Periodic Functions | Moosa Academy

Some functions repeat the same pattern forever: the tide, a heartbeat, a wheel turning. A function like that is called periodic, and the length of one repeat is its period.

Concept Periodic function and period

A function $f$ is periodic when $f(x + T) = f(x)$ for every $x$ . The period $T$ is the smallest positive value for which this holds.

For $\sin\theta$ and $\cos\theta$ the period is $360° = 2\pi$ radians. After a full turn around the unit circle the point is back where it started, so the values start over — and they carry on repeating in both directions, without end.

Example One cycle of the sine function

At 0° it starts at 0.

At 90° it reaches its maximum, 1.

At 180° it is back to 0.

At 270° it reaches its minimum, −1.

⟹ at 360° it returns to 0 — one full period is complete

Note Periodic is wider than circular

Circular functions — sin, cos, tan — come from the unit circle.

Periodic functions are any functions that repeat, such as this square wave.

Every circular function is periodic; not every periodic function is circular.

Reference Some periodic functions

Function	Period	Shape
sin θ	360° = 2π	smooth wave, starts at 0
cos θ	360° = 2π	smooth wave, starts at 1
Sawtooth	as defined	a ramp, then a jump back
Square wave	as defined	alternates between two values
Triangle wave	as defined	straight rise, straight fall

Note What changes the wave

In $y = A\sin(Bx)$ , the two constants do different jobs. A larger $B$ squeezes the wave and shortens the period; a smaller $B$ stretches it out.

Changing $A$ stretches the curve vertically without touching the period, and adding a shift slides the whole curve sideways without changing its shape.

Summary

A periodic function repeats the same pattern: f(x + T) = f(x).
The period T is the smallest positive value that satisfies that condition.
sin θ and cos θ are the best-known examples, both with period 360° = 2π.
Sawtooth, square and triangle waves are periodic too, but they are not circular functions.