Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
Definition – Angle
An angle is the union of two rays with a common endpoint. That endpoint is called the vertex of the angle.
Definition – Degree Measure
The degree measure of an angle is the number of degrees in the intercepted arc of a circle centered at the
vertex. The degree measure is positive if the rotation is counterclockwise and negative if the rotation is
clockwise.
Types of Angles
Zero angle
mA  0
Acute angle
0  mB  90
Right angle
mC  90
Obtuse angle
Straight angle
90  mD  180 mE  180
Reflex angle
180  mF  360
Definition – Coterminal Angles
Angles  and  in standard position are coterminal if and only if there is an integer k such that
m( )  m( )  k 360 .
Definition – Radian Measure
To find the radian measure of the angle  in standard position, find the length of the intercepted arc on the unit
circle. If the rotation is counterclockwise, the radian measure is the length of the arc. If the rotation is
clockwise, the radian measure is the opposite of the length of the arc.
Degree-Radian Conversion
Conversion from degrees to radians or radians to degrees is based on
180 degrees =  radians.
Theorem: Length of an Arc
The length s of an arc intercepted by a central angle of α radians on a circle of radius r is given by
s  r .
-1-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
Unit Circle: Angles
90,
120,
135,
150,
2
3

2
60,
3
4

3

60, 
60, 3
3
5
6
60,
45,

4

30,
3

6
180, 
210,
0, 0
7
6
330,
5
225,
4
315,
240,
4
3
300,
270,
3
2
11
6
7
4
5
3
-2-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
Definition: Linear Velocity and Angular Velocity
If a point is in motion on a circle of radius r through an angle of  radians in time t, then its linear velocity is
s
v ,
t
where s is the arc length determined by s   r , and its angular velocity is


t
.
Theorem – Linear Velocity in Terms of Angular Velocity
If v is the linear velocity of a point on a circle of radius r , and  is its angular velocity, then
v  r .
 Exercises:
Find two positive angles and two negative angles that are coterminal with each given angle.
2) 90
1) 45
0, 0
Name the quadrant in which each angle lies.
3) 110
4) 200
5) 205
6) 179
7) 980
Find the measure in degrees of the least positive angle that is coterminal with each given angle.
8) 540
9) 840
Convert each angle to decimal degrees. When necessary, round to four decimal places.
10) 456
11) 441932
Convert each angle to degrees-minutes-seconds. Round to the nearest whole number of seconds.
12) 39.4
13) 122.786
-3-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
 Exercises:
Convert each degree measure to radian measure. Give exact answers.
14) 45
15) 30
16) 225
17) 315
18) 48
Convert each degrees measure to radian measure. Use the value of  found on a calculator and round answers
to three decimal places.
19) 125.3
20) 9915
Convert each radian measure to degree measure. Use the value of  found on a calculator and round answers to
three decimal places.
21)
17
12
22) 0.452
Using radian measure, find two positive angles and two negative angles that are coterminal with each given
angle.
23)

4
24)
2
3
Find the measure in radians of the least positive angle that is coterminal with each given angle.
25)
19
42
26) 23.55
-4-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
 Exercises:
Name the quadrant in which each angle lies.
27)
32)
5
6
28)
11
4
29)
5
3
30) 
39
20
31) 23.1
Fill in the rest of the unit circle. An example has been given to you.
The format is Degrees ; Radians.
-5-
Math 170 - Cooley
Pre-Calculus
OCC
Section 5.1 – Angles and Their Measurements
 Exercises:
Find the length of the arc intercepted by the given central angle  in a circle of radius r.
33)   1, r  4 cm
35)   60, r  2 m
Find the radius of the circle in which the given central angle  intercepts an arc of the given length s.
36)   0.004, s  99 km
37)   360, s  8 m
38)
A surveyor sights her 6-ft 2 in. helper on a nearby hill. If the angle of sight between the helper’s feet and
head is 037 , then approximately how far away is the helper.
39)
What is the angular velocity in radians per minute for any point on a CD that is rotating at 10,350
revolutions per minute?
40)
If a car runs over a nail at 55 mph and the nail is lodged in the tire tread 13 in. from the center of the
wheel, then what is the angular velocity of the nail in radian per hour.
-6-