4.5 Graphing Sine and Cosine
Imagine taking the circumference of the unit circle and ‘peeling’ it off the circle
and straightening it out so that the radian measures from 0 to 2π lie on the x‐axis.
This is the framework we’re going to use to introduce the sine and cosine graphs.
ex) Complete the table of values listed below. Below the table is a stylized piece
of graph paper suited for plotting the points from the table.
x
0
π
π
π
π
6
4
3
2
3π
4
π
3π
2
y = sin(x)
This is the graph of y = sin(x) : Domain: ________
Range: _________
2π
ex) Complete the table of values listed below.
x
0
π
π
π
π
6
4
3
2
3π
4
π
3π
2
2π
y = cos(x)
This is the graph of y = cos(x) : Domain: ________
Range: _________
The trigonometric functions of sine and cosine are periodic functions. This means
they repeat the same output values over a specific interval on the x‐axis.
For sine and cosine the period is ________________
The sinusoidal (i.e. ‘wavy’) shape of the sine and cosine graphs can be used to
model any phenomenon with a periodic patter to it: motion of a pendulum, ocean
tide cycles, even some predator‐prey relationships.
The transformations we’ll perform on these graphs will affect the:
1. Amplitude (stretch or shrink vertically)
2. Period (stretch or shrink horizontally)
3. Phase Shift (shift the graphs left or right)
1. Amplitude Changes
Amplitude is the distance a wave travels above and below its equilibrium point.
For graphs of sine and cosine the amplitude can change by using this
transformation:
and
y = A sin( x )
y = A cos( x )
... where A is a vertical stretch or shrink factor.
ex) Graph two full periods (or cycles) of y = 3cos(x)
For graphs of sine and cosine you’ll need 5 critical points for each period. You’ll
need to label the critical points on the x‐axis as well as label the amplitude values
on the y‐axis to scale your graph.
ex) Graph two full periods of y = − 15 sin(x)
2. Period Changes
The period can be changed by causing a horizontal stretch or shrink. The
transformations which will change amplitude and period are:
and
y = A sin(Bx )
y = A cos(Bx )
... where B will cause the horizontal stretch or shrink.
To complete one full period the argument Bx needs to go from 0 to 2π .
Set up the inequality ... solve for x
0 ≤ Bx ≤ 2π
0 ≤ x ≤ 2Bπ
The period distance can always be calculated by
This is the period interval.
2π
B
.
When the graph doesn’t involve a horizontal phase shift (coming up next) the
period interval can be divided into 4 parts to label each critical point.
The increment between each critical point is always ¼ of the period.
ex) Graph two full periods of y = 2sin( 14 x ) .
Amplitude:
Period distance:
Increment:
ex) Graph two full periods of y = −8cos(4π x)
Amplitude:
Period distance:
Increment:
3. Phase Shift (Horizontal shifting)
The full transformations which will cause amplitude, period and phase shift
changes are:
and
y = A sin(Bx − C )
y = A cos(Bx − C )
... where the period interval’s endpoints can be found by using the same
inequality.
Set up the inequality 0 ≤ Bx − C ≤ 2π ... solve for x.
The initial period interval from [0, 2π ] will be changed to a new period interval.
After solving the inequality for x, the left value of the inequality will be the new
“0” starting point.
The period distance is still 2Bπ and you’ll need to remember to use addition to
locate the other critical points with the increment = ¼ of the period to label the x‐
axis critical points.
ex) Graph two full periods of y = −1.5cos(2 x − π )
When a phase shift is involved, it’s best to use the x‐axis shown below which has
the y‐axis omitted. Label the critical points first with the shifted “0” point in the
center and go one period right and left of this shifted center point.
Amplitude:
Period distance:
Phase Shift
Center point:
Increment:
On phase shift graphs, make sure
the phase shift center point is here
and graph one period left
and one period right of this point.
Make sure to label your critical points!
AND make sure to determine the
correct positioning of the y‐axis.
You need it to indicate amplitude.
ex) Graph two full periods of y = 23 sin( 12 x + π6 ) .
Amplitude:
Period distance:
Phase Shift
Center point:
Increment:
Don’t forget the shifted “0” point goes in the center!
Don’t forget to use the increment to label all critical points!
Don’t forget to locate the position of the y‐axis and label it for amplitude!