# Formula for Period

The formula for the period is used to calculate the time period of a wave. It is the time taken by a wave to reach from one peak to another. A periodic function is defined as a function that repeats its values at regular intervals or periods. The period of a function f(x) is p, if f(x + p) = f(x), for every x. Let us learn about the formula for the period with a few solved examples in the end.

## What is Formula for Period?

According to the definition of a period of a function, a function f(x) will be periodic with period p, so if we have f (x + p) = f (x), for every p > 0. The period of each of sin x, cos x, csc x, and sec x = 2π. The period of each of tan x and cot x = π. The period of the wave decreases as its frequency increases. Here is the formula for period (T) of a trigonometric function:

**Period, T = Period of parent function/ |Coefficient of x|**

**Frequency, F = 1/ Period**

Example: The period of tan 3x using the period formula is π / 3. You can observe this from the following graph also.

## Examples Using Formula for Period

**Example 1: Using the formula for period, find the period of the function f(x) = 2 sin (3x + 7) + 5.**

**Solution:**

We know that the period of the parent function, which is sin, is 2π.

The coefficient of x in the given function is 3.

Using the formula for period,

Period, T = (Period of parent function) / |Coefficient of x|

Period, T = 2π / |3| = 2π / 3

Therefore, The period of f(x) = 2π / 3.

**Example 2: Find the period of the function \(f(x) = 3 \tan \left( \dfrac{\pi}{2}\left(x+2\right) \right) - 7\).**

**Solution:**

We know that the period of the parent function, which is tan, is π.

The given function can be written as:

\(f(x) = 3 \tan \left( \dfrac{\pi}{2}x+\pi \right) - 7\)

The coefficient of x in the given function is π/2.

Using the formula for period,

Period, T = (Period of parent function) / |Coefficient of x|

\(\text{Period, }T =\dfrac{\pi}{\left|\dfrac{\pi}{2}\right| }=2\)

Therefore, the period of f(x) = 2.

**Example 3: Using the formula for period, find the period of the function f(x) = 2 sin (4x + 8) + 10.**

**Solution:**

We know that the period of the parent function, which is sin, is 2π.

The coefficient of x in the given function is 4.

Using the formula for period,

Period, T = (Period of parent function) / |Coefficient of x|

Period, T = 2π / |4| = 2π / 4 = π / 2

Therefore, The period of f(x) = π / 2

## FAQs on Formula for Period

### What is Meant by Formula for Period?

The formula for period is used to calculate the time interval taken by a wave to complete one cycle of vibration at a given point . A periodic function is defined as a function that repeats its values at regular intervals or periods. The period of a function f(x) is p, if f(x + p) = f(x), for every x. A function is said to be periodic if its value repeats after regular periods (intervals). The formula is Period, P = Period of parent function/ |Coefficient of x|

### What is the Formula to Find the Period?

a function f(x) will be periodic with period p, so if we have f (x + p) = f (x), for every p > 0. The period of each of sin x, cos x, csc x, and sec x = 2π. The period of each of tan x and cot x = π. Here is the formula for period (T) of a trigonometric function:

Period, P = Period of parent function/ |Coefficient of x|

### How to Find the Formula for Period?

Listed below are three main aspects to finding the formula for period:

- Find if it is a periodic function i.e. if the function repeats over at a constant period
- If the formula for period function is represented like f(x) = f(x + p), where p is the real number
- Period means the time interval between the two occurrences of the wave

### What Role does Amplitude play in Formula for Period?

On a graph, a period is when the function goes from one point to the next matching point. In amplitude helps in measuring the height of the function point measured from the highest to the lowest.

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