Period means the time interval between two waves. Periodic function is a function that repeats its values at regular intervals or periods. In other words, a function which repeats its values after every particular interval, is known as a **periodic function**. This particular interval is termed as the **period** of that function.

A function f is said to be periodic with period m, if we have

f (x + m) = f (x), For every m > 0.

It means that the function f(x) possess same values after an interval of "m". One can say that the function f repeats all its values after every interval of "m".**For example -** The sine function i.e. sin x has a period 2 $\pi$ because 2 $\pi$ is the smallest number for which sin (x + 2$\pi$) = sin x, for all x.

We may also calculate the period using the formula derived from the basic sine and cosine equations. The period for function y = A sin( B x - c ) and y = A cos( B x - c ) is $\frac{2 \pi}{B}$ radians. The reciprocal of period of a function is equal to its frequency. Frequency is defined as the number of cycles completed in one second. If the period of a function is denoted by P and f be its frequency, then -

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The fundamental period of a function is the period of the function which are of the form,${f}(x+k)={f}(x)$, then k is called the period of the function and the function f is called a periodic function.

Let us define the function h(t) on the interval [0,2] as follows:

$h(t) = \begin{cases}

0 & \text{ if } x= 0\leq t\leq 1 \text{or if} \frac{5}{3}\leq t\leq 2\\

3t-1 & \text{ if } x=\frac{1}{3}\leq t\leq \frac{2}{3} \\

1& \text{ if } x= \frac{2}{3}\leq t\leq \frac{4}{3}\\

-3t+5& \text{ if } x=\frac{4}{3}\leq t\leq\frac{5}{3}

\end{cases}$

If we extend the function h to all of R by the equation, ${h}(t+2)={h}(t)$

=> h is periodic with the period 2.

The graph of the function is shown below.

### Solved Examples

**Question 1: **Find the period of the given periodic function. Where f(x) = 9sin(6px7 + 5)

** Solution: **

**Question 2: **Find the period of the periodic function f(x) . Where f(x) = 9 Cos x

** Solution: **

As we are aware that sin (2$\pi$ + x) = sin x and cos (2$\pi$ + x) = cos x.., we see that the periods of sine and cosine functions are 2$\pi$..

Also, tan ($\pi$ + x) = tan x, hence the period of tan x is $\pi$

Let us graph the primary trigonometric functions.

The following graph shows the function**y = sin x**

Let us find the co-ordinates of the points to graph.

Period = 2 $\pi$

Axis: y = 0 [x-axis ]

Amplitude : 1

Maximum value = 1

Minimum value = -1

domain = { x : x $\in$ R }

Range = [-1, 1]

The following graph shows the trigonometric function**y = cos x**

In general we have three basic trigonometric functions like sin, cos and tan function, having 2$\pi$, 2$\pi$ and $\pi$ period respectively.### Solved Examples

**Question 1: **Graph of **y = 4 sin(x/2)**

** Solution: **

**Question 2: **Graph of **y = 4 sin2x**

** Solution: **

If we have a function f(x) = tan (xs), where s > 0, then the graph of the function makes s complete cycles between $-\frac{\pi}{2}$, 0 and $\frac{\pi}{2}$ and each of the function have the period of p = $\frac{\pi}{s}$.

### Solved Example

**Question: **Determine the period of the function $f(\theta ) = \tan \left ( \frac{3\theta }{2} \right )$

** Solution: **

For the trigonometric tan function the period p is given as follows:

P = $\frac{\pi}{s}$ = $\frac{\pi}{\frac{3}{2}}$ = $\frac{2\pi}{3}$

Period = $\frac{2\pi}{3}$

Axis y = 0

Amplitude: undefined

No maximum or minimum values.

### Solved Examples

**Question 1: **What is the period of the following periodic function?

Where f(x) = 6 sin 5x

** Solution: **
**Question 2: **The frequency of the periodic function is 2phi. Find the period of the periodic function.

** Solution: **
**Question 3: **Find the period, amplitude, horizontal shift, vertical shift and the equation of the axis for the following functions.

y = 3 cos(5x - 5$\pi$) + 2

** Solution: **

We have y = 3 cos(5x - 5 $\pi$) + 2 = 3 cos(5(x - $\pi$))+ 2

Comparing this with the general equation,

y = a cos (k(x - b)) + c

we get a = 3, k = 5, b = $\pi$, c = 2

Hence, the Period of the function is $\frac{2\pi}{\left | k \right |} = \frac{2\pi}{5}$

**Question 4: **The following trigonometric function
have the basic function y = sin x. Determine the equation of each of the
function a. The graph of the trigonometric function has period $\pi$ and
amplitude 12. The equation of the axis is y = -6.

** Solution: **

From the given data, a = 12, b = $\pi$, c = -6

Period = 2$\pi$/|k|

k = 2 $\pi$/period = 2 $\pi$/$\pi$ = 2

The general equation is, y = a sin( k (x-b) + c

Hence, our required**equation is**, **y = 12 sin(2x) ****-**** 6**

The graph of the function will be as below.

Let us define the function h(t) on the interval [0,2] as follows:

$h(t) = \begin{cases}

0 & \text{ if } x= 0\leq t\leq 1 \text{or if} \frac{5}{3}\leq t\leq 2\\

3t-1 & \text{ if } x=\frac{1}{3}\leq t\leq \frac{2}{3} \\

1& \text{ if } x= \frac{2}{3}\leq t\leq \frac{4}{3}\\

-3t+5& \text{ if } x=\frac{4}{3}\leq t\leq\frac{5}{3}

\end{cases}$

If we extend the function h to all of R by the equation, ${h}(t+2)={h}(t)$

=> h is periodic with the period 2.

The graph of the function is shown below.

If a function repeats over at a constant period we say that is a periodic function. Basically it is represented like f(x) = f(x + p), p is the real number and this is the period of the function. Period means the time interval between the two occurrences of the wave. To find the period of the periodic function we have to use the following formula, Where **Period = 2pb**, here b is the co - efficient of x

Sine and cosine functions have the forms of a periodic wave:

**Period**: It is represented as "T", Period is the distance among two repeating points on the graph function.

**Amplitude**: It is represented as "A" It is the distance between the middle point to highest or lowest point on the graph function.

$\sin (a \theta)$ = $\frac{2 \pi}{a}$

and $\cos (a \theta)$ = $\frac{2 \pi}{a}$

Given periodic function is f(x) = 9sin(6px7+ 5)

To find the period we have the formulas

period = 2pb

Where period of the periodic function = 2p(6p7) = 146 = 73

The given periodic function is f(x) = 9 Cos x

To find the period, we have the formula.

period = $\frac{2 \pi}{b}$

Where period of the periodic function = 2p $\times$ 1 = 2p

As we are aware that sin (2$\pi$ + x) = sin x and cos (2$\pi$ + x) = cos x.., we see that the periods of sine and cosine functions are 2$\pi$..

Also, tan ($\pi$ + x) = tan x, hence the period of tan x is $\pi$

Let us graph the primary trigonometric functions.

The following graph shows the function

Let us find the co-ordinates of the points to graph.

x | -2 π |
-3 π/2 |
- π |
- π/2 |
0 |
π/2 | π | 3 π/2 |
2π |
5 π/2 |

y | 1 | 1 | 0 | -1 | 0 |
1 |
0 |
-1 | 0 | 1 |

Period = 2 $\pi$

Axis: y = 0 [x-axis ]

Amplitude : 1

Maximum value = 1

Minimum value = -1

domain = { x : x $\in$ R }

Range = [-1, 1]

The following graph shows the trigonometric function

Let us prepare the table to values

x |
-2 π |
-3 π/2 |
- π | - π/2 |
0 | π/2 |
π | 3 π/2 | 2π |
5 π/2 |

y | 1 | 0 | -1 | 0 | 1 | 0 | -1 | 0 | 1 | 0 |

Period: 2 π

Axis: y = 0

Amplitude = 1

Maximum value = 1

Minimum value = -1

Domain: { x : x $\in$ R }

Range = [ -1, 1]

If we have a function f(x) = sin (xs), where s > 0, then the graph of the function makes s complete cycles between 0 and $2\pi$ and each of the function have the period, p = $\frac{2\pi}{s}$.

Now, lets discuss about some examples based on sin function:

Let us discuss the graph of y = sin 2x

Period = $\pi$

Axis: y = 0 [x-axis ]

Amplitude: 1

Maximum value = 1

Minimum value = -1

Domain: { x : x $\in$ R }

Range = [ -1, 1]

In general we have three basic trigonometric functions like sin, cos and tan function, having 2$\pi$, 2$\pi$ and $\pi$ period respectively.

Period = 4$\pi$

Axis: y = 0 [x-axis ]

Amplitude: 4

Maximum value = 4

Minimum value = -4

Domain: { x : x $\in$ R }

Range = [ -4, 4]

x |
- $\pi$ |
- 3$\pi$/4 | - $\pi$/2 | - $\pi$/4 | 0 |
$\pi$/4 | $\pi$/2 |
3$\pi$/4 |
$\pi$ |

y | 0 | 4 | 0 | -4 | 0 | 4 | 0 | -4 | 0 |

Period = $\pi$

Axis: y = 0 [x-axis ]

Amplitude: 4

Maximum value = 4

Minimum value = -4

Domain: { x : x $\in$ R }

Range = [ -4, 4]

From
the above examples, we observe that the period and amplitude of the
function changes as there is change in the numerical co-efficient and
the angle [argument].

Let us tabulate the argument and the amplitude and compare.

Function |
Amplitude |
Period |

y = sin x | 1 | 2 $\pi$ |

y = 4sin 2x | 4 | $\pi$ |

y = 4 sin(x/2) | 4 | 4 $\pi$ |

Hence, as the argument increases the period decreases and as the argument decreases the period increases.

Let us discuss the following general formula.

y = f(x) = a sin( k (x - b) + c and

y = f(x) = a cos ( k( x - b) ) + c

Since the primary functions sin x and cos x are of period 2 $\pi$, from the above general equation, the period = $\frac{2\pi}{\left | k \right |}$

Here, 'a' represents the amplitude

x = b, represents the horizontal shift

and y = c represents the axis.

For the trigonometric tan function the period p is given as follows:

P = $\frac{\pi}{s}$ = $\frac{\pi}{\frac{3}{2}}$ = $\frac{2\pi}{3}$

Below is the graph of the trigonometric function** y = tan x**, where x = 3$\theta$/2

Period = $\frac{2\pi}{3}$

Axis y = 0

Amplitude: undefined

No maximum or minimum values.

Given below are some of the examples on periodic functions.

Where f(x) = 6 sin 5x

The given periodic function is f(x) = 6 sin 5x

We have the formula for the period of a function.

period = $\frac{2 \pi}{b}$

Where period of the periodic function = $\frac{2 \pi}{5}$

The frequency of the periodic function will be f = $\frac{1}{p}$

So, period p = $\frac{1}{f}$

= $\frac{1}{2 \pi}$

y = 3 cos(5x - 5$\pi$) + 2

We have y = 3 cos(5x - 5 $\pi$) + 2 = 3 cos(5(x - $\pi$))+ 2

Comparing this with the general equation,

y = a cos (k(x - b)) + c

we get a = 3, k = 5, b = $\pi$, c = 2

Hence, the Period of the function is $\frac{2\pi}{\left | k \right |} = \frac{2\pi}{5}$

Amplitude of the function is a = 3

As b = $\pi$, the graph is shifted $\pi$ units to the right.

As c = 2, the equation of the axis is y = 2.

We can draw the graph as follows.

From the given data, a = 12, b = $\pi$, c = -6

Period = 2$\pi$/|k|

k = 2 $\pi$/period = 2 $\pi$/$\pi$ = 2

The general equation is, y = a sin( k (x-b) + c

Hence, our required

The graph of the function will be as below.

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