Digital Lesson
Graphs of Trigonometric
Functions
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that  1  y  1.
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range
over an x-interval of 2 .
6. The cycle repeats itself indefinitely in both directions of the
x-axis.
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2
Graph of the Sine Function
To sketch the graph of y = sin x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
sin x
0
2
1
0
-1
0
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = sin x
y
3

2



1

2
2

3
2
2
5
2
x
1
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3
Graph of the Cosine Function
To sketch the graph of y = cos x first locate the key points.
These are the maximum points, the minimum points, and the
intercepts.

3
x
0

2
2
cos x
1
2
0
-1
0
1
Then, connect the points on the graph with a smooth curve
that extends in both directions beyond the five points. A
single cycle is called a period.
y = cos x
y
3

2



1

2
2

3
2
2
5
2
x
1
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4
Example: Sketch the graph of y = 3 cos x on the interval [–, 4].
Partition the interval [0, 2] into four equal parts. Find the five key
points; graph one cycle; then repeat the cycle over the interval.
x
y = 3 cos x
y

(0, 3)
2
1

0
3
0

-3
x-int
min
2
max
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
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2
( 3 , 0)
2
3
4 x
5
The amplitude of y = a sin x (or y = a cos x) is half the distance
between the maximum and minimum values of the function.
amplitude = |a|
If |a| > 1, the amplitude stretches the graph vertically.
If 0 < |a| > 1, the amplitude shrinks the graph vertically.
If a < 0, the graph is reflected in the x-axis.
y
4
y = sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 2 sin x
y = 4 sin x
4
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6
The period of a function is the x interval needed for the
function to complete one cycle.
For b  0, the period of y = a sin bx is 2 .
b
For b  0, the period of y = a cos bx is also 2 .
b
If 0 < b < 1, the graph of the function is stretched horizontally.
y
y  sin 2
period: 2
period: 
y  sin x x


2
If b > 1, the graph of the function is shrunk horizontally.
y
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
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7
Use basic trigonometric identities to graph y = f (–x)
Example 1: Sketch the graph of y = sin (–x).
The graph of y = sin (–x) is the graph of y = sin x reflected in
the x-axis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
y = sin x
x

2
Example 2: Sketch the graph of y = cos (–x).
The graph of y = cos (–x) is identical to the graph of y = cos x.
y
Use the identity
x
cos (–x) = – cos x

2
y = cos (–x)
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8
Example: Sketch the graph of y = 2 sin (–3x).
Rewrite the function in the form y = a sin bx with b > 0
y = 2 sin (–3x) = –2 sin 3x
Use the identity sin (– x) = – sin x:
2  2
period:
amplitude: |a| = |–2| = 2
=
3
b
Calculate the five key points.
x
0
y = –2 sin 3x
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3
(0, 0)
2

(  ,-2)
2
2
3

2
5
6

x
(  , 0) 2
3
( , 0)
3
6
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9
Graph of the Tangent Function
sin x
To graph y = tan x, use the identity tan x 
.
cos x
At values of x for which cos x = 0, the tangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = tan x
1. domain : all real x

x  k  k   
2
2. range: (–, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2

2
 3
2
3
2
x

2
period: 
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10
Example: Find the period and asymptotes and sketch the graph
 y

1
x


x

of y  tan 2 x
4
4
3
1. Period of y = tan x is  .

 Period of y  tan 2 x is .
3
 1
2


, 

8
2
 8 3
x
2. Find consecutive vertical
 1
asymptotes by solving for x:
 3 1 
 , 
 , 


 8 3
 8 3
2x   , 2x 
2
2


Vertical asymptotes: x   , x 
4
4


 3
3. Plot several points in (0, )

x
0
2
8
8
8
1
1
1
1

y  tan 2 x 
0
4. Sketch one branch and repeat.
3
3
3
3
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11
Graph of the Cotangent Function
cos x
To graph y = cot x, use the identity cot x 
.
sin x
At values of x for which sin x = 0, the cotangent function is
undefined and its graph has vertical asymptotes.
y
Properties of y = cot x
y  cot x
1. domain : all real x
x  k k  
2. range: (–, +)
3. period: 
4. vertical asymptotes:
x  k k  
vertical asymptotes
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
3
2
 

2
x  
x0

 3
2
2
x 
x
2
x  2
12
Graph of the Secant Function
1
sec
x

The graph y = sec x, use the identity
.
cos x
At values of x for which cos x = 0, the secant function is undefined
and its graph has vertical asymptotes.
y
y  sec x
Properties of y = sec x
1. domain : all real x

x  k  (k  )
2
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:

x  k  k   
2
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4
y  cos x
x



2
2

3
2
2
5
2
3
4
13
Graph of the Cosecant Function
1
To graph y = csc x, use the identity csc x 
.
sin x
At values of x for which sin x = 0, the cosecant function
is undefined and its graph has vertical asymptotes.
y
Properties of y = csc x
4
y  csc x
1. domain : all real x
x  k k  
2. range: (–,–1]  [1, +)
3. period: 
4. vertical asymptotes:
x  k k  
where sine is zero.
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x



2
2

3 2
2
5
2
y  sin x
4
14