Trigonometric Graphs
1.6
Day 1
Let’s first make a chart of this function:
y = sin x
x
sin x
0
30
60
90
0
0.5 0.87 1
120 150 180 210 240 270 300 330 360
0.87 0.5
0
-0.5 -0.87 -1 -0.87 -0.5 0
Now let’s plot
1
0.5
0
30
-0.5
-1
60
90
120
150
180
210
220
270
300
330 360
y = sinx
1
0
90
180
270
360
-1
Maximum Value = 1
Minimum Value = -1
Domain: All Reals
Range: -1 ≤ y ≤ 1
Let’s first make a chart of this function:
y = cos x
x
cos x
0
1
30
60
0.87 0.5
90
120 150 180 210 240 270 300 330 360
0
-0.5 -0.87 -1 -0.87 -0.5 0
0.5
0.87 1
y = cosx
1
0
90
180
270
360
-1
Maximum Value = 1
Minimum Value = -1
Domain: All Reals
Range: -1 ≤ y ≤ 1
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.
7
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 = 2sin x

2

3
2
2
x
y = 1 sin x
2
y = sin x
y = – 4 sin x
reflection of y = 4 sin x
y = 4 sin x
4
8
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
Shrink Horizontally:
y
y = sin 2x
period: 2π
period: 
y  sin x


x
2
Stretch Horizontally:
y
1
y  cos x
2 
period: 4 
y  cos x

9
2
3
4
period: 2π
x
Y = 7sinx
7
0
90
180
-7
Maximum Value = 7
Minimum Value = -7
270
360
4
Y = 4cosx
0
90
180
270
360
-4
Maximum Value = 4
Minimum Value = -4
What is the
equation of
this graph?
Y = - 8sinx
8
0
90
180
270
360
-8
“Opposite” to Sin x
Maximum Value = 8
Minimum Value = -8
What is the
equation of
this graph?
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 xaxis.
y = sin (–x)
y
Use the identity
sin (–x) = – sin x
x

y = sin 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
cos (–x) = – cos x
x

y = cos (–x)
13
2
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
2
period: 2  = 
3
b
Use the identity sin (– x) = – sin x:
amplitude: |a| = |–2| = 2
Calculate the five key points.
x
0
y = –2 sin 3x
y

2
6
0


6
3
2
2
3
–2
0
2
0
( , 2)


6
3
2
( , -2)
6
14
2
3

2
(  , 0)
(0, 0)
2

3
( 2 , 0)
3
5
6

x
Graph of the Tangent Function
To graph y = tan x, use the identity
sin
x
.
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 

2
k  

2
2. range: (–, +)
3. period: 
 3
2
4. vertical asymptotes:
x  k 

2

2
k  
15
period: 
3
2 x
Example: Find the period and asymptotes and sketch the graph
of
 y

1
x


x

y  tan 2 x
4
4
3
.
1. Period of y = tan x is
 Period of y  tan 2 x is

2
.

2. Find consecutive vertical
asymptotes by solving for x:
2x  

2
, 2x 
3
8
 1
 , 
 8 3

2
 1
 , 
 8 3

 3 1 
 , 
 8 3
2


Vertical asymptotes: x   , x 
4
4
3. Plot several points in

(0, )
2
x


8
1
1
y  tan 2 x 
3
3
4. Sketch one branch and repeat.
16
0
0

8
1
3
3
8
1

3
x
Graph of the Cotangent Function
To graph y = cot x, use the identity
cos
x
.
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

3
2
 

2

 3
2
2
2
x  k k  
x  
vertical asymptotes
17
x0
x 
x  2
Graph of the Secant Function
sec x 
The graph y = sec x, use the identity
1
.
cos x
At values of x for which cos x = 0, the secant function is undefined and its
graph has vertical asymptotes.
y  sec x
y
Properties of y = sec x
4
1. domain : all real x
x  k 

2
( k  )
y  cos x
x
2. range: (–,–1]  [1, +)

3. period: 


2
2
4. vertical asymptotes:
x  k 

2
k  
4
18

3
2
2
5
2
3
Graph of the Cosecant Function
To graph y = csc x, use the identity
.1
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
y  csc x
4
1. domain : all real x
x  k k  
2. range: (–,–1]  [1, +)
x
3. period: 

4. vertical asymptotes:


2
2

3 2
2
5
2
y  sin x
x  k k  
4
where sine is zero.
19
Y = -9sinx
9
0
90
180
-9
“Opposite” to Sin x
270
360
Y = sin 2x
1
0
90
180
270
360
-1
Period of graph is 1800
There are 2 cycles between 00 and 3600
Combining these rules
Draw y = 6sin2x
Max 6
2 cycles
Min -6
Period = 360 ÷ 2 = 1800
6
Y = 6sin 2x
0
90
-6
180
270
360
Recognising Graph
Y = 8cos4x
Max 8
4 cycles
Cosine
Min -8
8
0
90
-8
180
270
360
Combining our two rules
Draw y = 8sin2x
Max 8
2 cycles
Min -8
Period = 360 ÷ 2 = 1800
8
Y = 8sin 2x
0
90
-8
180
270
360