Trig – Section 2
The Unit Circle
Essential Questions:
• What are the values for special right
triangles?
• How do you generate the unit circle?
• Define the six trig ratios in terms of x
and y.
45-45-90 Right Triangles
In a 45-45-90 right triangle, the
hypotenuse is 2 times as long as
the leg.
2
45
1
45
1
Ex.1 Find the missing sides.
a)
b)
8 2
45
3
6
c)
45
45
30-60-90 Right Triangle
In a 30-60-90 triangle, the hypotenuse is twice as long
as the shorter leg, and the longer leg is 3
times as long as the shorter leg.
30
2
3
60
1
Ex 2. Find the missing sides.
a)
b)
30
6 3
60
4
c)
60
18
Class Work
Special Right Triangles Wkst
The Unit Circle
A circle where the radius equals 1.
On your special triangles worksheet, convert all
of the radii to unit 1 and find the new length of
each side.
Discuss what you notice about 30, 45, 60 with
your shoulder partner.
Constructing
Sine:
Cosine:
Tangent:
45
Constructing Sine, Cosine, Tangent:
a b c
2
2
a b 1
2
2a  1
1
2
a 
2
2
a
2
2
2
2
45
1
ba
a
radius  1
Constructing Sine, Cosine, Tangent:
2
sin 
2
2
cos 
2
45
1
2
2
tan  1
2
2
What is the coordinate for this point?
 2 2
,


 2 2 
y
2
2
45
2
2
x
Pass out Unit Circle worksheet
Fill in degrees only for each circle on back
Go over answers
Pass out Hand made Circle worksheet and find point of each circle
That is bold. Transfer information onto large unit circle
Stop for today!
What is the circumference
Of a circle with a radius of 1?
C  2 r 3
C  2 4
3

4

2

5

4

4
r 1
5
4
3
2
1

4
0, 2
7

4
7
4 3  2
4
These are called
Radians: 3
4

2

4
0, 2

5
4
3
2
7
4
Radians
A radian is another form of measuring angles.
The radian measure of an angle drawn in
standard position in the plane is equal to the
length of the arc on the unit circle subtended by
that angle.
Radian Measure
What would be the degrees of these radian measures?
3

135,
4

,90
2
180, 
5
225,
4

, 45
4
0, 2
0,360
3
,270
2
7
,315
4
What will be the radian measure of these arcs?
2
3

2

3
5
6
2
 1
6 
6

6
0, 2

7
6
4
3
3
2
11
6
5
3
What will be the degree measure of these?
2
120,
3
5

150,
6
180, 
7
210,
6
4
240,
3

,90
2
 ,60
3

,30
6
2,360
11 ,330
6
3
, 270
5 ,300
2
3
R E L AT I O N S H I P B E T W E E N D EG R E ES
AND RADIANS
180   radians
1 radian 
180

1 

180
radians

To convert degrees to radians, multiply by
180
To convert radians to degrees, multiply by
180

Ex 3. Find the degrees or radians of
each:
a) 30   
180 6
 180
b) 
 60
3 
e) 210   7
180
6
4 180
f)

 240
3 
c) 120   2
180 3
5 180
 150
d) 
6 
g ) 300   5
180 3
11 180
 330
h)

6


, 45
4
What is the coordinate for this point?
 2 2
,


 2 2 
2
2
45
2
2

2 2
,
 

 2 2 
2
2
45
2

2
Label all the coordinates. 
3
4
2

4
0, 2

5
4
3
2
7
4
What are the coordinates of

?
6
 3 1
, 

 2 2 
,30
60 6
1
1
2
30
3
2
What are the coordinates of

?
3
 ,60
3
1 , 3


2 2 
1
60
30
3
2
1
2
Fill in the Unit Circle
( x, y )
We define cos as the 'x' coordinate.
We define sin as the 'y' coordinate.
HW #2
Unit Circle Worksheet