### 4.1

```Basic Graphs of Sine
and Cosine Functions
4.1
JMerrill, 2009
(contributions by DDillon)
Sine Function
x 0
y
0

6

4

3

2
2
3
1
2
2
2
3
2
1
3
2
3 5
4 6

7 5
6 4
4 3 5 7 11
3 2 3 4 6
2
1
2
0
 1  2  3 -1  3  2  1
2 2 2
2 2 2
0
Notice the sine function
has origin symmetry. (If
you rotate it 180° about
the origin, the graph looks
the same.)
This means that
the sine function
is odd.
sin (-x) = - sin x
2
2
Period: Sine Function
x 0

6

4

3

2
2
3
y
1
2
2
2
3
2
1
3
2
0
This one piece of the sine
function repeats over and
over, causing the sine
function to be periodic.
The length of this piece is
called the period
of the function.
3 5
4 6

7 5
6 4
4 3 5 7 11
3 2 3 4 6
2
1
2
0
 1  2  3 -1  3  2  1
2 2 2
2 2 2
0
2
2
Cosine Function
x 0
y
1

6

4

3
3
2
2
2
1
2

2
2
3
0
1  2  3
2
2
2
Notice the cosine function
has y-axis symmetry. (If
you reflect it across the yaxis, the graph looks the
same.)
This means that
the cosine
function is even.
cos (-x) = cos x
3 5
4 6

7 5
6 4
4 3 5 7 11
3 2 3 4 6
-1
 3  2 1
2
2
2
0
1
2
2
2
3
2
2
1
Period: Cosine Function
x 0
y
1

6

4

3
3
2
2
2
1
2

2
2
3
3 5
4 6
0
1  2  3
2
2
2
This one piece of the cosine
function repeats over and
over, causing the cosine
function to be periodic. The
length of this piece is called
the period of the function.

7 5
6 4
4 3 5 7 11
3 2 3 4 6
-1
 3  2 1
2
2
2
0
1
2
2
2
3
2
2
1
Period
The period of a normal sine or cosine function is 2π.
To change the period of a sine or cosine function,
you would need to horizontally stretch or shrink the
function.
The period is found by: period =
y  sin Bx
2
B
y  cos Bx
Period
Examples of f(x) = sin Bx
• The period of the
sin(x) (parent) is 2π
• The period of
sin2x is π. p= 2  2  
b
2
• If B > 1, the graph
shrinks.
• This graph is
happening twice as
often as the original
wave.
Period
Examples of f(x) = sin Bx
• The period of the sinx
(parent) is 2π
• The period of sin ½ x
is 4π. p= 2
1
2
 2 2   4
• If b < 1, the graph
stretches.
• This graph is
happening half as
often as the original
wave.
What is the period?
Examples
y  sin
1
x
2
y  cos 3x
2
y  sin
x
3
y  cos

2
x
Horiz. stretch by ½
period  2  2  4
Horiz. shrink by 3
2
period  2  3 
3
Horiz. shrink by 2π/3
2
period  2 
3
3
Horiz. shrink by π/2
period  2 

2
4
Amplitude: Sine Function
x 0

6

4

3

2
2
3
y
1
2
2
2
3
2
1
3
2
0
The maximum height of
the sine function is 1. It
goes one unit above and
one unit below the x-axis,
which is the center of it’s
graph. This maximum
height is called
the amplitude.
3 5
4 6

7 5
6 4
4 3 5 7 11
3 2 3 4 6
2
1
2
0
 1  2  3 -1  3  2  1
2 2 2
2 2 2
0
2
2
1
1
Amplitude: Cosine Function
x 0
y
1

6

4

3
3
2
2
2
1
2

2
2
3
0
1  2  3
2
2
2
The maximum height of
the cosine function is 1.
It goes one unit above
and one unit below the xaxis, which is the center
of it’s graph. This
maximum
height is called
the amplitude.
3 5
4 6

7 5
6 4
4 3 5 7 11
3 2 3 4 6
-1
 3  2 1
2
2
2
0
1
2
2
2
3
2
1
1
2
1
Amplitude
The amplitude of the normal sine or cosine function is 1.
To change the amplitude of a sine or cosine function, you
would need to vertically stretch or shrink the function.
amplitude = |A|
(Choose the line that is dead-center of the graph. The
amplitude has the same height above the center line (axis of
the wave) as the height below the center line.
y  A cos x
y  A sin x
What is the amplitude?
Examples
y  3sin x
y
y
1
cos x
2

4
sin x
Vert. stretch by 3
Vert. shrink by ½
Vert. shrink by π/4
amp.  3
1
amp. 
2
amp. 

4
Examples: y = A sin Bx
y = A cos Bx
• Give the amplitude and period of each funtion:
 Y = 4 cos 2x
 A = 4,
p 
2

2
 y= -4 sin 1/3 x
 A = 4,
p
2
 2 (3)  6
1
3
3

sin x
2
2
3
2
2

A  ,p 
 2    4

2
 
2
 y 
Can You Write the
Equation?
•
•
•
•
Sine or cosine?
Amplitude? 2
Period? 24
b? 1
12
• Equation?
1
y  2cos x
12
Equation?
•
•
•
•
Sine or Cosine?
Amplitude? 2
Period? 8

b?
4
• Equation:
y  2 sin

4
x
Harmonic Motion
•
•
•
•
3 Types:
Simple – unvarying period motion
Damped – motion decreases with time
Resonance – motion increases with
time
Weight on Spring
video
A weight is at rest hanging from a spring. It is then pulled down 6
cm and released. The weight oscillates up and down, completing
one cycle every 3 seconds.
Sketch
Distance above/below resting point, in cm
6
3
-6
Time, in
seconds
Equation
Amplitude = 6
6
A=6
3 = 2π/B
B = 2π/3
3
-6
2
y  6sin
( x  1.5)
3
Positions
Determine the position of the weight at 1.5 seconds.
Let x = 1.5; plug into equation for function.
y = 0 cm (back at original position)
Use the graph to find the time when y = 3.5 for the first time.
Graph y1 = equation you wrote; graph y2 = 3.5.
Find intersection.
x = 1.797 seconds
3.5 is the 3.5 cm distance above the original position of the weight.
```