Periodicity of Functions

Definition

How Many Periods Does a Periodic Function Have?

If $T$ is a period, then for any positive integer $n$, $nT$ is also a period:

and by induction, $f(x + nT) = f(x)$ for every positive integer $n$. So a periodic function has infinitely many periods.

Can a Period Be Negative?

Yes. If $T$ is a period, then $-T$ is also a period: letting $y = x - T$, we have $f(y + T) = f(y)$, i.e. $f(x) = f(x - T)$.

For example, $y = \sin x$ repeats every $2\pi$:

That is, $\sin(x + 2\pi) = \sin x$ for all $x$.

Fundamental Period

Since any integer multiple $nT$ of a period is also a period, a periodic function has infinitely many positive and negative periods. For convenience we usually care about the fundamental (least positive) period.

Is the Fundamental Period Unique?

Yes—if a periodic function has a fundamental period, it is unique.

Proof sketch (by contradiction): suppose $T_1$ and $T_2$ are two different fundamental periods with $T_1 > T_2$. Since both are periods, $T_1 - T_2$ is also a period (because $f(x + (T_1 - T_2)) = f((x + T_1) - T_2) = f(x)$). But $0 < T_1 - T_2 < T_1$, contradicting that $T_1$ is the smallest positive period. Hence the fundamental period is unique.

Is a Constant Function Periodic?

Yes. For $f(x) = c$, every positive $T$ satisfies $f(x + T) = c = f(x)$.

However, a constant function has no fundamental period: for any proposed $T_0$, there is a smaller positive number that is also a period. So not every periodic function has a fundamental period.

Common Periodic Functions

The fundamental periods of the common ones:

• $\sin x$, $\cos x$: fundamental period $2\pi$
• $\tan x$, $\cot x$: fundamental period $\pi$
• $\sin^2 x$, $\cos^2 x$: fundamental period $\pi$

Important Properties

Arithmetic of Periodic Functions

Property: If $f(x)$ and $g(x)$ are periodic with periods $T_1$ and $T_2$, and they have a common period $T$ (for example, when $T_1/T_2$ is rational), then:

1. The sum $f(x) + g(x)$ is periodic
2. The difference $f(x) - g(x)$ is periodic
3. The product $f(x) \cdot g(x)$ is periodic
4. The quotient $\frac{f(x)}{g(x)}$ (where defined and the denominator is nonzero) is periodic

Examples:

• $\sin x + \cos x$: both have period $2\pi$, so the sum has period $2\pi$
• $\sin x + \sin(2x)$: $\sin x$ has period $2\pi$, $\sin(2x)$ has period $\pi$; they share the period $2\pi$, so the sum has period $2\pi$

Caution: if $T_1$ and $T_2$ are incommensurable, the result of arithmetic may not be periodic at all—for example, $\sin x+\sin(\sqrt{2}x)$ is not periodic.

Composition of Periodic Functions

Property: If $f(x)$ is periodic with period $T$, the periodicity of $f(g(x))$ depends on the inner function $g(x)$.

• Case 1: if $g(x) = kx + b$ ($k \neq 0$) is linear, then $f(kx + b)$ is periodic with period $\frac{T}{|k|}$.
• Case 2: if $g(x)$ itself is periodic with period $T_g$, then $f(g(x))$ has $T_g$ as a period; its fundamental period needs further analysis.

Examples:

• $f(x) = \sin(3x)$: since $\sin x$ has period $2\pi$, $\sin(3x)$ has period $\frac{2\pi}{3}$
• $f(x) = \cos(x + \frac{\pi}{4})$: translation does not change the period, so it is $2\pi$

Integration of Periodic Functions

Property: If $f(x)$ is periodic with period $T$ and integrable on $[a, a + T]$, then its integral over any interval of length one period is the same:

Meaning: the integral of a periodic function over any full period is identical—useful when computing averages or power of periodic signals.

Example: for $f(x) = \sin x$ with period $2\pi$,

for every $a$.

How to Decide

Deciding whether a function is periodic, and finding its period, takes different methods. The following examples walk through the main ones.

Example 1

Decide whether $f(x) = \sin x$ is a periodic function; if it is, find its period.

Analysis:

By the definition of a periodic function, we need a nonzero constant $T$ such that $f(x + T) = f(x)$ for every $x$ in the domain.

That is, we need $T \neq 0$ with $\sin(x + T) = \sin x$ for all $x$.

From the definition of the sine function (on the unit circle, the sine of an angle is the vertical coordinate of the point at that angle), the angles $x$ and $x + 2\pi$ pick out the same point of the unit circle (a rotation by $2\pi$ returns to the same point), so their sines agree: $\sin(x + 2\pi) = \sin x$.

Verification: take $T = 2\pi$. For every $x$,

and $2\pi \neq 0$, so $f(x) = \sin x$ is a periodic function and $2\pi$ is one of its periods.

Method summary: this is the definition method, which uses the definition of a periodic function directly. Solve $f(x + T) = f(x)$, or simply verify a candidate $T$, to decide whether such a $T$ exists.

Example 2

Read the period of the sine function $y = \sin x$ and the tangent function $y = \tan x$ off their graphs.

Analysis:

Look at the graph of the sine function $y = \sin x$:

The values wave up and down, and the same pattern repeats every $2\pi$. Therefore the sine function has period $2\pi$.

Look at the graph of the tangent function $y = \tan x$:

The values jump periodically, and the same pattern repeats every $\pi$. Therefore the tangent function has period $\pi$.

Method summary: this is the graph method, which decides periodicity by looking for repetition in the graph. The graph method is intuitive and shows the geometric meaning of periodicity, but it needs an accurate drawing or a careful reading of the graph.

Example 3

Find the period of each of the following functions:

1. $f(x) = \sin(3x)$
2. $f(x) = \tan(\frac{x}{2})$
3. $f(x) = \sin x + \cos x$

Analysis:

(1) $f(x) = \sin(3x)$

The period of $\sin x$ is $2\pi$, and for $\sin(kx)$ ($k \neq 0$) the period is $\frac{2\pi}{|k|}$.

Therefore the period of $\sin(3x)$ is $\frac{2\pi}{3}$.

(2) $f(x) = \tan(\frac{x}{2})$

The period of $\tan x$ is $\pi$, and for $\tan(kx)$ ($k \neq 0$) the period is $\frac{\pi}{|k|}$.

Therefore the period of $\tan(\frac{x}{2})$ is $\frac{\pi}{\frac{1}{2}} = 2\pi$.

(3) $f(x) = \sin x + \cos x$

Both $\sin x$ and $\cos x$ have period $2\pi$.

If two periodic functions have a common period, their sum has that common period as well.

Since $\sin x$ and $\cos x$ both have period $2\pi$, the period of $\sin x + \cos x$ is $2\pi$.

Method summary: this is the formula method, which decides periodicity from known periodic functions and their properties. The usual rules are:

• $\sin(kx)$, $\cos(kx)$ ($k \neq 0$): period $\frac{2\pi}{|k|}$
• $\tan(kx)$, $\cot(kx)$ ($k \neq 0$): period $\frac{\pi}{|k|}$
• for $f(kx + b)$ ($k \neq 0$), if $f(x)$ has period $T$, then $f(kx + b)$ has period $\frac{T}{|k|}$
• if $f(x)$ and $g(x)$ have a common period $T$, then $f(x) + g(x)$ and $f(x) - g(x)$ both have period $T$

The formula method is fast and accurate, and it covers the common periodic functions.

Beyond the Formula

1. Periodic Phenomena in Nature

Periodic phenomena are everywhere in nature, and periodic functions are the natural way to describe them:

Celestial motion:

• Day and night: the Earth’s rotation makes day and night cycle with a period of about 24 hours
• Phases of the moon: the Moon’s orbit around the Earth has a period of about 29.5 days
• Seasons: the Earth’s revolution around the Sun produces the seasons, with a period of 1 year (about 365.25 days)

Biological rhythms:

• Biological clocks: many organisms have physiological cycles with a 24-hour period (the circadian rhythm)
• Sleep cycles: the human sleep-wake cycle and the flowering time of plants are both periodic

Wave phenomena:

• Water waves: the motion of waves on the surface has a clear periodicity
• Sound waves: sound travels as periodic vibrations, and the frequency decides how high the pitch is

Mathematical modeling:

These natural phenomena can be described with trigonometric functions, for example:

• the daily temperature swing can be approximated by $T(t) = A\sin(\omega t + \phi) + B$ (where $A$ is the amplitude, $\omega$ is the angular frequency, and the period is $\frac{2\pi}{\omega}$)
• the phases of the moon can be modeled with a sine or cosine function

Understanding periodic functions helps us describe and predict these natural phenomena better.

Exercises

Exercise 1

Find the period of the function $f(x) = \sin(3x) + \cos(2x)$.

Exercise 2

Determine whether the function $f(x) = \sin^2 x + \cos^2 x$ is a periodic function. If so, find its period.

Exercise 3

Find the fundamental period of $f(x) = \tan\left(2x + \frac{\pi}{3}\right)$.

Summary

Symbols Used in This Article