Periodic Functions and Trigonometric Graphs

Periodic functions repeat the same pattern indefinitely. The trigonometric functions are the most important periodic functions in mathematics, appearing in models of waves, oscillations, and rotations. Understanding their graphs and periods is essential for sketching, computing limits, and evaluating integrals.

Function	Period
$\sin x$ , $\cos x$	$2 π$
$\tan x$ , $\cot x$	$π$
$A \sin (k x + B)$ , $A \cos (k x + B)$	$\frac{2 π}{k}$ ( $k > 0$ )
$A \tan (k x + B)$ , $A \cot (k x + B)$	$\frac{π}{k}$ ( $k > 0$ )

Periodic Functions

The graph of a function $f$ may have a pattern that repeats forever in both directions. Such functions are called periodic.

A periodic function with period T, showing the repeating pattern shifted by T units

A periodic function with period $T$ . Periodic functions have translational symmetry.

A function $f$ is periodic with period $T$ ( $T \neq 0$ ) if

f (x + T) = f (x);

that is, adding $T$ to the input does not change the output.

If $f$ is periodic with period $T$ , it is also periodic with periods $2 T$ , $3 T$ , $- T$ , $- 2 T$ , etc. This is because $f (x + 2 T) = f ((x + T) + T) = f (x + T) = f (x)$ , and $f (x - T) = f (x - T + T) = f (x)$ .

The smallest positive period of a periodic function (if it exists) is called the fundamental period of the function.

Periods of Trigonometric Functions

The most important periodic functions are the trigonometric functions.

Sine and cosine are defined as the $y$ and $x$ coordinates of the intersection point of the terminal side of an angle $θ$ with the unit circle. Since one revolution corresponds to $2 π$ radians, the same point is reached for $θ + 2 π k$ (integer $k$ ). Therefore: $\sin (θ + 2 π) = \sin θ, \cos (θ + 2 π) = \cos θ .$ The fundamental period of both $\sin x$ and $\cos x$ is $2 π$ .
The tangent and cotangent functions have the smaller fundamental period $π$ : $\tan (θ + π) = \tan θ, \cot (θ + π) = \cot θ .$

Graphs of sine, cosine, tangent, and cotangent over two full periods, showing their characteristic shapes

The fundamental periods of sine and cosine are $2 π$ , and the fundamental periods of tangent and cotangent are $π$ .

Transformed Trigonometric Functions

The fundamental period of $y = A \sin (k x + B)$ and $y = A \cos (k x + B)$ is $\frac{2 π}{k}$ (for $k > 0$ ). The constant $A$ is the amplitude and $B$ is the phase shift.
The fundamental period of $y = A \tan (k x + B)$ and $y = A \cot (k x + B)$ is $\frac{π}{k}$ (for $k > 0$ ).

Frequently Asked Questions

What does the amplitude of

A \sin (k x + B)

tell you?

The amplitude

| A |

is the maximum distance the function reaches from zero. The function oscillates between

- | A |

and

| A |

. If

A < 0

, the graph is a vertical reflection of the standard sine curve. Note that

\tan x

and

\cot x

have no amplitude because they are unbounded.

How do I find the period of

y = \cos (3 x - π)

Identify

k = 3

(the coefficient of

x

). The period is

\frac{2 π}{k} = \frac{2 π}{3}

. The phase shift is determined from

k x + B = 0

, giving

x = - B / k = π / 3

(shift right by

π / 3

). The amplitude is 1.

Why does tangent have a smaller period than sine and cosine?

Because

\tan θ = \sin θ / \cos θ

depends on the ratio of the coordinates, not the coordinates themselves. The ratio completes one full cycle of values every half-revolution (

π

radians), rather than every full revolution (

2 π

radians). You can verify:

\tan (θ + π) = \frac{\sin (θ + π)}{\cos (θ + π)} = \frac{- \sin θ}{- \cos θ} = \tan θ