4.5 Graphs of Sine and Cosine
FUNctions
How can I sketch the graphs of sine
and cosine FUNctions?
Basic Sine and Cosine Curves
y = sin x
y = cos x
5
4
gx = cos x
5
3
4
2
3
2
1
1
-4
-2
2
4
6
8
10
12
14
-6
-4
-2
2
-1
-1
-2
-2
-3
-4
-3
-4
-5
4
6
8
10
12
Let’s look at the five key points on one
period of a sine graph
5
4
Maximum:
(π/2,1)
3
2
Intercept:
(π,0)
1
-4
-2
2
4
6
8
10
12
-1
Intercept:
(0,0)
-2
-3
-4
Minimum:
(3π/2,-1)
Intercept:
(2π,0)
14
Let’s look at the five key points on one
period of a cosine graph
gx = cos x
5
Maximum: Intercept: Maximum:
(0,1)
(3π/2,0) (2π,1)
4
3
2
1
-6
-4
-2
2
4
-1
Intercept:
(π/2,0)
-2
-3
-4
Minimum:
(π,-1)
6
8
10
12
For the rest of this section, we will
apply the concepts we learned way
back in section 1.5 to the
trigonometric FUNctions.
Amplitude and Period of Sine
FUNctions
amplitude (new word, old concept) – the
maximum displacement from equilibrium.
For y = asinx and y = acosx, the amplitude
is a.
Let’s graph y = 2sinx.
fx = sinx
• First, we will graph y = sinx.
2.5
2
1.5
1
0.5
-1
1
-0.5
-1
-1.5
-2
-2.5
2
3
4
5
6
7
8
• Next, we will label the five new key points,
and graph the new FUNction.
Maximum:
(π/2,2)
Intercept:
(π,0)
2sin
ffxx == sin
xx
2.5
2.5
22
1.5
1.5
11
0.5
0.5
-1
-1
1
1
22
33
44
55
66
77
-0.5
-0.5
Intercept:
(0,0) Minimum:
(3π/2,-2)
-1
-1
-1.5
-1.5
-2
-2
-2.5
-2.5
Intercept:
(2π,0)
88
Please graph: y = 2cosx
fx = cos x
gx = 2cos x
2.5
2
1.5
1
0.5
-1
1
-0.5
-1
-1.5
-2
-2.5
-3
2
3
4
5
6
7
8
Please graph: y = .4sinx
Notes:
• The period of a FUNction is 2π/b.
• y = asinbx and y = acosbx complete one
period from 0/b to 2π/b.
Let’s graph y = cos(2x).
• First, we will graph y = cosx.
fx = cos x
2.5
2
1.5
1
0.5
-1
1
-0.5
-1
-1.5
-2
-2.5
-3
2
3
4
5
6
7
8
• Next, we will label the five new key points,
and graph the new FUNction.
Maximum:
(0,1)
Maximum:
Intercept: (π,1)
(3π/4,0)
qfxx==cos
cos2x
x 
2.5
2.5
22
1.5
1.5
11
0.5
0.5
-1-1
11
22
3 3
-0.5
-0.5
-1-1
-1.5
-1.5
Intercept:
(π/4,0)
Minimum:
(π/2,-1)
-2-2
-2.5
-2.5
-3-3
4 4
5 5
6 6
7 7
8 8
Translations of Sine and Cosine Curves
• Sine and cosine graphs of the form
have a directed horizontal shift of c/b
• In other words, to find the right and left xcoordinates of your key points for one cycle,
please solve
bx – c = 0
bx – c = 2π
Let’s graph y = 2sin(2x + π/2)
• Our y-values are easy. We just take the original ycoordinates and multiply by 2. We’ll work on the
y’s.
• Okay…. Let’s find our new endpoints:
2x + π/2 = 0
2x + π/2 = 2π
x = -π/4
x = 3π/4
• Now, we will divide the interval into four equal
parts (HINT: Find the average of the x-coordinates
of your endpoints, then average that value with
each endpoint.)
Key Points
•
•
•
•
•
(-π/4,0)
(0,2)
(π/4,0)
(π/2,-2)
(3π/4,0)
The graph!
fx = sinx
gx = 2sin
2.5
 

2x+
2
2
1.5
1
0.5
-1
1
-0.5
-1
-1.5
-2
-2.5
2
3
4
5
6
7
Good news!
• Vertical shifts are easy! We just shift up or
down after we are finished with everything
else.
• In other words, find your five key points, then
change the y-values to account for the shift.
Try one on your own!
y = 1 – 0.5sin(0.5x – π)
fx = sinx
5
gx = 1-0.5sin0.5x-
4
3
2
1
-2
2
-1
-2
-3
-4
-5
4
6
8
10
12
14
16
Mathematical Modeling
• The average monthly temperatures of a
certain southern city can be modeled by
T = 74.6 + 12.87sin(0.52t – 2.09).
where T is the average monthly temperature
and t is the month with t = 1 corresponding to
January. Please use this model to predict the
average monthly temperature in June.
T ≈ 85.63°