### 5.1 Radian and Degree Measure Measuring Angles

```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
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
208º
II
I
-75º
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
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.
A second way to measure angles is in radians.
e c tio
n 4 .1 , F igof
u rea 4central
.5 , Illu s tra
tio n o fthat intercepts
measure
angle
e n g th , p g . 2 4 9
arc s equal in lengthA rc
to Lthe
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
2 radians corresponds to 360 
2  6.28
 radians corresponds to 180 
  3.14


2
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
R a d ia n A n g le s , p g . 2 4 9
15

1.
To convert degrees to radians, multiply degrees by
2.
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
1. 54
2. -300
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
```