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.)

Period of a Function - dCode

Tag(s) : Functions

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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.)

The period $ t $ of a **periodic function** $ f(x) $ is the value $ t $ such that $$ f (x+t) = f(x) $$

Graphically, its curve is repeated each period, by translation. The function is equal to itself all the lengths $ t $.

The value of the period $ t $ is also called the periodicity of the 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 $.

If the period is equal to 0, then the 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.

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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