How Do You Find the Period of a Function?

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A guide for finding the periods of sine, cosine & tangent functions

Co-authored by Sophie Burkholder, BA Reviewed by Joseph Meyer

Last Updated: June 6, 2024 Fact Checked

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What is a periodic function?

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Finding the period of a trigonometric function is an essential piece of the pre-calculus puzzle—so how do you do it? The period of a function is the distance between each repeating interval on a graph, or the distance between the peaks of each wave. To learn how to calculate the period of any function, follow the equations and examples below and ace your next math test!

Finding the Period of a Sinusoidal Function

Periodic functions are functions that repeat continuously, and a period is the distance between each repetition. Find the period for a sine function $f(x)=A\sin(Bx+C)+D$ and a cosine function $f(x)=A\cos(Bx+C)+D$ with the formula $2\pi /|B|={\text{Period}}$ .

Section 1 of 4:

Finding the Period of a Sine or Cosine Function

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Find the coefficient B in your sine or cosine function. Your function should follow this function formula: $f(x)=A\sin(Bx+C)+D$ or $f(x)=A\cos(Bx+C)+D$ . Identify the value of B in your formula. ^{[1]
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- None of the other coefficients will affect the period of the sine wave, so you can disregard them for now.
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Plug B into the formula ${\text{Period}}=2\pi /|B|$ . This formula is used for both sine and cosine functions, so you can follow the same steps for either. Divide $2\pi$ by the absolute value of your coefficient B. The absolute value of B is the non-negative value of B. For example, the absolute value is 3 if B equals 3 or -3. ^{[2]
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- Suppose your sine function is $f(x)=\sin \left(2x\right)$ . In this case, $B=2$ .
- Plug 2 into the formula to get ${\text{Period}}=2\pi /|2|=2\pi /2=\pi$ .
- In this case, the period of the sine function is $\pi$ .
- The same steps are used to find the period of a secant function $y=A\sec(Bx)$ and a cosecant function $y=A\csc(Bx)$ .
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If your sine or cosine function is $\sin(x)$ or $\cos(x)$ , the period is always $2\pi$ . $2\pi$ is the period for a standard sine or cosine curve. If no other coefficients or variables are introduced to change the period, the period of $\sin(x)$ or $\cos(x)$ will always be $2\pi$ . ^{[3]
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Section 2 of 4:

Finding the Period of a Tangent Function

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Find coefficient B in tangent function $y=A\tan(Bx)$ . If your tangent function is $y=\tan(2x)$ , then $B=2.$ Ignore any other whole numbers in the tangent function; i.e. if the function is $y=8\tan(3x)$ , you only need to pay attention to 3, which represents coefficient B . ^{[4]
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Substitute the value of B into the formula $P=\pi /|B|$ . Divide $\pi$ by the absolute value of B to find the period of the tangent function. For example, suppose your tangent function is $y=\tan(2x)$ . ^{[5]
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- Plug B (2) into the formula to obtain $P=\pi /|2|$ .
- The absolute value of 2 is 2, which leads us to $P=\pi /2$ .
- Solve the equation to get $P=\pi /2$ .
- The period of this tangent function is $\pi /2$ .
- The same steps are used to find the period of a cotangent function $y=A\cot(Bx)$ .
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If the tangent function is $\tan(x)$ , the period is always $\pi$ . A standard tangent function will always have a period of $\pi$ . Therefore, you don’t have to calculate the period for function $\tan(x)$ . ^{[6]
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Section 3 of 4:

Periodic Function Practice Problems

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Example Problem 1: Find the period of the periodic function $\sin(4x+5)$ . ^{[7]
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- Solution:
 - $B=4.$
 - ${\text{Period}}=2\pi /|4|$ → $2\pi /4$ → $\pi /2$ .
 - The period of $\sin(4x+5)$ is $\pi /2$ .
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Example Problem 2: Find the period of the periodic function $\sin(-8x)$ .
- Solution:
 - $B=-8.$
 - ${\text{Period}}=2\pi /|-8|$ → $2\pi /8$ → $\pi /4$ .
 - The period of $\sin(-8x)$ is $\pi /4$ .
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Example Problem 3: Find the period of the periodic function $\tan \left(3x\right)$ .
- Solution:
 - $B=3.$
 - ${\text{Period}}=\pi /|3|$ → $\pi /3$ .
 - The period of $\tan \left(3x\right)$ is $\pi /3$ .
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Example Problem 4: Find the period of the periodic function $\sec \left(2x+1\right)$ .
- Solution:
 - $B=2.$
 - ${\text{Period}}=2\pi /|2|$ → $2\pi /2$ → $\pi$ .
 - The period of $\sec \left(2x\right)+1$ is $\pi$ .
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Example Problem 5: Find the period of the periodic function $5\cos \left(12x\right)$ .
- Solution:
 - $B=12.$
 - ${\text{Period}}=2\pi /|12|$ → $2\pi /12$ → $\pi /6$ .
 - The period of $5\cos \left(12x\right)$ is $\pi /6$ .
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Example Problem 6: Find the period of the periodic function $\cot(x)$ .
- Solution:
 - The period of $\cot(x)$ is $\pi$ because it is a standard cotangent function.

Section 4 of 4:

What is a periodic function?

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A periodic function is a function that repeats itself at regular intervals. The period of a function is the distance between each repetition. Periodic functions are represented by the formula $f(x+p)=f(x)$ , where $p$ is the period of the function and $f$ is the periodic function. ^{[8]
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- A periodic function is also defined by having a positive real number (i.e., a number greater than zero) to represent $p$ . Therefore, $f(x+p)=f(x)$ is true for $x$ being all real numbers.
- The fundamental period of a function is the lowest possible value of the positive real number $p$ , or the period in which a function repeats itself.
- The sine wave is a classic example of a periodic function. The sine wave graph looks like the same wave shape repeated over and over again.
- The distance between the peaks (or valleys) of each subsequent wave on the graph is the period of the function , as it represents the distance between each repetition.
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Periodic functions are also defined by amplitude, phase/vertical shift, and frequency. The common form for graphing periodic functions is $f(x)=A\sin(Bx+C)+D$ or $f(x)=A\cos(Bx+C)+D$ , where A = amplitude , B = frequency, –C/B = phase shift , and D = vertical shift . While the period of a function defines the distance between each repetition of the curve, these other coefficients define other dimensions of the graph. ^{[9]
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- The amplitude of a periodic function is the height or highest point of each peak in a curve pattern. Amplitude can be found by graphing the function, identifying the minimum and maximum values of the function, then halving that difference. That final number is the amplitude. ^{[10]
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- The frequency of a periodic function is the number of repeated wave patterns within a certain interval. For sine and cosine functions, this interval is often from 0 - 2pi. For tangent functions, it’s typically from 0 - pi.
- A phase shift is when a curve shifts horizontally on a graph to the left or right of its normal position. The phrase shift does not affect the other variables in a periodic function, i.e. the frequency, period, or amplitude.
- A vertical shift is when a curve shifts vertically on the graph to be higher or lower than its usual position. If D is negative, the function will shift vertically DOWN the y -axis—if positive, the function will shift upward.

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There are two ways to be sure of the periodicity of a function: if its range repeats itself at regular intervals, and if the function is in the form $f(x+p)=f(x).$

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How Do You Find the Period of a Function?

Finding the Period of a Sinusoidal Function

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Finding the Period of a Sine or Cosine Function

Finding the Period of a Tangent Function

Periodic Function Practice Problems

What is a periodic function?

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