4.5
Graphs of Sine and Cosine
Functions
In this lesson you will learn to graph functions of the form
y = a sin bx and y = a cos bx where a and b are positive constants
and x is in radian measure. The graphs of all sine and cosine
functions are related to the graphs of y = sin x and y = cos x
which are shown below.
y = sin x
y = cos x
Fill in the chart.
x
0

2

3
2
2
Sin x
Cos x
These will be key points on the
graphs of y = sin x and y = cos x.
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.
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
1

3
2
2
5
2
x
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
1

3
2
2
5
2
x
Before sketching a graph, you need
to know:
• Amplitude – Constant that gives vertical
stretch or shrink.
• Period – 2 .
b
• Interval – Divide period by 4
• Critical points – You need 5.(max., min.,
intercepts.)
Amplitudes and Periods
The graph of y = A sin Bx has
amplitude = | A|
2
period =
B
To get your critical points (max, min, and intercepts) just
take your period and divide by 4.
Example: y  3 cos x
2
Period 
 2
1
Period 2 


4
4
2
So critical points will

3
come at 0, ,  ,
, 2
2
2
Interval


2


2
 

2
2
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
Notice that since
all these graphs
have B=1, so the
period doesn’t
change.
y = 2sin x

2
y=
1
2

3
2
2
x
sin x
y = – 4 sin x
reflection of y = 4 sin x
4
y = sin x
y = 4 sin x
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 b > 1, the graph of theyfunction is shrunk horizontally.
y  sin 2 x
period: 2
y  sin x x
period: 


2
If 0 < b < 1, the graph of
y the function is stretched horizontally.
y  cos x
1
y  cos x
period: 2
2 
2
3
4

x
period: 4
Example 1: 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

0
3
max
2
0

-3
x-int min
y

(0, 3)
2
1
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
2
( 3 , 0)
2
3
4 x
Example 2
Determine the amplitude of y = 1/2 sin x. Then graph
y = sin x and y = 1/2 sin x for 0 < x < 2.
y
y = sin x
1
y = 1/2sinx
1
2
1

2
-1
2
2˝

2

˝
3
2
x
Example 3
x
Sketch the graph of y  sin .
2
Example 3
x
Sketch the graph of y  sin .
2
For the equations
y = a sin(bx-c)+d and y = a cos(bx-c)+d
• a represents the amplitude. This constant acts as a
scaling factor – a vertical stretch or shrink of the
original function.
• Amplitude = a .
• The period is the sin/cos curve making one complete
cycle.
2
• Period = b .
• c makes a horizontal shift.
• d makes a vertical shift.
• The left and right endpoints of a one-cycle interval can
be determined by solving the equations bx-c=0 and
bx-c= 2 .
Example 4
1 

Sketch the graph of y  sin  x  .
2 
3
Example 4
1 

Sketch the graph of y  sin  x  .
2 
3
Example 6
Sketch the graph of y  2  3 cos 2 x.
Example 6
Sketch the graph of y  2  3 cos 2 x.
Tides
•
Throughout the day, the
depth of the water at the end
of a dock in Bangor,
Washington varies with the
tides. The tables shows the
depths (in feet) at various
times during the morning.
(a) Use a trig function to model
the data.
(b) A boat needs at least 10 feet
of water to moor at the dock.
During what times in the
evening can it safely dock?
y  a cos(bt  c)  d
Time
Midnight
2 a.m.
4 a.m
6 a.m.
8 a.m.
10 a.m.
Noon
Depth, y
3.1
7.8
11.3
10.9
6.6
1.7
0.9
Homework
• Page 304-305
• 3-13 odd, 15-21 odd, 31 – 34 all, 47-57
odd, 65, 67