Transcript Radian and Degree Measure

Radian and Degree Measure
Section 4.1
Radian and Degree Measure
 We will begin our study of precalculus by
focusing on the topic of trigonometry
 Literal meaning of trigonometry
 The measurement of triangles
 Thus, we will be spending a lot of time working
with and studying different triangles
Radian and Degree Measure
 To begin our study on trigonometry, we first start with
angles and their measures
 An angle is determined by rotating a ray (half-line) about its
endpoint.
Vertex
Radian and Degree Measure
 Standard Position
 An angle in standard position has 2 characteristics:
1) Initial side lies on the x-axis
2) Vertex is at the origin
y
x
Radian and Degree Measure
 Standard Position
 An angle in standard position has 2 characteristics:
1) Initial side lies on the x-axis
2) Vertex is at the origin
y
x
Radian and Degree Measure
 Standard Position
 An angle in standard position has 2 characteristics:
1) Initial side lies on the x-axis
2) Vertex is at the origin
y
x
Radian and Degree Measure
 Positive Angles
 Rotate clockwise
 In standard position, start by going up
 Negative Angles
 Rotate counterclockwise
 In standard position, start by going down
Radian and Degree Measure
Negative Angle
Radian and Degree Measure
Positive Angle
Radian and Degree Measure
 Angles can be measured in one of two units:
1) Degrees
2) Radians

One full revolution of a central angle would be equal to:
1) 360º
2) 2π radians (or 6.28 radians)
Radian and Degree Measure
 In radians, there are common angles that will need to be

2
memorized

= 180º

= 90º
2
3
= 270º
2

3
2
Radian and Degree Measure
 In addition to our quadrant angles, there are 3 more angles
that we will be using throughout the year.

= 30º
6

= 45º
4

= 60º
3

3

4

6
Radian and Degree Measure
 Coterminal Angles
 Two angles that have the same:
 Vertex
 Initial Side
 Terminal Side
 All angles have an
infinite number of
coterminal angles
Radian and Degree Measure
 Finding Coterminal angles
 To find a positive coterminal angle
 Add 2π (or 360º) to the given angle
 To find a negative coterminal angle
 Subtract 2π (or 360º) from the given angle
Radian and Degree Measure
 Graph the following angle and determine two coterminal
angles, one positive and one negative.
5
6
5
17
 2 
6

6
6 
6
5
7
 2  
6
6
3
26
0
39
26
12
6
Radian and Degree Measure
 Graph the following angles and find two coterminal angles,
one positive and one negative.

a)
4
7
b)
6
4
c) 
3
d) 

2
Radian and Degree Measure

a)
4
2
4
4
4

9
pos :  2 
4
4

7
neg :  2  
4
4
0
6
4
Radian and Degree Measure
7
b)
6
3
6
6
6
7
19
pos :
 2 
6
6
7
5
neg :
 2  
6
6
0
9
6
Radian and Degree Measure
4
c) 
3
4
2
pos : 
 2 
3
3
4
11
neg : 
 2  
3
3
Radian and Degree Measure
d) 

2
pos : 
neg : 

2

2
 2
 2
3

2
5

2
Radian and Degree Measure
Section 4.1
Radian and Degree Measure
 Graph the following angles and find two coterminal angles,
one positive and one negative.
11
1)
4
2
2) 
3
Radian and Degree Measure
11
1)
4
2
2) 
3
19 3
P:
,
4 4
5
N :
4
4
P:
3
8
N :
3
Radian and Degree Measure
 Yesterday we covered:
 Angles in degrees and radians
 Coterminal angles
 Today we are going to cover:
 Complementary and supplementary angles
 Converting between degrees and radians
 Converting minutes & seconds to degrees
Radian and Degree Measure
 Complementary Angles

 Two positive angles whose sum is
(or 90º)
2
 Supplementary Angles
 Two positive angles whose sum is π of (180º)
Radian and Degree Measure
 Find the complement and supplement to the following
angle.
2
5

2
complement:

2 5
supplement:


10
3
2


5
5
Radian and Degree Measure
 Find the complement and supplement of the
following angles:
7
b)
8

a)
3
comp :

6
2
sup :
3
comp :

sup :
8
3
c)
7
comp :

14
4
sup :
7
Radian and Degree Measure
 Conversions between degrees and radians
1. To convert degrees to radians, multiply degrees by:
 rad
180
o
2. To convert radians to degrees, multiply radians by:
o
180
 rad
Radian and Degree Measure
Degrees to Radians
135
) 
135 (
o
o
180
180
o
 rad
3

4
Radians to Degrees
7 180
7 (180 )
o
(
) 
 210
6  rad
6
o
Radian and Degree Measure
 Convert the following from degrees to radians.
270
3
) (
) 
a)270 (
o
o
180
180
2
o
 rad
540
) (
)  3
b)540 (
o
o
180
180
o
 rad
225
5
)  (
)
c)  225 (
o
o  
180
180
4
o
 rad
Radian and Degree Measure
 Convert the following from radians to degrees.
 180o
180
a)  (
) 
 90o
2  rad
2
o
11(180 )
180
11
o

(
)
 330
b)
6
6  rad
o
180
360
o
) 
c)2 rad (
 114.59
 rad

Radian and Degree Measure
Section 4.1
Radian and Degree Measure
 Find the complement, supplement, and two coterminal
angles of the following angle.
4
9

5 22
14
 ,
,
,
18
9
9
9
 Convert the angle above to degrees.
o
80
Radian and Degree Measure
 So far in this section, we have:
Graphed angles in both radians & degrees
b) Found positive and negative coterminal angles
c) Found complementary and supplementary angles
d) Converted between radians and degrees
a)

Today we are going to apply this to different word
problems (arc length, linear speed, angular speed)
Radian and Degree Measure
 Arc Length
 The distance along the circumference of a circle with a central
angle of θ
 Given by the formula: s = r θ
Where: s = the arc length
r = the radius of the circle
θ = the central angle in radians

Radian and Degree Measure
 A circle has a radius of 4 inches. Find the length of the
arc intercepted by a central angle of 240º.
1)
Convert the angle to radians.
2)
 rad
4
)
240 (
o
180
3
o
Apply the formula.
4
S = (4) =16.7 inches
3
Radian and Degree Measure
 On a circle with a radius of 9 inches, find the length of the
arc intercepted by a central angle of 140º.
 rad
7
)
140 (
o
180
9
o
7
S = (9)
= 22 inches
9
Radian and Degree Measure
 Linear & Angular Speed
 Linear speed measures how fast a particle is moving
along the circular arc of a circle with radius r
 Given by the formula:
arc length
s

time
t
Radian and Degree Measure
 Linear & Angular Speed
 Angular speed measures how fast the angle changes
 Given by the formula:
central angle  
t
time
Radian and Degree Measure
 The second hand of a clock is 11 inches long. Find the linear
speed of the tip of this second hand as it passes around the
clock face.
In one revolution, how far does the tip travel?
r = 11 inches
s = 2π r
= 22π inches
What is the time required to travel this distance?
t = 60 seconds
Radian and Degree Measure
 The second hand of a clock is 11 inches long. Find the linear
speed of the tip of this second hand as it passes around the
clock face.
s = 22π inches
Linear Speed =
t = 60 seconds
s
22 inches

60 seconds
t
 1.15 in/sec
Radian and Degree Measure
 A car is moving at a rate of 65 mph, and the diameters of its
wheels is 2.5 feet.
a)
Find the number of revolutions per minute the wheels are
rotating.
b)
Find the angular speed of the wheels in radians per minute.
Radian and Degree Measure
 A car is moving at a rate of 65 mph, and the diameters of its
wheels is 2.5 feet.
a) Find the number of revolutions per minute the wheels are
rotating.
Find the arc length for one revolution:
S = r θ = (1.25) (2π) = 2.5 π feet per revolution
How many feet per hour is the car traveling?
(65 mph ) (5,280 feet) = 343,200 feet/hour
= 5,720 feet/min
= 728.3 revolutions
Radian and Degree Measure
 A car is moving at a rate of 65 mph, and the diameters of its
wheels is 2.5 feet.
b) Find the angular speed of the wheels in radians per minute.
(728.3 revolutions) (2π) = 4,576 radians
Angular Speed    4,576 rad.
t
1 min.
 4,576 rad/min
Radian and Degree Measure
 A car is moving at a rate of 35 mph, and the radius of its
wheels is 2 feet.
a)
Find the number of revolutions per minute the wheels are
rotating.
245.1 revolutions per minute
b)
Find the angular speed of the wheels in radians per minute.
 1,540 rad/min
Radian and Degree Measure