Objective: Students will use radian angle
measures to model real life problems.
2015
• Describe angles
• Use radian measure
• Use degree measure and convert between
degree and radian measure
• Use angles to model and solve real-life
problems
As
derived from the Greek language, the word
trigonometry means “measurement of triangles.”
Initially, trigonometry dealt with relationships
among the sides and angles of triangles and was
used in the development of astronomy, navigation,
and surveying.
With
the development of calculus and the physical
sciences in the 17th century, a different perspective
arose—one that viewed the classic trigonometric
relationships as functions with the set of real
numbers as their domains.
Consequently,
the applications of trigonometry
expanded to include a vast number of physical
phenomena involving rotations and vibrations,
including the following.
◦ sound waves
◦ light rays
◦ planetary orbits
◦ vibrating strings
◦ pendulums
◦ orbits of atomic particles
• An angle is determined by rotating a ray about its
endpoint.
• The measure of an angle is determined by the amount of
rotation from the initial side, to the terminal side.
• The shared initial point of the two rays is called the vertex
of the angle.
• An angle is in standard position if its vertex is at the
origin of the rectangular coordinate system and the
initial side lies along the positive x-axis.
• If the rotation of the angle is in the counterclockwise
direction, then the angle is said to be positive. If the
rotation is clockwise, then the angle is negative.
Angles
are labeled with Greek letters such as 
(alpha),
 (beta), and (theta), as well as uppercase letters such
as A,B, and C. In Figure 4.4, note that angles  and 
have the same initial and terminal sides. Such angles
are coterminal.
Figure 4.4
Radian Measure
The
measure of an angle is determined by the
amount of rotation from the initial side to the
terminal side. One way to measure angles is in
radians.
This
type of measure is
especially useful in calculus.
To define a radian, you can
use a central angle of a circle,
one whose vertex is the center
of the circle, as shown in
Figure 4.5.
Figure 4.5
r
r
How many radii fit around a circle?
2
3
1
?
4
5
6
6
1
4
Is there a more accurate way
to calculate the number of radii?
2
3
1
?
4
5
6
Circumference measures the distance
around a circle
2
3
1
?
4
5
6
C  2 r
C   2  r
C  6.28 r
One radian is the measure of a central angle " " that intercepts
an arc “s” equal in length to the radius of the circle.
Since the circumference of a circle is C  2r , it takes 2
radians to get completely around the circle once. Therefore,
it takes  radians to get halfway around the circle.
The
four quadrants in a coordinate system are
numbered I, II, III, and IV. Figure 4.8 shows which
angles between 0 and 2 lie in each of the four
quadrants. Note that angles between 0 and 2 and
are acute and that angles between 2 and  are
obtuse.

Figure 4.8
y

2

2
3
3
3 
90 60
120
4
4
45
135

5
30 6
6 150
 180
0 0
7 210
330 11
6
6
315
5 225
7
240
300
4 4
270
5 4
3
3
3
2
x
Two angles in standard position that have the same
terminal side are said to be coterminal.
140
y
Coterminal Angles
Quadrant II
140, 220,500
x
220
Find one positive and one negative
angle coterminal with the given angles:
a)
20
b)
135
Find one positive and one negative
angle coterminal with the given angles:
a)

6
b) 

2
Two
positive angles  and  are
complementary (complements of each other)
when their sum is 90 (or 2)
Two positive angles are supplementary
(supplements of each other) when their sum is
180 (or  ).
Complementary angles
Supplementary angles
Figure 4.14
Find the complement and supplement of

.
5
Degree Measure
Another
way to measure angles is in terms of
degrees, denoted by the symbol . A measure of
one degree (1) is equivalent to a rotation of
of a complete revolution about the vertex.
Figure 4.12
Figure 4.13
So,
a full revolution (counterclockwise)
corresponds to 360 a half revolution to 180,
a quarter revolution to 90 and so on.
Because
2 radians corresponds to one
complete revolution, degrees and radians are
related by the equations
360
= 2 rad
and
180 =  rad.
Since
1


=
180 =  radians, it follows that:
radians
1 radian =
which lead to the following conversion rules.
To convert degrees to radians, multiply degrees
by  radians
180

and
To convert radians to degrees, multiply radians
180
by
 radians
11
1. 220   radians   11 radians


9
 180 
9
4   radians  4
2. 80 
radians

9
 180 
9
36
2  180 
1.
 72


5   radians 
6  180 
2.
 154.3


7   radians 
r
arc length
s
central
angle

1 2
Area: A  r 
Arc Length: s  r
2
(angle must be in radians) (angle must be in radians)
 
A   r2 
 2
 1 2
 r 
 2
  60
60


3
10
Arc Length: S 
3
1
2 
Area: A  10   
2
3
50

3
Find the length of the arc that subtends a central
angle with measure 120° in a circle with radius 5
inches.
Consider a particle moving at a constant speed along a circular
arc of radius r. If s is the length of the arc traveled in time t,
then the linear speed of the particle is
Linear speed =
arclength s

time
t
measures how fast the particle moves
Moreover, if  is the angle (in radians measure) corresponding
to the arc length, s, then the angular speed of the particle is
Angular speed =
central angle 

time
t
measures how fast the angle changes
A lawn roller with a 10-inch radius makes 1.2
revolutions per second.
Find the angular speed of the roller in radians per
second.
 1.2  2

 2.4 radians per second.
t 1second
Find the speed of the tractor that is pulling the roller.
s r 10  2.4  inches


 24 or 75.4 inches per second.
t
t
1second
The
second hand of a clock is 10.2 centimeters
long, as shown in Figure 4.16. Find the linear
speed of the tip of this second hand.

Figure 4.16
In one revolution, the arc length traveled is
s  r
s  10.2  2  20.4  centimeters
Linear Speed is:
s 20.4 cm.

 1.068centimeters per second
t
60sec


4.1 pg. 255 13-63,odd 77-87 odd
Find one positive and one negative
angle co-terminal with the given angle:
3
8