This document discusses measuring angles in degrees and radians. It defines an angle, describes quadrant classification of angles, and conversions between degree and radian measure. Key concepts covered include: one radian is the measure of a central angle that intercepts an arc equal to the radius; to convert degrees to radians multiply by π/180; to convert radians to degrees multiply by 180/π.

1. Angles and Arcs in the Unit Circle
1

2. 2
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

3. 3
Angles
Trigonometry: measurement of triangles
Angle Measure
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–2
Section 4.1, Figure 4.1, Terminal and
Initial Side of an Angle , pg. 248

4. 4
Standard Position
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–3
Section 4.1, Figure 4.2, Standard
Position of an Angle, pg. 248
Vertex at origin
The initial side of an angle
in standard position is always located
on the positive x-axis.

5. 5
Positive and negative angles
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4
Section 4.1, Figure 4.3, Positive and
Negative Angles, pg. 248
When sketching angles,
always use an arrow to
show direction.

6. 6
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 of one
revolution.
1
360

7. 7
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
Classifying Angles
Quadrant 1
Quadrant 3
Quadrant 4

8. 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. 9
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–6
Section 4.1, Figure 4.5, Illustration of
Arc Length, pg. 249
s
r
 
In general,

10. 10
Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 90
2






2 6.28
3.14
1.57
2







11. 11
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



12. a) 60
b) 30
c) -54
d) -118
e) 45

13. a)
b)
c)
d)
e)
6

2

11
18


9
