Graphs of Trigonometric Functions
Graph of y = sin x (x is measured in 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
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)
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.
Graph of y = cos x (x is measured in degrees)
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.
Graph of y = tan x (x is measured in degrees)
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.
Graph of y = cosec x (x is measured in 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.
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 secx 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)
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`\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)
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`
