1. What Is Periodicity?
A function is called periodic if its values repeat after a fixed interval. This interval is called the period of the function.
In mathematical form, a function \(f(x)\) is periodic with period \(T\) if:
\( f(x + T) = f(x) \text{ for all } x \)
Trigonometric functions are naturally periodic because they are based on circular motion.
2. Periodicity of sin x and cos x
Both sine and cosine repeat their values every 2π units.
\( \sin(x + 2\pi) = \sin x \)
\( \cos(x + 2\pi) = \cos x \)
Their waves look identical after each interval of \(2\pi\).
2.1. Reason for Their Period
Sine and cosine are based on the coordinates of points on the unit circle. A full rotation around the circle measures 2π radians, so returning to the same coordinate repeats the value.
3. Periodicity of tan x and cot x
The tangent and cotangent functions repeat more frequently.
\( \tan(x + \pi) = \tan x \)
\( \cot(x + \pi) = \cot x \)
Their period is π because the slope of the terminal side repeats after a half-turn.
3.1. Why Their Period Is π
In terms of the unit circle, the slope of a line through the origin repeats after a 180° rotation (π radians), giving tan and cot a shorter cycle.
4. Periodicity of sec x and cosec x
These functions are reciprocals of cos x and sin x respectively. Therefore, their periods match the periods of their parent functions:
\( \sec(x + 2\pi) = \sec x \)
\( \csc(x + 2\pi) = \csc x \)
5. Summary Table of Periods
| Function | Period |
|---|---|
| sin x | 2π |
| cos x | 2π |
| tan x | π |
| cot x | π |
| sec x | 2π |
| cosec x | 2π |
6. Examples to Understand Periodicity
Example 1: Show that sin 30° = sin 390°.
Since 390° = 30° + 360°, and 360° = 2π radians:
\( \sin(30° + 360°) = \sin 30° \)
Example 2: tan 45° = tan 225° because:
\( 225° = 45° + 180° \)
\( \tan(45° + 180°) = \tan 45° \)