Convert between degrees and radians Find the length of an intercepted arc Find the area of a sector

1. 6.1 Radian Measure
Chapter 6 Circular Functions and Their Graphs

2. Concepts and Objectives
 Convert between degrees and radians
 Calculate the length of an arc intercepted by a given
angle
 Calculate the area of a sector

3. Radian Measure
 Up until now, we have measured angles in degrees.
Another unit of measure that mathematicians use is
called radian measure.
An angle with its vertex at
the center of a circle that
intercepts an arc on the
circle equal in length to the
radius of the circle has a
measure of 1 radian.
r
r
x
y

 = 1 radian

4. Radian Measure (cont.)
 You should recall that the circumference of a circle is
given by C = 2r, where r is the radius of the circle. An
angle of 360°, which corresponds to a complete circle,
intercepts an arc equal to the circumference.
 360 2 radians
 180 radians



180
1 radian

 1 radians
180
If no unit of angle measure is specified, then the
angle is understood to be measured in radians.

5. Radians and Degrees
 Converting between radians and degrees is just like
converting between any other type of units:
 Put the unit you are converting to on the top, and
the unit you are converting from on the bottom.
 Example: Convert 120° to radians.
120120°
180° 180
 2
3
= =
1
radians

6. Radians and Degrees (cont.)
 Example: Convert 57° 48' to radians
 Example: Convert radians to degrees
3
5

7. Radians and Degrees (cont.)
 Example: Convert 57° 48' to radians
 Example: Convert radians to degrees
3
5
 
  
57.8
180
    
48
57 48' 57 57.8
60


57.8
180
 0.321 1.01 radians

8. Radians and Degrees (cont.)
 Example: Convert 57° 48' to radians
 Example: Convert radians to degrees
3
5
 
  
57.8
180
    
48
57 48' 57 57.8
60


57.8
180
 0.321 1.01 radians
 
  
3 180
5
 3 36  108

9. The Unit Circle
0 0

 90
2
  180


3
270
2
  360 2

10. The Unit Circle
0 0

 45
4

 90
2

 
3
135
4
  180

 
5
225
4


3
270
2

 
7
315
4
  360 2

11. The Unit Circle
0 0

 45
4

 60
3

 90
2

 
2
120
3

 
3
135
4
  180

 
5
225
4

 
4
240
3 

3
270
2

 
5
300
3

 
7
315
4
  360 2

12. The Unit Circle
0 0

 30
6

 45
4

 60
3

 90
2

 
2
120
3

 
3
135
4

 
5
150
6
  180

 
7
210
6

 
5
225
4

 
4
240
3 

3
270
2

 
5
300
3

 
7
315
4


11
330
6
  360 2

13. Arc Length of a Circle
 The length of an arc is proportional to the measure of its
central angle.
•
r
s

The length s of the arc on
a circle of radius r created
by a central angle
measuring  is given by
( must be in radians)
s r

14. Arc Length of a Circle (cont.)
 Example: Find the length, to the nearest hundredth of a
foot, of the arc intercepted by the given central angle and
radius. 
 
5
1.38 ft,
6
r 

15. Arc Length of a Circle (cont.)
 Example: Find the length, to the nearest hundredth of a
foot, of the arc intercepted by the given central angle and
radius.
 Remember—the angle measure must be in radians. If
you are given an angle measure in degrees, you must
convert it into radians.

 
5
1.38 ft,
6
r 
 
 
  
 
5
1.38
6
s 3.61 ft

16. Arc Length of a Circle (cont.)
 Example: Find the length to the nearest tenth of a meter
of the arc intercepted by the given central angle and
radius.
  2.9 m, 68r 

17. Arc Length of a Circle (cont.)
 Example: Find the length to the nearest tenth of a meter
of the arc intercepted by the given central angle and
radius.
  2.9 m, 68r 
  
 
    
2.9 68
180
s
3.5 m

18. Using Latitudes
 Example: Erie, Pennsylvania is approximately due north
of Columbia, South Carolina. The latitude of Erie is
42° N, while that of Columbia is 34° N. If the Earth’s
radius is 6400 km, find the north-south distance
between the two cities.

19. Using Latitudes
 Example: Erie, Pennsylvania is approximately due north
of Columbia, South Carolina. The latitude of Erie is
42° N, while that of Columbia is 34° N. If the Earth’s
radius is 6400 km, find the north-south distance
between the two cities.
The measure of the central angle between Erie and
Columbia is 42° ‒ 34° = 8°. Therefore,
 6400 8 894 km
180
s
 
   
 

20. Area of a Sector of a Circle
 Recall that a sector is the portion of the interior of the
circle intercepted by a central angle. The area of the
sector is proportional to the area of the circle.
r

•
The area A of a sector of a
circle of radius r and
central angle  is given by
( must be in radians)
 21
2
A r 

21. Area of a Sector of a Circle
 Example: Find the area of a sector of a circle having the
given radius r and central angle  (round to the nearest
kilometer). 
 
2
59.8 km,
3
r 

22. Area of a Sector of a Circle
 Example: Find the area of a sector of a circle having the
given radius r and central angle  (round to the nearest
kilometer). 
 
2
59.8 km,
3
r 
   
  
 
21 2
59.8
2 3
A  2
3745 km

23. Classwork
 College Algebra
 Page 565: 26-52, page 539: 14-28, page 513: 74-80
(all evens)