A circle with center at (0, 0) and radius 1 is called a unit circle.
Remember that the equation of this circle would be
x  y 1
2
2
(0,1)
(-1,0)
(1,0)
(0,-1)
So points on this circle must satisfy this equation.
You should
memorize
this. This is
a great
reference
because
you can
figure out
the trig
functions of
all these
angles
quickly.
1
3
 ,

2
2 

So if I want a trig function for  whose terminal side contains a
point on the unit circle, the y value is the sine, the x value is
the cosine.
 2 2


,
 2 2 


(-1,0)
(0,1)

(0,-1)
1
3
 ,

2 2 


(1,0)
sin  
cos
2
2
2

2
1
3
 ,

2
2 


We divide the unit circle into various pieces and learn the point
values so we can then from memory find trig functions.
Now let’s look at the unit circle to compare trig functions
of positive vs. negative angles.
What is cos
1
2

3
?
 
What is cos   ?
 3
1
2
1
3
 ,

2

2


What is sin

3
?
3
2
 
What is sin    ?
 3
3

2
1
3
 ,

2
2 

GRAPHS OF
Key Vocabulary
y = A sin(Bx)
Amplitude: The value A affects the amplitude. The
amplitude (half the distance between the maximum and
minimum values of the function) will be |A|, since
distance is always positive. Increasing or decreasing the
value of A will vertically stretch or shrink the graph
Period: The horizontal length of one repeating pattern of
the function (the wave length), the number of cycles it
completes in an interval of from 0 to 360 (2∏). The value
B affects the period. The period of sine and cosine is
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)
Remember from Unit 3 that an odd function
(which the sine is) is symmetric with
respect to the origin as can be seen here
Knowing the Unit Circle, what is our domain
and range of Sin Function?
Domain: All Real Numbers
Range: [-1, 1]
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 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
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
Knowing Unit Circle, what is domain and range
of cosine function?
Domain: All Real Numbers
Range: [-1, 1]
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)
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