Table of Contents
Understanding Periodicity in Trigonometric Functions
Trigonometric functions are classic examples of periodic functions. In this chapter, we focus on what “periodic” means and how to work with the periods of sine, cosine, tangent, and their transformed versions.
What It Means for a Function to Be Periodic
A function $f$ is called periodic if there exists a positive number $T$ such that, for every $x$ in its domain,
$$
f(x + T) = f(x).
$$
Any such number $T$ is called a period of $f$.
The fundamental period (or least period) is the smallest positive number $T$ for which this equation holds.
Key ideas:
- The graph of a periodic function repeats itself over and over along the $x$-axis.
- Once you know the graph on any interval of length $T$ (for example, from $0$ to $T$), you can reproduce the whole graph by repeating that “pattern” to the left and right.
Not every function is periodic. A straight line like $f(x) = 2x + 1$ is not periodic because its values never repeat in a regular cycle.
Basic Periods of Sine, Cosine, and Tangent
You should already be familiar with the definitions and basic graphs of $\sin x$, $\cos x$, and $\tan x$, including their values on the unit circle. Here, we focus only on how those definitions lead to periodicity.
Sine and Cosine: Period $2\pi$
By the geometry of the unit circle, adding $2\pi$ radians corresponds to going once around the circle and landing at the same point.
This leads to:
$$
\sin(x + 2\pi) = \sin x, \qquad \cos(x + 2\pi) = \cos x
$$
for all real $x$. Therefore:
- $\sin x$ is periodic with period $2\pi$.
- $\cos x$ is periodic with period $2\pi$.
Moreover, $2\pi$ is the fundamental period of both sine and cosine. There is no smaller positive number $T$ for which $\sin(x + T) = \sin x$ holds for all real $x$ (and similarly for cosine).
Graphically:
- One complete “wave” of $\sin x$ or $\cos x$ occurs over an interval of length $2\pi$.
- The pattern from $0$ to $2\pi$ repeats from $2\pi$ to $4\pi$, from $-2\pi$ to $0$, and so on.
Tangent: Period $\pi$
The tangent function is defined (where it exists) by
$$
\tan x = \frac{\sin x}{\cos x}.
$$
Because both sine and cosine repeat every $2\pi$, it is not surprising that tangent is also periodic. However, tangent repeats sooner, after only $\pi$ radians:
$$
\tan(x + \pi) = \tan x
$$
for all $x$ where both sides are defined (i.e., where $\cos x \neq 0$ and $\cos(x+\pi) \neq 0$).
Thus:
- $\tan x$ is periodic with period $\pi$.
- $\pi$ is the fundamental period of $\tan x$.
Graphically:
- One full cycle of the tangent graph, including its repeating shape and asymptotes, occurs over any interval of length $\pi$, for example from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ or from $0$ to $\pi$ (excluding the points where $\tan x$ is undefined).
Other Basic Trig Functions
Without going into their detailed properties:
- $\cot x$ has period $\pi$.
- $\sec x$ and $\csc x$ have period $2\pi$.
These follow from their definitions in terms of sine and cosine.
Period of Transformed Trigonometric Functions
In practice, we often work with transformed functions like
$$
y = A\sin(Bx + C) + D, \quad
y = A\cos(Bx + C) + D, \quad
y = A\tan(Bx + C) + D.
$$
Here $A$, $B$, $C$, and $D$ are constants.
Each parameter affects the graph in a different way (such as vertical stretch, shift, etc.), which is treated in more detail in graph-focused chapters. Here we care only about the effect on the period.
Sine and Cosine: Period $\dfrac{2\pi}{|B|}$
The basic functions $\sin x$ and $\cos x$ have period $2\pi$. When you replace $x$ by $Bx$ (or $Bx+C$), the graph is “compressed” or “stretched” horizontally.
For:
$$
y = \sin(Bx), \quad y = \cos(Bx),
$$
the period becomes
$$
\text{Period} = \frac{2\pi}{|B|}.
$$
The same period holds for
$$
y = A\sin(Bx + C) + D, \quad
y = A\cos(Bx + C) + D.
$$
- $A$ (vertical stretch/flip) does not change the period.
- $C$ (horizontal shift, or phase shift) does not change the period length, just where a cycle starts.
- $D$ (vertical shift) also does not affect the period.
So, for any nonzero $B$:
- The function repeats every $\dfrac{2\pi}{|B|}$ along the $x$-axis.
Example:
- For $y = 3\sin(2x - \pi)$, the period is
$$
\frac{2\pi}{|2|} = \pi.
$$
Tangent (and Cotangent): Period $\dfrac{\pi}{|B|}$
The basic tangent function $\tan x$ has period $\pi$. As with sine and cosine, replacing $x$ by $Bx$ changes the horizontal scale.
For:
$$
y = \tan(Bx), \quad y = \cot(Bx),
$$
the period is
$$
\text{Period} = \frac{\pi}{|B|}.
$$
And for
$$
y = A\tan(Bx + C) + D, \quad
y = A\cot(Bx + C) + D,
$$
the period is still
$$
\frac{\pi}{|B|}.
$$
Again, $A$, $C$, and $D$ do not change the length of one cycle; they only stretch vertically or shift the graph.
Example:
- For $y = \tan\left(\frac{x}{3}\right)$, $B = \frac{1}{3}$, so the period is
$$
\frac{\pi}{|1/3|} = 3\pi.
$$
How to Find the Period From the Equation
To find the period of a basic transformed trig function, look for the coefficient of $x$ inside the sine, cosine, or tangent.
- Write the function in a standard form:
- $y = A\sin(Bx + C) + D$
- $y = A\cos(Bx + C) + D$
- $y = A\tan(Bx + C) + D$
- Identify $B$ (the coefficient of $x$).
- Use:
- For sine or cosine:
$$
\text{Period} = \frac{2\pi}{|B|}.
$$ - For tangent:
$$
\text{Period} = \frac{\pi}{|B|}.
$$ - Ignore $A$, $C$, and $D$ when calculating period; they do not affect the period length.
A useful way to remember:
- Start with the basic period (sine/cosine: $2\pi$, tangent: $\pi$) and always divide by $|B|$.
Relating Periodicity to the Unit Circle
Every trigonometric function comes from moving around the unit circle:
- One full turn around the circle is $2\pi$ radians.
- After going around $2\pi$ radians, the point on the circle is exactly where it started, so its coordinates repeat.
- Because $\sin x$ and $\cos x$ are defined using those coordinates, they naturally repeat every $2\pi$.
Tangent is based on the ratio $\sin x / \cos x$. This ratio repeats after $\pi$ radians, even though the point on the circle is not back to its original location; the angle has turned halfway around, and both sine and cosine have changed signs, leaving the ratio unchanged.
This geometric picture explains why:
- Sine, cosine, secant, and cosecant have period $2\pi$.
- Tangent and cotangent have period $\pi$.
Multiple Periods and Repetition
Once you know the fundamental period $T$ of a function $f$, other periods are just multiples of $T$:
- If $f$ has fundamental period $T$, then
$$
f(x + nT) = f(x) \quad \text{for every integer } n.
$$ - Numbers like $2T$, $3T$, $5T$ are also periods, but they are not smaller than $T$, so they are not fundamental periods.
On a graph, this means:
- You can draw one cycle on any interval of length $T$, then tile the entire $x$-axis with copies of that cycle shifted left and right by $nT$.
For trigonometric functions, this repetition is what produces their wave-like or repeating-asymptote patterns.
Periodicity in Degree Measure
So far, periods were described in radians. If you use degrees instead:
- $\sin x$ and $\cos x$ have period $360^\circ$.
- $\tan x$ has period $180^\circ$.
For transformed functions like $\sin(Bx)$ where $x$ is in degrees, you can think similarly:
- Basic period (in degrees) $\div |B|$ gives the new period.
For example:
- $y = \sin(3x)$, with $x$ in degrees, has period
$$
\frac{360^\circ}{3} = 120^\circ.
$$
The same ideas about repetition and cycles apply; only the units (radians vs degrees) change.
Why Periodicity Matters
Periodicity tells you that:
- Trigonometric functions model repeating or cyclical behavior (waves, rotations, oscillations).
- You can understand their entire behavior by studying one interval of length equal to the period.
- When solving equations like $\sin x = \frac{1}{2}$ or $\tan x = 1$, the periodic nature explains why there are infinitely many solutions, repeating every period (or sometimes every half-period, depending on the function and equation).
A solid grasp of periods makes it much easier to:
- Sketch accurate trig graphs.
- Describe oscillations in physics, engineering, and other applications.
- Work confidently with trigonometric equations and identities that exploit repetition.