If you have not watched the
PowerPoint on the unit circle you
should watch it first. After you’ve
watched that PowerPoint you are
ready for this one.
If you watched it, just click to
begin this part of the section.
Let’s think about the function y = sin x
What is the domain? (remember domain means the “legal”
things you can put in for x ).
You can put in anything you want
so the domain is all real
numbers.
What is the range? (remember range means what you get out
of the function).
The range is: -1  sin x  1
(0, 1)
Let’s look at the unit circle to
answer that. What is the
lowest and highest value
you’d ever get for sine?
(sine is the y value so what
is the lowest and highest y
value?)
(1, 0)
(-1, 0)
(0, -1)
Let’s think about the function y = cos x
What is the domain? (remember domain means the “legal”
things you can put in for x).
You can put in anything you want
so the domain is all real
numbers.
What is the range? (remember range means what you get out
of the function).
The range is: -1  cos x  1
(0, 1)
Let’s look at the unit circle to
answer that. What is the
lowest and highest value
you’d ever get for cosine?
(cosine is the x value so
what is the lowest and
highest x value?)
(-1, 0)
(1, 0)
(0, -1)
Look at the unit circle and determine sin 420°.
In fact sin 780° = sin 60°
since that is just another
360° beyond 420°.
Because the sine
values are equal for
coterminal angles that
are multiples of 360°
added to an angle, we
say that the sine is
periodic with a period
of 360° or 2.
1
3
 ,

2

2


All the way around is 360° so we’ll need more than that. We
see that it will be the same as sin 60° since they are
coterminal angles. So sin 420° = sin 60°.
The cosine is also periodic with a period of 360° or 2.
1
1
3
 ,

2
2 

GRAPHS OF
We are interested in the graph of y = f(x) = sin x
Start with a "t" chart and let's choose values from our
unit circle and find the sine values.
plot these points
y
x
y = sin x
0

6

2
5
6
0
1
2
1
1
2
1
x
-1
We are dealing with x's and y's on the unit circle
to find values. These are completely different
from the x's and y's used here for our function.
choose more values
x

7
6
3
2
y = sin x
plot these points
0
join the points
1

2
y
1
1
11
1

6
2
2
y = f(x) = sin x
0
-1

6

2 x
If we continue picking values for x we will start
to repeat since this is periodic.
Here is the graph y = f(x) = sin x showing
from -2 to 6. Notice it repeats with a
period of 2.
2
2
2
2
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the sine function)
From College Algebra recall that an odd
function (which the sine is) is symmetric
with respect to the origin as can be seen
here
What are the x intercepts?
Where does sin x = 0?
…-3, -2, -, 0, , 2, 3, 4, . . .

7
2
 3

3
2
 2  

5
2
2
0

2
3
4
Where is the function maximum? Where does sin x = 1?
7
3  5

,
, ,

2
2 2 2
Where is the function minimum? Where does sin x = -1?
5
 3 7

, ,
,

2
2 2
2

7
2


3
2
2
 3  2  

5
2
5
2


2
0

2
3
2
3
7
2
4
Thinking about transformations that you learned
in College Algebra and knowing what y = sin x
looks like, what do you suppose y = sin x + 2
looks like?
y = 2 + sin x This is often written
with terms traded
places so as not to
confuse the 2 with
part of sine function
The function value
(or y value) is just
moved up 2.
y = sin x
Thinking about transformations that you've
learned and knowing what y = sin x looks like,
what do you suppose y = sin x - 1 looks like?
y = sin x
The function value
(or y value) is just
moved down 1.
y = - 1 + sin x
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = sin (x + /2) looks like?
y = sin x
This is a horizontal
shift by - /2
y = sin (x + /2)
Thinking about transformations that you learned
and knowing what y = sin x looks like, what do
you suppose y = - sin (x )+1 looks like?
y = 1 - sin (x )
This is a reflection about
the x axis (shown in
green) and then a
vertical shift up one.
y = - sin x
y = sin x
What would the graph of y = f(x) = cos x look like?
We could do a "t" chart and let's choose values from our
unit circle and find the cosine values.
plot these points
x
y = cos x
0

3

2
2
3
1
1
2
0
1

2
y
1
-1

6
x
We could have used the same values as we did
for sine but picked ones that gave us easy
values to plot.
y = f(x) = cos x
Choose more values.
x
y = cos x

4
1
1

2
3
plot these points
y
1
3
2
0
5
3
1
2
-1
2
1
cosine will then repeat as you go another loop
around the unit circle

6
x
Here is the graph y = f(x) = cos x showing
from -2 to 6. Notice it repeats with a
period of 2.
2
2
2
2
It has a maximum of 1 and a minimum of -1 (remember
that is the range of the cosine function)
Recall that an even function (which the cosine is)
is symmetric with respect to the y axis as can be
seen here
Where does cos x = 0?
What are the x intercepts?
3
  3 5

, , ,
,

2
2 2 2 2
 2
3  

2 2
2
0

2
3
2
5
2
Where is the function maximum? Where does cos x = 1?
…-4, -2, , 0, 2, 4, . . .
Where is the function minimum? Where does cos x = -1?
…-3, -, , 3, . . .
 2
3  

2 2
 3

2
0

2

3
2
5
2
3
4
You could graph transformations of the cosine function the
same way you've learned for other functions.
moves up 3
moves right /4
Let's try y = 3 - cos (x - /4)
reflects over x axis
y = - cos x
y = cos x
y = 3 - cos x
y = 3 - cos (x - /4)
unit circle
y = sin t
x = cos t
value of angle t
These graphs illustrate how as you go around the unit circle and plot the
y value (upper left) or the x value (lower right) you generate the sine and
cosine graphs. The lower left shows the value of the angle t at any given
time. Notice the axis for cosine are reversed here so you can see how
the x value moves but you can rotate this graph to have t horizontal and x
vertical and see the cosine graph like it is traditionally graphed.
What would happen if we multiply the function by a
constant?
All function values would be twice as high
y = 2 sin x
amplitude
of this
graph is 2
amplitude is here
y = 2 sin x
y = sin x
The highest the graph goes (without a vertical shift) is
called the amplitude.
For y = A cos x and y = A sin x, A  is the amplitude.
What is the amplitude for the following?
y = 4 cos x
y = -3 sin x
amplitude is 4
amplitude is 3
absolute value of this
is the amplitude
y  A cos  x  C   D
y  A sin  x  C   D
This is the phase shift
(horizontal translation)
remember it is opposite in sign
This is the vertical
translation
Given this graph, let’s see if we can find it’s equation in the form
y = A sin (x – C)+ D

y = 2 sin (x) + 3



So what is A?
A=2
A









There is no
horizontal shift so
C = 0 and we have
our equation.
D




Now let’s
determine the
vertical shift D.

So what is D?

First let’s find the vertical center of the graph and
then we can determine the amplitude.
D=3