In mathematics, the Period is the length of time or distance it takes for a repeating pattern (a cycle) to finish one complete repetition. Functions that repeat themselves at regular intervals are called Periodic Functions.
[Image of periodic function graph showing one full cycle]You encounter periods every day: the hands of a clock return to the same spot every 12 hours, the seasons repeat every year, and your heart beats in a regular rhythm.
1. Defining the Period
Mathematically, a function $f(x)$ is periodic if there is a number $P$ such that:
The smallest positive value of $P$ for which this is true is called the Fundamental Period.
2. Period of Sine and Cosine
The standard Sine and Cosine functions describe circular motion. Since a full circle is $360^{\circ}$ or $2\pi$ radians, these functions repeat every $2\pi$.
- Parent Function: $y = \sin(x)$
- Standard Period: $2\pi$ (approx. 6.28)
3. Calculating Period from an Equation
When the variable $x$ is multiplied by a coefficient $B$, the speed of the cycle changes. This stretches or compresses the wave horizontally.
For functions like $y = \sin(Bx)$ or $y = \cos(Bx)$, the formula for the period ($P$) is:
[Image of formula for period of sine and cosine]Example:
Find the period of $y = \cos(4x)$.
- Here, $B = 4$.
- $P = 2\pi / 4 = \pi / 2$.
- This wave repeats 4 times faster than the standard wave.
4. Period of Tangent
The Tangent function is unique. It repeats twice as often as Sine or Cosine. Its standard period is just $\pi$.
5. Period vs. Frequency
These two concepts are inverses of each other.
- Period: How long it takes for one cycle to occur (Time per Cycle).
- Frequency: How many cycles occur in one unit of time (Cycles per Time).
For example, if a pendulum swings back and forth every 2 seconds (Period = 2s), its frequency is 0.5 swings per second (0.5 Hz).
Conclusion
Understanding the Period allows us to predict the future behavior of cyclic phenomena. Whether calculating the tides for a harbor or tuning a radio frequency, mastering the period is essential for analyzing the rhythms of the world.