# Periodic Formulas

AÂ functionÂ isÂ describedÂ asÂ aÂ relationÂ thatÂ hasÂ oneÂ outputÂ valueÂ forÂ everyÂ permissibleÂ orÂ possibleÂ inputÂ value. The number “T” is the period of a function. If the number is fixed andÂ
$$\begin{array}{l}\theta\end{array}$$
is an angle we have the following periodic formula.Â 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 yÂ =Â

$$\begin{array}{l}cos\end{array}$$
$$\begin{array}{l}\frac{x}{3}\end{array}$$

Using the period formula,

T =

$$\begin{array}{l}\frac{2\pi}{\omega}\end{array}$$

The multiple of x =Â

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

T =

$$\begin{array}{l}\frac{2\pi}{\frac{1}{3}}\end{array}$$
Â =Â 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
$$\begin{array}{l}\pi\end{array}$$
.