)=tan(x). Has vertical asymptotes. Unbounded.
Non-Periodic
Linear, quadratic, cubic, exponential, log: none are periodic. They do not repeat. Constant f(x)=c is periodic.
Period of Sum
For sum of periodic functions, the period is the LCM of individual periods. Requires rational ratio of frequencies.
Teaching Example: f(x)=sin(2x). Period = 2pi/|2| = pi. Check: sin(2(x+pi)) = sin(2x+2pi) = sin(2x). Period is pi, not 2pi.
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