# Graphs of Trigonometric Functions

## 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

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

## 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

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

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

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

## 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

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

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.

## 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.

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

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.

## 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.

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

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.

## 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.

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

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\nen\pi , n \epsilon Z

Range: -\infty < y <+\infty

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

## 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

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