Periodic Functions And Trigonometric Graphs

A periodic function repeats its values at regular intervals. Sine, cosine, tangent, and their transformed variations form the archetype of periodic behavior in mathematics and physics.

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Function	Fundamental Period	Domain	Range
$\sin x$	$2 π$	$(- \infty, + \infty)$	$[- 1, 1]$
$\cos x$	$2 π$	$(- \infty, + \infty)$	$[- 1, 1]$
$\tan x$	$π$	$x \neq \frac{π}{2} + k π$	$(- \infty, + \infty)$
$\cot x$	$π$	$x \neq k π$	$(- \infty, + \infty)$

Definition of Periodic Functions

Waveform showing repeating pattern with period T — A periodic function repeats its shape continuously every T units.

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

f (x + T) = f (x) for all x

The smallest positive period T of a periodic function is its fundamental period.

Trigonometric Periods and Graphs

Periodicity of fundamental trigonometric functions:

Sine and cosine repeat every $2 π$ radians: $\sin (θ + 2 π) = \sin θ$ and $\cos (θ + 2 π) = \cos θ$ . Fundamental period is $2 π$ .
Tangent and cotangent repeat every $π$ radians: $\tan (θ + π) = \tan θ$ and $\cot (θ + π) = \cot θ$ . Fundamental period is $π$ .

Graphs of sine, cosine, tangent, and cotangent over several periods — Fundamental periods of the basic trigonometric functions.

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Definition of Periodic Functions

Trigonometric Periods and Graphs

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