Period and Frequency Calculator


Instructions: Use this Period and Frequency Calculator to find the period and frequency of a given trigonometric function, as well as the amplitude, phase shift and vertical shift when appropriate. Please type in a periodic function (For example: f(x)=3sin(πx)+4f(x) = 3\sin(\pi x)+4)

Enter the trigonometric funcion you want to analzye (Ex. '3sin(pi*x+3)-2', or '4cot(2(x-1))', etc)

Lower Limit (Optional. Ex. 1, 2/3, etc) =
Upper Limit (Optional. Ex. 1, 2/3, etc) =

Period and Frequency Calculator

When dealing with periodic functions, there are some crucial parameters that need to be computed, and these are the period (PP) and the frequency (ff).

The period PP of a periodic function corresponds to the number that satisfies the following property:

f(x+P)=f(x)f(x+P) = f(x)

for all values of xx. Observe that not all functions have a period. Those who do are called periodic functions .

Period of some common functions

Trigonometric functions are examples of periodic functions. For example, if we consider function, f(x)=sinxf(x) = \sin x, its period is 2π2\pi, as shown in the graph below:

Period Calculator

For cosx\cos x we also have the the period is 2π2\pi. Check out the graph below:

Cosine - Example of period calculation

Period of Other Trigonometric Functions

Recall that the cosecant function cscx\csc x is the inverse of sinx\sin x, this is cscx=1sinx\csc x = \frac{1}{\sin x}, so then the period of cscx\csc x is also 2π2\pi.

Similarly, the secant function secx\sec x is the inverse of cosx\cos x, this is secx=1cosx\sec x = \frac{1}{\cos x}, so then the period of secx\sec x is 2π2\pi as well.

How about the tangent? The tangent function tanx\tan x is slightly different because its period is π\pi. Indeed, its graph looks different than those of the sine and cosine, but tangent is also periodic. One difference is that tanx\tan x has discontinuities. Check it out:

Tangent function - example of period calculation

Similarly as before, the cotangent function cotx\cot x is the inverse of tanx\tan x, with cotx=1tanx\cot x = \frac{1}{\tan x}, so then the period of cotx\cot x is also π\pi.

Calculation of the Frequency

Another important element to consider for periodic function is the frequency (ff), which is calculated in terms of the period PP as:

f=1Pf = \frac{1}{P}

So the frequency is the inverse of the period. And vice-versa, the period is the inverse of the frequency.

For example, what is the frequency of sinx\sin x? Following the above formula, since we know that for sine the period is P=2πP = 2\pi:

f=1P=12π0.1592f = \frac{1}{P} = \frac{1}{2\pi} \approx 0.1592

This calculator will also compute the amplitude, phase shift and vertical shift if the function is properly defined. Those parameters pretty determine the behavior of trigonometric function.

If you need to graph a trigonometric function, you should use this trigonometric graph maker .

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