The Unit Circle
Part II
(With Trig!!)
MSpencer
Multiples of 90°,

90°,

2
2
180°, 
0°, 0 360°, 2
270°,
3
2
The Quadrants (with Angles)

90°,
QII
180°, 
2
QI
90° <  < 180° 0° <  < 90°


0
<

<
2
2 <<
QIII
QIV
180° <  < 270° 270° <  < 360°
 <  < 3  2 3  2 <  < 2
270°,
3
2
0°, 0 360°, 2
The Unit Circle
Remember it is
called a unit
circle because
the radius is one
unit.
r=1
So let’s add in
ordered pairs to
the unit circle.
Multiples of 90°,

2
2
(0, 1)
r=1
90°,

180°, 
(1, 0)
r=1
r=1
r=1
270°,
3
2
(0, 1)
0°, 0
(1, 0)
45°,

4
45°, 4
45°
45°
Notice that 45°

or 4 forms
one of the two
special right
triangles from
geometry.
45°,

4
Let’s review this triangle
from geometry.
45°
45°
Opposite the congruent,
45° angles are congruent
sides.
These sides are the legs
of the right triangle. So
the triangle is an
isosceles right triangle.
45°,

4
Let’s call the two
congruent legs s.
s 2
45°
45°
s
s
The hypotenuse is the
length of either leg, s,
times 2 ; thus, s 2 .
45°,
s 2  1 45°
s

4
Lastly, now remember that
the hypotenuse is the radius
of the unit circle, which
means it must equal one.
45°
s
Solve for s.
s 2 1
1
2
2
s


2 2
2
45°,

4
 2 2
45°, 4  2 , 2 


1
45°
2
2
45° 2
2
The distance across
the bottom side of
the triangle
represents the xcoordinate while
the right, vertical
side represent y.
Signs and Quadrants

90°,
180°, 
2
Q II
QI
(, +)
(+, +)
Q III Q IV
(, )
(+, )
270°,
3
2
The signs of each
ordered pair
follow the signs of
x and y for each
quadrant.
0°, 0
Multiples of 45°,
 2 2
3
,

 135°, 4
2 
 2
2 45°
2
45°
 2 45°
 2
2
2 45°
 2  2
5
,

 225°, 4
2 
 2

4

45°,
 2 2
4 2 , 2 


45° 2
2
45°
2
2
 2  2
7 
315°, 4  2 , 2 
60°,

3
60°, 3
30°
60°
Notice that 60°

or 3 forms
the other
special right
triangle from
geometry.
60°,

3
Let’s review this triangle
from geometry.
2s
60°
s
30°
s 3
Call the the smallest side
opposite 30° s.
The hypotenuse is twice
the smallest side, or 2s.
The medium side
opposite 60° is 3 times
the smallest side, or s 3.
60°,
2s = 1
60°
s

3
The hypotenuse is the radius
of the unit circle, which
means it must equal one.
30°
s 3
Solve for s.
2s  1
1
s
2
The medium side
opposite 60° is 3
2
60°,

60°,
3
2
1
2
x

3
1 3
3  2, 2 


y
Notice that since
the triangle is taller
than it is wide, that
the y-coordinate is
larger than the xcoordinate.
Multiples of 60°,
 1 3 
 ,

 2 2 
2
120°,

60°,
3
3
2
1
2
 1  3 
 ,

2
2


4
240°,
3

3
1 3
3  2, 2 


3
2
1
2
300°,
5
1  3
3  2, 2 


30°,

6

60°
30°,
1
2
30°
3
2
x
 3 1
6  2 ,2


y
Notice this is the same
special right triangle
as for 60° except the x
and y coordinates are
switched.
Multiples of 30°,
 3 1
, 

 2 2
5
150°,
6

6
  3 1 
,  210°,7 6

2 
 2

30°,
30°
3
2
60° 1
2
 3 1
,
6  2 2


 3 1 
11
330°, 6  2 , 2 


Ordered Pairs and Trig
From geometry, recall SOHCAHTOA,
which defines sine, cosine, and tangent.
sine (Sin) =
Opposite
Hypotenuse
cosine(Cos)
Adjacent
= Hypotenuse
tangent (Tan) =
Opposite
Adjacent
Ordered Pairs and Trig
 3 1
30°, 6  2 , 2 


3
1
60°
Adjacent
2

2 Cos 30° = Hypotenuse
1
3
cos 30° = 2

30°
3
2
Notice that the cosine
of the angle is simply
the x-coordinate!
Ordered Pairs and Trig

30°,
30°
3
2
60° 1
2
 3 1
6  2 ,2


Sin 30° =
1
Opposite
 2
Hypotenuse 1
sin 30° =
1
2
Notice that the sine
of the angle is simply
the y-coordinate!
Ordered Pairs: Cosine & Sine
 (x, y) (cos , sin )

And this is true
for ANY angle,
often called .
cos  = x
sin  = y
Signs for Cosine and Sine

90°,
180°, 
2
Q II
QI
(, +)
(+, +)
Q III Q IV
(, )
(+, )
270°,
3
2
The “signs” of
cosine and “sine”
follow the signs of
x and y in each
quadrant.
0°, 0
So in QII, for
instance, cosine
is negative while
sine is positive.
The Whole Unit Circle Together (Grouped)
 1 3 
 ,
 120°,2
3
 2 2 
 2 2
,
3


2  135°, 4
 2
 3 , 1 
5

2 2  150°, 6

(1, 0) 180°, 
  3 , 1 

2
2 

7
210°,
(0, 1)
1 3


90°, 2 60°, 3  2 , 2 



  2 , 2 
45°, 4  2 2 
  3 , 1 
30°, 6  2 2 
0°, 0 (1, 0)
11  3 , 1 
330°, 6  2 2 
7  2 ,  2 
315°, 4  2 2 
6
 2  2
5
,

 225°, 4
2 
 2
4
 1  3 
240°,
3
 ,

 2 2 

3
5  1  3 
270°, 2 300°, 3  2 , 2 


(0, 1)

The Whole Unit Circle Together (In
Ascending Order)
 1 3 
 ,
 120°,2
3
 2 2 
 2 2
,
3


2  135°, 4
 2
 3 , 1 
5

2 2  150°, 6

(1, 0) 180°, 
  3 , 1 

2
2 

7
210°,
(0, 1)
1 3


90°, 2 60°, 3  2 , 2 



  2 , 2 
45°, 4  2 2 
  3 , 1 
30°, 6  2 2 
0°, 0 (1, 0)
11  3 , 1 
330°, 6  2 2 
7  2 ,  2 
315°, 4  2 2 
6
 2  2
5
,

 225°, 4
2 
 2
4
 1  3 
240°,
3
 ,

 2 2 

3
5  1  3 
270°, 2 300°, 3  2 , 2 


(0, 1)
