Periodic Formulas

function is described as relation that has one output value for every permissible or possible input value. The number “T” is the period of a function. If the number is fixed and 
\(\begin{array}{l}\theta\end{array} \)
is an angle we have the following periodic formula. The function sine and cosine have a period 2
\(\begin{array}{l}\pi\end{array} \)
. Some of the trigonometry periodic functions are given below: \[\large Sin\left(\omega\theta\right)\Rightarrow T=\frac{2\pi}{\omega}\] \[\large Cos\left(\omega\theta\right)\Rightarrow T=\frac{2\pi}{\omega}\] \[\large Tan\left(\omega\theta\right)T=\frac{\pi}{\omega}\]

Solved Example

Question: Find the period of the function

\(\begin{array}{l}cos\end{array} \)
\(\begin{array}{l}\frac{x}{3}\end{array} \)
.

Solution: Let y = 

\(\begin{array}{l}cos\end{array} \)
\(\begin{array}{l}\frac{x}{3}\end{array} \)

Using the period formula,

T =

\(\begin{array}{l}\frac{2\pi}{\omega}\end{array} \)

The multiple of x = 

\(\begin{array}{l}\frac{1}{3}\end{array} \)
=
\(\begin{array}{l}{\omega}\end{array} \)

T =

\(\begin{array}{l}\frac{2\pi}{\frac{1}{3}}\end{array} \)
 = 6
\(\begin{array}{l}\pi\end{array} \)

Hence, the period of the function

\(\begin{array}{l}cos\end{array} \)
\(\begin{array}{l}\frac{x}{3}\end{array} \)
is 6
\(\begin{array}{l}\pi\end{array} \)
.

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