Angles and Arcs in the Unit Circle
1
In this section, we will study the following
topics:
Terminology used to describe angles
 Degree measure of an angle
 Radian measure of an angle
 Converting between radian and degree
measure
 Finding coterminal angles

2
Angles
Trigonometry: measurement of triangles
Angle
S e c tio n 4 .1 , F ig u re 4 .1 , T e rm in a l a n d
Measure
In itia l S id e o f a n A n g le , p g . 2 4 8
C o p yrig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
D ig ita l F ig u re s , 4 – 2
3
Standard Position
S e c tio n 4 .1 , F ig u re 4 .2 , S ta n d a rd
P o s itio n o f a n A n g le , p g . 2 4 8
Vertex at origin
C o p yrig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
The initial side of an angle
in standard position is always located
on the positive x-axis.
D ig ita l F ig u re s , 4 – 3
4
Positive and S
negative
angles
Figure 4.3, P ositive and
ection 4.1,
N egative A ngles, pg. 248
When sketching angles,
always use an arrow to
show direction.
C opyrig ht © H o ug hto n M ifflin C om pany. A ll rig hts re serve d.
D igital F ig ures, 4–4
5
Measuring Angles
The measure of an angle is determined by the amount of
rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees
and in radians.
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of
1
of one
360
revolution.
6
S e c tio n 4 .1 , F ig u re 4 .1 3 , C o m m o n D e g re e
Measuring Angles
M e a s u re s o n th e U n it C irc le , p g . 2 5 1
1
360
C o p yrig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
D ig ita l F ig u re s , 4 – 9
7
Classifying Angles
Angles are often classified according to the
quadrant in which their terminal sides lie.
Ex1: Name the quadrant in which each angle
lies.
50º
Quadrant 1
208º
Quadrant 3
II
I
-75º
Quadrant 4
III
IV
8
Classifying Angles
Standard position angles that have their terminal side
on one of the axes are called quadrantal angles.
For example, 0º, 90º, 180º, 270º, 360º, … are
quadrantal angles.
9
Coterminal Angles
e c tio
n same
4 .1 , Finitial
ig u re
4 .4
, C o tesides
rm inare
al
Angles thatShave
the
and
terminal
A n g le s , p g . 2 4 8
coterminal.
Angles  and  are coterminal.
C o p yrig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
D ig ita l F ig u re s , 4 – 5
10
Example of Finding Coterminal Angles
You can find an angle that is coterminal to a given angle  by
adding or subtracting multiples of 360º.
Ex 2:
Find one positive and one negative angle that are
coterminal to 112º.
For a positive coterminal angle, add 360º : 112º + 360º = 472º
For a negative coterminal angle, subtract 360º: 112º - 360º = -248º
11
Ex. 3: Find one positive and one negative angle
that is coterminal with the angle  = 30 o in
standard position.
Ex. 4: Find one positive and one negative angle
that is coterminal with the angle  = 272 in
standard position.
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
e c tio
n 4 .1 , F igof
u rea 4central
.5 , Illu s tra
tio n o fthat intercepts
One radian isSthe
measure
angle
e n g th , p g . 2 4 9
arc s equal in lengthA rc
to Lthe
radius r of the circle.
In general,
 
s
r
C op yrig h t © H ou gh ton M ifflin C om p any. All righ ts reserved .
D ig ita l F ig u re s , 4 –6
13
Radian Measure
2 radians corresponds to 360 
2  6.28
 radians corresponds to 180 
  3.14


2
radians corresponds
S e c tio n 4 .1 ,to
F ig90
u re4 .6 , Illu s tra tio n o f
S ix R a d ia n L e n g th s , p g . 2 4 9
C o p yrig ht © H o ug hto n M ifflin C o m p a n y. A ll rig hts re se rve d .
 1.57
2
D ig ita l F ig u re s , 4 – 7
14
S e c tio n 4 .1 , F ig u re 4 .7 , C o m m o n
Radian Measure
R a d ia n A n g le s , p g . 2 4 9
15
Conversions Between Degrees and Radians

1.
To convert degrees to radians, multiply degrees by
2.
To convert radians to degrees, multiply radians by
180
180

16
a)
60
b)
30
c)
-54
d)
-118
e)
45
a)

b) 6
c)

d) 
2
11
18
e) 
9

3
Ex 8: Find one positive and one negative angle that
is coterminal with the angle  = 7  in standard
position.
5
Degree and Radian Form of “Special” Angles
90 ° 
 120 °
60 ° 
 135 °
45 ° 
 150 °
30 ° 

0° 
 180 °
360 ° 
 210 °
330 ° 
 225 °
315 ° 
 240 °
300 ° 
270 ° 
20
Convert from degrees to radians.
1. 54
2. -300
Convert from radians to degrees.
3.
11
4.
3

1 3
12
Find one postive angle and one negative angle
in standard position that are coterminal with
the given angle.
5. 135
6.
11
6