Aim: What is the unit circle?
Do Now: given the equation of a circle
x2 + y2 = 1. Write the center
and radius.
HW: p.366 # 4,6,8,10,18,20
p.367 # 2,4,6,8
A circle with center at (0, 0) and radius 1 is
called a unit circle.
(0,1)
(-1,0)
(1,0)
(0,-1)
The points on the circle must satisfy x2 + y2 = 1
Let's pick a point on the circle. We'll choose a point where the
x is 1/2. If the x is 1/2, what is the y value?
You can see there
2
2
x  y 1
are two y values.
2
They can be found
x
=
1/2
1
 
2
by putting 1/2 into

y
1
 
the equation for x
(0,1)  1 3 
2
,


and solving for y.
3
2
2
2


y 
4
3
y
2
(-1,0)
(1,0)
1
3
 ,

2

2

(0,-1) 
We'll look at a
larger version
of this and
make a right
triangle.
We know all of the sides of this triangle. The bottom leg is just
the x value of the point, the other leg is just the y value and
the hypotenuse is always 1 because it is a radius of the circle.
(-1,0)
3
opp
(0,1)
 1 3  sin  
3
 2
 ,


hyp
2 2 
1
2


1
3
1
adj
(1,0)
1
2
2
cos





hyp
1
1 2
3
2
opp
tan  
 2  3
1
adj
(0,-1)
2
Notice the sine is just the y value of the unit circle point and
the cosine is just the x value.
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 and y/x is the tangent.
 2 2


,
 2 2 


(-1,0)
(0,1)

1
3
 ,

2 2 


sin  
(1,0)
tan 
(0,-1)
1
3
 ,

2
2 


cos
2
2
2

2
2
 2  1
2

2
We divide the unit circle into various pieces and learn the point
values so we can then from memory find trig functions.
Unit Circle
90 (0, 1)
(-1, 0) 180
0
90
(0, -1)
(1, 0)
Quadrantal Angles
• Definition. A
quadrantile angle is an
angle whose initial side
lies on one of the

coordinates axes.
180

270
90
270

180 
• How do we find trig
values in this case?

360

• Examples.
90
90

360

270
Here is the unit circle divided into 8 pieces. Can you figure
out how many degrees are in each division?
These are
0,1
easy to
 2 2
 2 2
90°
memorize




,
 2 2 
 2 , 2 

 135°


since they
45°
all have the
2
sin 225  2
same value
with
180°
45°
 1,0
0° 1,0 different
signs
depending
225°
on the
 2
2
315°


2
2




,
 2 , 2 
quadrant.
270°



2
2 


We can label this all the way around
0,1 with how many degrees an
angle would be and the point on the unit circle that corresponds
with the terminal side of the angle. We could then find any of the
trig functions.
Here is the unit circle divided into 12 pieces. Can you figure
out how many degrees are in each division?
You'll need
 1 3  0,1
1 3



,
to
 , 
 2 2 


cos 330  23


2 2 
90°
memorize
120°
60°
 3 1
these too
 , 
 3 1
 2 2  150°
 , 
but you


 2 2


30°
can see
the pattern.
180°
30°
 1,0
0° 1,0
210°
 3 1
330°  3 , 1 


,

 2 2
 2 2
240°




270° 300°
sin 240  
3
2
 1
3
  , 
 2 2 


0,1
1
3
 , 
2 2 


We can again label the points on the circle and the sine is the y
value, the cosine is the x value and the tangent is y over x.
You should
memorize
this. This is
a great
reference
because
you can
figure out
the trig
functions of
all these
angles
quickly.
In unit circle
sin θ = y
cos θ = x
1
3
 ,

2
2 

Look at the unit circle and determine sin 420°.
In fact sin 780° = sin 60°
since that is just another
360° beyond 420°.
All of these would
have a sine value that
is the y value of the
point on the unit circle
here.
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°.
Now let’s look at the unit circle to compare trig functions
of positive vs. negative angles.
What is cos 60 ?
1
2
What is cos 60?
1
2
Remember negative
angle means to go
clockwise
1
3
 ,

2

2


What is sin 60 ?
3
2
What is sin 60?
3

2
1
3
 ,

2
2 

What is tan 60 ?
3
What is tan 60?
 3
1
3
 ,

2
2 

You can memorize the unit circle easily because you know the points
on the axis are 0’s and 1’s with negatives in appropriate places.
All of the multiples of 45 that don’t land on an axis have the value
for both x and y with the appropriate sign for the quadrant.
2
2
All of the multiples of 30  have the values
appropriate signs for the quadrant.
3 1
or
2
2
You can tell which one of these comes first because
than 1
2
so when x is bigger put it first.
with
3 is bigger
2
For angles that are multiples of 30 they only have one of 2 possible
values for x and y if they don’t land on an axis. If the x is further
you know that one is the square root of 3 over 2, if it is not as far
over as the y is up or down you know it is ½.
 1 3
 , 
 2 2 


120°
 3 1
 , 
 2 2


 1,0
0,1
90°
In unit circle
sin θ = y
cos θ = x
1 3
 , 
2 2 


60°
150°
30°
 3 1
 , 
 2 2


0° 1,0
180°
210°
 3 1


 2 , 2 


330°
240°
 1
3
  , 
 2 2 


270°
300°
0,1
 3 1
 , 
 2 2


1
3
 , 
2 2 

