This document discusses radian and degree measure of angles. It covers terminology used to describe angles, converting between radian and degree measure, finding coterminal angles, and classifying angles by quadrant. Radian measure is defined as the measure of an angle whose terminal side intercepts an arc of length r on a circle of radius r. Common conversions between degrees and radians are also provided.

1. 1
Radian and Degree Measure
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
 Find coterminal angles

2. 2
Bellwork

3. 3
Radian and Degree Measure
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
6.1 Radian and Degree Measure
Standard Position:
An angle is in standard position when its vertex is at the origin
and its initial side lies on the positive x-axis.
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
Radian and Degree Measure
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

7. 7

8. 8
6.1 Radian and Degree Measure
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

9. 9
Radian and Degree Measure
Measuring Angles
1
360
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–9
Section 4.1, Figure 4.13, Common Degree
Measures on the Unit Circle, pg. 251

10. 10
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
Radian and Degree Measure
Classifying Angles
Quadrant 1
Quadrant 3
Quadrant 4

11. 11
Radian and Degree Measure
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.

12. 12
Radian and Degree Measure
Coterminal Angles
Angles that have the same initial and terminal sides are
coterminal.
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–5
Section 4.1, Figure 4.4, Coterminal
Angles, pg. 248
Angles  and  are coterminal.

13. Find one positive angle and one negative angle
that are coterminal with (a) −45° and (b) 395°.
There are many such angles, depending on what multiple of 360° is
added or subtracted.
13

14. 14
Radian and Degree Measure
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

15. Ex 3. Find one positive and one negative angle that is
coterminal with the angle  = 30° in standard position.
Ex 4. Find one positive and one negative angle that is
coterminal with the angle  = 272 in standard position.

16. 16
Radian and Degree Measure
Radian Measure
A second way to measure angles is in radians.
Definition of Radian:
One radian is the measure of an angle in standard position
whose terminal side intercepts an arc of length r.
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,

17. 17
Radian and Degree Measure
Radian Measure
2 radians corresponds to 360
radians corresponds to 180
radians corresponds to 90
2






2 6.28
3.14
1.57
2






Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–7
Section 4.1, Figure 4.6, Illustration of
Six Radian Lengths, pg. 249

18. 18
Radian and Degree Measure
Radian Measure
Section 4.1, Figure 4.7, Common
Radian Angles, pg. 249

19. 19

20. 20
Radian and Degree Measure
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



21. Ex 5. Convert the degrees to radian measure.
a) 60
b) 30
c) -54
d) -118
e) 45

22. Ex 6. Convert the radians to degrees.
a)
b)
c)
d)
6

2

11
18


9


23. Bellwork
Convert the degrees to radian measure.
1. 30
2. -54
Convert the radians to degrees.
3.
4.
23
9

11
18



24. 24

25. Ex 7. Find one positive and one negative angle that is
coterminal with the angle  = in standard position.
Ex 8. Find one positive and one negative angle that is
coterminal with the angle  = in standard position.
7
5

3


26. 26
0° 
360 ° 
30 ° 
45 ° 
60 ° 
330 ° 
315 ° 
300 ° 
 120 °
 135 °
 150 °
 240 °
 225 °
 210 °
 180 °
90 ° 
270 ° 

Degree and Radian Form of “Special” Angles

27. Find one postive angle and one negative
angle in standard position that are
coterminal with the given angle.
5. 135
6.
11
6


28. Bellwork
Convert from degrees to radians.
1. 54
2. -300
Convert from radians to degrees.
3.
4.
11
3

13
12



29. A sector is a region of a circle that is bounded by two radii
and an arc of the circle. The central angle θ of a sector is
the angle formed by the two radii. There are simple
formulas for the arc length and area of a sector when the
central angle is measured in radians.
29

30. A softball field forms a sector with the dimensions
shown. Find the length of the outfield fence and
the area of the field.
30

31. In Exercises 33–38, use a calculator to
evaluate the trigonometric function.
31