Tool to compute the period of a function. The period of a function is the lowest value t such that the function repeats itself : f(x+t)=f(x-t)=f(x), that is the case for trigo functions (cos, sin, etc.)

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Tool to compute the period of a function. The period of a function is the lowest value t such that the function repeats itself : f(x+t)=f(x-t)=f(x), that is the case for trigo functions (cos, sin, etc.)

Answers to Questions

How to find the period of a function?

To find the period \( t \) of a function \( f(x) \), demonstrate that $$ f(x+t)=f(x) $$

Example: The trigonometric function \( \sin(x + 2\pi) = \sin(x) \) so \( \sin(x) \) is periodic of period \( 2\pi \)

Trigonometric functions are usually periodic period, to guess the period, try multiples of pi for value \( t \).

The value of the period found is also called the periodicity of the function.

If the period is equal to 0, then the function is not periodic.

How to prove that a function is not periodic?

If \( f \) is periodic, then it exists a real not null such as $$ f(x+t)=f(x) $$ Demonstration consists in proving that it is impossible. For example with a reductio ad absurdum or performing a calculation that leads to a contradiction.

What are usual periodic functions?

The most common periodic functions are trigonometric functions based on sine and cosine functions (which have a period of 2 Pi).

Function

Period

Sine \( \sin(x) \)

\( 2\pi \)

Cosine \( \cos(x) \)

\( 2\pi \)

Tangent \( \tan(x) \)

\( \pi \)

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