Tool to compute the period of a function: the 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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The period $ t $ of a periodic function $ f(x) $ is the smallest value $ t $ such that $$ f(x+t) = f(x) $$
Graphically, its curve is repeated over the interval of each period. The function is equal to itself for every cycle of length $ t $ (it presents a pattern/graph that is repeated by translation).
The value of the period $ t $ is also called the periodicity of the function or fundamental period.
To find the period $ t $ of a signal or a function $ f(x) $, demonstrate that $$ f(x+t)=f(x) $$
Trigonometric/sinusoidal functions are usually periodic, with a period $ 2\pi $, to guess the period, try multiples of pi for value $ t $.
If the period is equal to 0, then the function is not periodic.
Any periodic function of period $ t $ repeats every $ t $ values. To predict the cycle value of a periodic function, for a value $ x $ calculate $ x_t = x \mod t $ (modulo t) and find the known value of $ f(x_t) = f(x) $
Example: The function $ f(x) = \cos(x) $ has a period of $ 2\pi $, the value for $ x = 9 \pi $ is the same as for $ x \equiv 9 \pi \mod 2 \pi \equiv \pi \mod 2 \pi $ and therefore $ \cos(9 \pi) = \cos(\pi) = -1 $
The amplitude is the absolute value of the non-periodic part of the function.
Example: $ a \sin(x) $ has for amplitude $ | a | $
The demonstration of the existence of a period $ t $ for a function $ f $ consists in calculating if the equation $ f(x+t)=f(x) $ is true.
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).
|Sine $ \sin(x) $||$ 2\pi $|
|Cosine $ \cos(x) $||$ 2\pi $|
|Tangent $ \tan(x) $||$ \pi $|
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Period of a Function on dCode.fr [online website], retrieved on 2022-10-05,