Graphs of Trigonometric Functions

Graph of y = sin x (x is measured in radians)


radians

Amplitude & Period:


Amplitude: The maximum ordinate (y) value in a graph is called its ``amplitude``.

Period: "Period" of a trigonometric function is the smallest +ve positive number which, when added to the original circular measure of the angle, gives the same value of the function.

From the graph of sine function shown above,

For y = sin x :

Amplitude = 1

Period = 2`\pi`

Domain: -`\infty` <x <+`\infty`

Range: -1 `\leq` y `\leq` +1

one-period

The figure shown here depicts the graph of y = sin x for one complete period.


Graph of y = sin x (x is measured in degrees)


measured

General Graphs of the Sine Function:


For a sine function of the form: y = a sin bx

Amplitude = | a |

Period = ` (2\pi)/ b `


Example 1

Find the amplitude and period of the function y = `2 sin( x /3)`

Solution: For the given function y = 2 sin `( x /3)` = 2 sin `( 1/3x )`

Amplitude = | a | = 2

Period = `(2\pi)/b` = `(2\pi)/(( 1 )/3)` = 6`\pi`

amplitude

Graph of y = cos x (x is measured in radians)


measured

From the graph of cosine function shown above,

For y = cos x :

Amplitude = 1

Period = 2`\pi`

Domain: -`\infty` <x <+`\infty`

Range: -1 `\leq` y `\leq` +1

cosine

The figure on the right shows the graph of y = cos x for one complete period.


Graph of y = cos x (x is measured in degrees)


Graph

General Graphs of the Cosine Function:


For a cosine function of the form:

y = a cos bx

Amplitude = | a |

Period = `(2\pi)/b`


Example 2:

Find the amplitude and period of the function y = `( 1 )/2` cos 3x

Solution: For the given function y = `( 1 )/2` cos 3x

Amplitude = | a | = `( 1 )/2`

Period = ` (2\pi)/b` = `(2pi)/3`

amplitude

Graph of y = tan x (x is measured in radians)


tan-x tan-x1

The graph of `y= tan x = (sin x)/cos x` has been shown above. From the function definition, we can see that the value of tangent function is undefined for all values of x for which `cos x=0` (division by zero is undefined). Therefore the domain of tangent function excludes odd integer multiples of ` (\pi)/2 ` . Vertical asymptotes can be observed on all these values of x.

Note:

As evident from the graph, the tangent function has a period of pi (`\pi`). That is tan (`\pi`+ `\theta`)=`tan (\theta)`

From the graph shown above,

For y = tan x :

Amplitude = none; there is no maximum value for tangent function.

Period = `\pi` (Tangent function completes one cycle between ` (-\pi)/2 to (\pi)/2) `

Domain: -`\infty` < x < +`\infty` , x`\neq((2n+1)\pi)/2` , n `\epsilon` Z

Range: -`\infty` <y < +`\infty`


The figure on the right shows the graph of y = tan x for one complete period.


Graph of y = tan x (x is measured in degrees)


tan-x2

General Graphs of the Tangent Function:


For a tangent function of the form:

y = a tan bx

Period = ` \pi/b `


Example 3:

Graph y = `( 1 )/2` tan 3x

Solution: For the given function y = `( 1 )/2` tan 3x

Period = `\pi/3`

The dashed lines indicate the vertical asymptotes that occur at

x = `\pi/6 \pm (n \pi)/3` where n is an integer.

Tangent-Function

Graph of y = cosec x (x is measured in radians)


radians1 radians2

The graph of y=cosec x = `( 1)/ sin x ` has been shown above. From the function definition, we can see that the value of cosec function is undefined for all values of x for which `sin x=0` (division by zero is undefined). Therefore the domain of cosec function excludes integer multiples of `\pi`. Vertical asymptotes exist on all these values of x.

From the graph shown above,

For y = cosec x :

Amplitude = none; there is no maximum value for cosec function.

Period = `2\pi`

Domain: -`\infty` < x < +`\infty` , x`\ne n\pi` , n `\epsilon` Z

Range: y `\le` -1 or y `\ge` 1

The figure on the right shows the graph of y = cosec x for one complete period.


Graph of y = cosec x (x is measured in degrees)


degrees

General Graphs of the Cosec Function:


For a cosec function of the form:

y = cosec bx

Period = `(2\pi)/b`


Example 4

Graph y = cosec 2x

Solution:For the given function y = cosec 2x

Period = `(2\pi)/2` = `\pi`

The dashed lines indicate the vertical asymptotes that

occur at x = ` (n\pi)/2` where n is an integer.

cosce

Graph of y = sec x (x is measured in radians)


radians4 radians3

The graph of y=sec x = `( 1)/ cos x` has been shown in red above. From the function definition, it can be seen that the value of y=sec x is undefined for all values of x for which `cos x=0` (division by zero is undefined). Therefore the domain of sec⁡x excludes odd integral multiples of `(\pi)/2 ` . Vertical asymptotes exist on all these values of x.

From the graph shown above,

For y = sec x :

Amplitude = none; there is no maximum value for sec function.

Period = 2`\pi`

Domain: -`\infty` < x <+`\infty` , x`\ne` (2n+1) `\pi` )/2 , n `\epsilon` Z

Range: y `\le` -1 or y `\ge` 1

The figure on the right shows the graph of y = sec x for one complete period.


Graph of y = sec x (x is measured in degrees)


degrees1

General Graphs of sec Function:


For a secant function of the form:

y = sec bx

Period = `(2\pi)/b `


Example 5:

Graph y = sec 3x

Solution:For the given function y = sec 3x

Period = ` (2\pi)/3 `

The dashed lines indicate the vertical asymptotes that occur at

x = ` \pi/6 \pm (n\pi)/3` where n is an integer.

vertical

Graph of y = cot x (x is measured in radians)


radians5 radian6

The graph of y=cot x = `( cos x )/sin x ` has been shown above. From the function definition, we see that the value of (y=cot x) function is undefined for all values of x for which `sin x=0` (division by zero is undefined). Therefore the domain of tangent function excludes integer multiples of `\pi` . Vertical asymptotes can be observed on all these values of x.

Note:

Cot function has a period of `pi` (`\pi`). That is cot (`\pi`+ `\theta`)=cot (`\theta`) as evident from the graph as well.

For y = cot x :

Amplitude = none; there is no maximum value for cot function.

Period = `\pi` (Tangent function completes one cycle between `(-\pi )/2 to (\pi)/2)`

Domain: -`\infty` < x <+`\infty` , x`\ne`n`\pi` , n `\epsilon` Z

Range: -`\infty` < y <+`\infty`

The figure on the right shows the graph of y = cot x for one complete period.


Graph of y = cot x (x is measured in degrees)


degrees2

General Graphs of the Cotangent Function:


For a cotangent function of the form:

y = a cot bx

Period = `\pi/b`


Example 6:

Graph y = cot `( x )/2`

Solution: For the given function y = cot` ( x )/2`

Period = `(\pi)/(( 1)/2)` = `2\pi`

graph1





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