Formula For Period
The formula for period is used to calculate the time interval between two waves, called period. A periodic function is defined as the 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). Let us learn about the formula for period with a few solved examples in the end.
What is the 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. What if the coefficient of x is not 1? Here is the formula for period (T) of a trigonometric function:
Period, P = Period of parent function/ Coefficient of x
Example: The period of tan 3x using the period formula is π / 3. You can observe this from the following graph also.
Let us have a look at a few solved examples to have a better understanding of the formula for period.

Example 1: Using the formula for period, find the period of the function f(x) = 2 sin (3x + 7) + 5.
Solution:
To find: The period of the given function.
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
Answer: 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:
To find: The period of the given function.
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
\(\text{Period, }T =\dfrac{\pi}{\left\dfrac{\pi}{2}\right }=2\)
Answer: The period of f(x) = 2.