Transcript No Slide Title

Chapter 3
Radian Measure and
Circular Functions
Copyright © 2005 Pearson Education, Inc.
3.1
Radian Measure
Copyright © 2005 Pearson Education, Inc.
Measuring Angles




Thus far we have measured angles in degrees
For most practical applications of trigonometry
this the preferred measure
For advanced mathematics courses it is more
common to measure angles in units called
“radians”
In this chapter we will become acquainted with
this means of measuring angles and learn to
convert from one unit of measure to the other
Copyright © 2005 Pearson Education, Inc.
Slide 3-3
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. (1 rad)
Copyright © 2005 Pearson Education, Inc.
Slide 3-4
Comments on Radian Measure


A radian is an amount of rotation that is independent of
the radius chosen for rotation
For example, all of these give a rotation of 1 radian:



radius of 2 rotated along an arc length of 2
radius of 1 rotated along an arc length of 1
radius of 5 rotated along an arc length of 5, etc.
r1
r1
Copyright © 2005 Pearson Education, Inc.
r2 1 rad
r2
Slide 3-5
More Comments on Radian Measure



As with measures given in degrees, a
counterclockwise rotation gives a measure expressed
in positive radians and a clockwise rotation gives a
measure expressed in negative radians
Since a complete rotation of a ray back to the initial
position generates a circle of radius “r”, and since the
circumference of that circle (arc length) is 2r , there are
2 radians in a complete rotation
Based on the reasoning just discussed:
2 rad  3600
0
 rad  180 0
180
0
 57.3
1 rad 
Copyright © 2005 Pearson Education, Inc.

 rad  1800
 rad
0
180
1
Slide 3-6
Converting Between Degrees and Radians




From the preceding discussion these ratios both
equal “1”:  rad 1800

1
0
180
 rad
To convert between degrees and radians:
 rad
Multiply a degree measure by
0 and simplify
180
to convert to radians.
1800
Multiply a radian measure by
and simplify
 rad
to convert to degrees.
Copyright © 2005 Pearson Education, Inc.
Slide 3-7
Example: Degrees to Radians


Convert each degree measure to radians.
a) 60
60  60 
0

0
 rad
180
0


rad
3
b) 221.7
221.7  221.7 
0
Copyright © 2005 Pearson Education, Inc.
0
 rad
180
0
 3.896 rad
Slide 3-8
Example: Radians to Degrees

Convert each radian measure to degrees.

a) 11 rad
4
11 rad 11 rad 1800


 4950
4
4
 rad

b) 3.25 rad
3.25 rad 1800
3.25 rad 

 186.20
1
 rad
Copyright © 2005 Pearson Education, Inc.
Slide 3-9
Equivalent Angles in Degrees and Radians
Degrees
Radians
Degrees
Exact
Approximate
0
0
0
90
30

6
.52
180
45

4

3
.79
1.05
60
Copyright © 2005 Pearson Education, Inc.
Radians
Exact
Approximate

2

1.57
270
3
2
4.71
360
2
6.28
3.14
Slide 3-10
Equivalent Angles in Degrees and Radians
continued
Copyright © 2005 Pearson Education, Inc.
Slide 3-11
Finding Trigonometric Function Values of
Angles Measured in Radians




All previous definitions of trig functions still apply
Sometimes it may be useful when trying to find a trig
function of an angle measured in radians to first convert
the radian measure to degrees
When a trig function of a specific angle measure is
indicated, but no units are specified on the angle
measure, ALWAYS ASSUME THAT UNSPECIFIED
ANGLE UNITS ARE RADIANS!
When using a calculator to find trig functions of angles
measured in radians, be sure to first set the calculator to
“radian mode”
Copyright © 2005 Pearson Education, Inc.
Slide 3-12
Example: Finding Function Values of
Angles in Radian Measure



Find exact function value:
4
a) tan
3
Convert radians to
degrees.
4
4

tan
 tan   180 
3
3

 tan 240  tan600 

4
b) sin
3
4
sin
 sin 2400
3
3
0
  sin 60  
2
 3
Copyright © 2005 Pearson Education, Inc.
Slide 3-13
Homework

3.1 Page 97
All: 1 – 4, 7 – 14, 25 – 32, 35 – 42, 47 – 52,
61 – 72

MyMathLab Assignment 3.1 for practice

MyMathLab Homework Quiz 3.1 will be due for a
grade on the date of our next class meeting

Copyright © 2005 Pearson Education, Inc.
Slide 3-14
3.2
Applications of Radian Measure
Copyright © 2005 Pearson Education, Inc.
Arc Lengths and Central Angles of a Circle




Given a circle of radius “r”, any angle with vertex at the
center of the circle is called a “central angle”
The portion of the circle intercepted by the central angle
is called an “arc” and has a specific length called “arc
length” represented by “s”
From geometry it is know that in a specific circle the
length of an arc is proportional to the measure of its
central angle
For any two central angles,  1 and 2 , with
s2
corresponding arc lengths s1 and s 2 : s1
1
Copyright © 2005 Pearson Education, Inc.

2
Slide 3-16
Development of Formula for Arc Length

Since this relationship is true for any two central
angles and corresponding arc lengths in a circle
of radius r: s1  s2
1

2
Let one angle be  rad with corresponding arc
length s and let the other central angle be
a whole rotation, 2 rad with arc length 2r
s
s
2r
s  r
r


 rad 2 rad
 in radians!
Copyright © 2005 Pearson Education, Inc.
Slide 3-17
Example: Finding Arc Length

A circle has radius 18.2
cm. Find the length of the
arc intercepted by a
central angle having the
following measure:
3

8
s  r
 3 
s  18.2 
 cm
 8 
54.6
s
cm  21.4cm
8
Copyright © 2005 Pearson Education, Inc.
Slide 3-18
Example: Finding Arc Length continued


For the same circle with
r = 18.2 cm and  = 144,
find the arc length
convert 144 to radians
  
144  144 

 180 
4

radians
5
Copyright © 2005 Pearson Education, Inc.
s  r
 4 
s  18.2 
 cm
 5 
72.8
s
cm  45.7cm
5
Slide 3-19
Note Concerning Application Problems
Involving Movement Along an Arc

When a rope, chain, belt, etc. is attached to a
circular object and is pulled by, or pulls, the
object so as to rotate it around its center, then
the length of the movement of the rope, chain,
belt, etc. is the same as the length of the arc
s
l
sl
Copyright © 2005 Pearson Education, Inc.
Slide 3-20
Example: Finding a Length


A rope is being wound
around a drum with
radius .8725 ft. How
much rope will be wound
around the drum if the
drum is rotated through
an angle of 39.72?
Convert 39.72 to radian
measure.
Copyright © 2005 Pearson Education, Inc.
s  r

  
s  .8725 39.72 
   .6049 ft.
 180  

Slide 3-21
Example: Finding an Angle Measure


Two gears are adjusted
so that the smaller gear
drives the larger one, as
shown. If the smaller gear
rotates through 225,
through how many
degrees will the larger
gear rotate?
The motion of the small
gear will generate an arc
length on the small gear
and an equal movement
on the large gear
Copyright © 2005 Pearson Education, Inc.
Slide 3-22
Solution

Find the radian measure of the angle and then
find the arc length on the smaller gear that
determines the motion of the larger gear.
   5
225  225 

 180  4
 5  12.5 25
s  r  2.5   

cm.
4
8
 4 

This same arc length will occur on the larger
gear.
Copyright © 2005 Pearson Education, Inc.
Slide 3-23
Solution continued

An arc with this length on the larger gear
corresponds to an angle measure , in radians
where
s  r
25
 4.8
8
125
  (Angle measuredin radians)
192

Convert back to degrees.
Copyright © 2005 Pearson Education, Inc.
125  180 

  117
192   
Slide 3-24
Sectors and Central Angles of a Circle



The pie shaped portion of the interior of circle
intercepted by the central angle is called a
“sector”
From geometry it is know that in a specific circle
the area of a sector is proportional to the
measure of its central angle
For any two central angles,  1 and 2 , with
corresponding sector areas A1 and A2 :
A1
1
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
A2
2
Slide 3-25
Development of Formula for Area of Sector

Since this relationship is true for any two central
angles and corresponding sectors in a circle of
A1 A2
radius r:
1


2
Let one angle be  rad
with corresponding
sector area A and let the other central angle be
2
a whole rotation, 2 rad with sector area r
A
r

 rad 2 rad
2
Copyright © 2005 Pearson Education, Inc.
A
2
r

 2
r
A
2
2
 in radians!
Slide 3-26
Area of a Sector

The area of a sector of a circle of radius r and
central angle  is given by
r
A
2
2
1 2
A  r ,
2
Copyright © 2005 Pearson Education, Inc.
 in radians.
Slide 3-27
Example: Area


Find the area of a sector with radius 12.7 cm
and angle  = 74.
Convert 74 to radians.
  
74  74 
  1.292 radians
 180 

Use the formula to find the area of the sector of
a circle.
1 2
1
2
2
A  r   (12.7) 1.292  104.193 cm
2
2
Copyright © 2005 Pearson Education, Inc.
Slide 3-28
Homework

3.2 Page 103
All: 1 – 10, 17 – 23, 27 – 42

MyMathLab Assignment 3.2 for practice

MyMathLab Homework Quiz 3.2 will be due for a
grade on the date of our next class meeting

Copyright © 2005 Pearson Education, Inc.
Slide 3-29
3.3
The Unit Circle and
Circular Functions
Copyright © 2005 Pearson Education, Inc.
Circular Functions Compared with
Trigonometric Functions






“Circular Functions” are named the same as trig
functions (sine, cosine, tangent, etc.)
The domain of trig functions is a set of angles
measured either in degrees or radians
The domain of circular functions is a set of real
numbers
The value of a trig function of a specific angle in its

1
domain is a ratio of real numbers  sin 300  sin rad  
6
2

The value of circular function of a real number “x” is the
same as the corresponding trig function of “x radians”
Example: sin 23  sin 23 rad  .84622
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Slide 3-31
Circular Functions Defined




The definition of circular
functions begins with a unit
circle, a circle of radius 1 with
center at the origin
Choose a real number s, and
beginning at (1, 0) mark off
arc length s counterclockwise
if s is positive (clockwise if
negative)
Let (x, y) be the point on the
unit circle at the endpoint of
the arc
Let  be the central angle for
the arc measured in radians
Copyright © 2005 Pearson Education, Inc.


Since s=r  , and r = 1, s  
Define circular functions of s to
be equal to trig functions of 
x, y 
s

1,0
Slide 3-32
Circular Functions
y y
sin s  sin     y
r 1
x x
cos s  cos    x
r 1
y
tan s  tan  x  0
x
r 1
csc s  csc    y  0
y y
r 1
sec s  sec   x  0
x x
x
cot s  cot   y  0
y
x, y 
s

1,0
Copyright © 2005 Pearson Education, Inc.
Slide 3-33
Observations About Circular Functions



If a real number s is
represented “in standard
position” as an arc length
on a unit circle,
the ordered pair at the
endpoint of the arc is:
(cos s, sin s)
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cos s, sin s 
s
1,0
Slide 3-34
Further Observations About Circular
Functions


Draw a vertical line
through (1,0) and draw a
line segment from the
endpoint of s, through the
origin, to intersect the
vertical line
The two triangles formed
are similar
sin s t
tan s 
 t
cos s 1
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cos s, sin s 
s
1
1,0
t
tan s
Slide 3-35
Unit Circle with Key Arc Lengths,
Angles
and
2
Ordered Pairs Shown
5
3

6
2
5
cos
  .87
6
cos
1
3
2
tan
 2  3
1
3

2
2
tan
  1 .7
3
1
2
2
sin 3150   .71
sin 3150  
Copyright © 2005 Pearson Education, Inc.
2
Slide 3-36
Domains of the Circular Functions



Assume that n is any integer and s is a real
number.
Sine and Cosine Functions: (, )

 

Tangent and Secant Functions: s | s  2n  1 
2 




s
can'
t
be
any
odd
multiple
of


2


Cotangent and Cosecant Functions:
s can't be any multipleof  
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s | s  n 
Slide 3-37
Evaluating a Circular Function



Circular function values of real numbers are
obtained in the same manner as trigonometric
function values of angles measured in radians.
This applies both methods of finding exact
values (such as reference angle analysis) and to
calculator approximations.
Calculators must be in radian mode when
finding circular function values.
Copyright © 2005 Pearson Education, Inc.
Slide 3-38
Example: Finding Exact Circular
Function Values





7
7
7
Find the exact values of sin , cos , and tan .
4
4
4 7
Evaluating a circular function of the real number
7
4
is equivalent to evaluating a trig function for 4
radians.
7 180
0
Convert radian measure to degrees: 4    315
What is the reference angle? 450
Using our knowledge of relationships between
trig functions of angles and trig functions of
reference angles: sin 7  sin 315o   sin 45o   2
cos
7
2
 cos315o  cos 45o 
4
2
Copyright © 2005 Pearson Education, Inc.
4
2
tan
7
 tan 315 o   tan 45 o  1
4
Slide 3-39
Example: Approximating Circular Function
Values with a Calculator




Find a calculator approximation to four decimal
places for each circular function. (Make sure the
calculator is in radian mode.)
a) cos 2.01   .4252
b) cos .6207  .8135
For the cotangent, secant, and cosecant
functions values, we must use the appropriate
reciprocal functions.
c) cot 1.2071
1
cot1.2071 
 .3806
tan1.2071
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Slide 3-40
Finding an Approximate Number Given its
Circular Function Value

Approximate the value of s in the interval
given that: cos s  .9685

With calculator set in radian mode use the
inverse cosine key to get:
cos1 .9685 .2517
 
0, 2 


Is thisin theintervalspecified?



 Yes, it's between 0 and  1.574
2


Copyright © 2005 Pearson Education, Inc.
Slide 3-41
Finding an Exact Number Given its Circular
Function Value


 3 
Find the exact value of s in the interval  , 
 2
given that: tan s  1
What known reference angle has this exact
tangent value? 45 0  
4


Based on the interval specified, in what quadrant
must the reference angle be placed? III
The exact real number we seek for “s” is:
s  

4

Copyright © 2005 Pearson Education, Inc.
5
4
Slide 3-42
Homework

3.3 Page 113
All: 3 – 6, 11 – 18, 23 – 32, 49 – 60

MyMathLab Assignment 3.3 for practice

MyMathLab Homework Quiz 3.3 will be due for a
grade on the date of our next class meeting

Copyright © 2005 Pearson Education, Inc.
Slide 3-43
3.4
Linear and Angular Speed
Copyright © 2005 Pearson Education, Inc.
Circular Motion



When an object is traveling in a circular path,
there are two ways of describing the speed
observed:
We can describe the actual speed of the object
in terms of the distance it travels per unit of time
(linear speed)
We can also describe how much the central
angle changes per unit of time (angular speed)
Copyright © 2005 Pearson Education, Inc.
Slide 3-45
Linear and Angular Speed

Linear Speed: distance traveled per unit of time
(distance may be measured in a straight line or
along a curve – for circular motion, distance is
an arc length)
distance
s
speed =
or v  ,
time
t

Angular Speed: the amount of rotation per unit of
time, where  is the angle of rotation measured
in radians and t is the time.


t
Copyright © 2005 Pearson Education, Inc.
Slide 3-46
Formulas for Angular and Linear Speed
Angular Speed


t
(  in radians per unit
time,  in radians)
Linear Speed
s
v
t
r
v
t
v  r
Memorize these formulas!
Copyright © 2005 Pearson Education, Inc.
Slide 3-47
Example: Using the Formulas

Suppose that point P is on a circle with radius 20

cm, and ray OP is rotating with angular speed
18
radians per second.
a) Find the angle generated by P in 6 sec.
b) Find the distance traveled by P along the
circle in 6 sec.
c) Find the linear speed of P.
r  20 cm
O
Copyright © 2005 Pearson Education, Inc.
P
Slide 3-48
Solution: Find the angle generated by P in 6

seconds.
r  20 cm,   rad/sec, t  6 sec
18


Which formula includes the unknown angle and
other things that are known?   
t
Substitute for  and t to find 


18


t

6
6 

 radians.
18 3
Copyright © 2005 Pearson Education, Inc.
Slide 3-49
Solution: Find the distance traveled by P in 6
seconds r  20 cm,    rad/sec, t  6 sec    rad
18


3
The distance traveled is along an arc. What is
the formula for calculating arc length? s  r
The distance traveled by P along the circle is
s  r
   20
 20   
cm.
3
3
Copyright © 2005 Pearson Education, Inc.
Slide 3-50
Solution: Find the linear speed of P
r  20 cm,  

18
rad/sec, t  6 sec


3
rad
20
s
cm
3
There are three formulas for linear speed. You
can use any one that is appropriate for the
s
v
information that you know:
t
r
Bett er way:
v
t
 Linear speed:
v  r
v  r
20
 10
s
v  20  
cm/sec
3
v 
18
9

t
6
20
20 1 10

6 
 
cm per sec
3
3 6
9
Copyright © 2005 Pearson Education, Inc.
Slide 3-51
Observations About Combinations of
Objects Moving in Circular Paths


When multiple objects, moving in circular paths, are
connected by means of being in contact, or by being
connected with a belt or chain, the linear speeds of all
objects and any connecting devices are all the same
In this same situation, angular speeds may be different
and will depend on the radius of each circular path
Everypointin red will be movingat thesame linearspeed
v
Angular speeds may be different :  
r
Copyright © 2005 Pearson Education, Inc.
Slide 3-52
Observations About Angular Speed



Angular speed is sometimes expressed in units such as
revolutions per unit time or rotations per unit time
In these situations you should convert to the units of
radians per unit time by normal unit conversion methods
before using the formulas
Example: Express 55 rotations per minute in terms of
angular speed units of radians per second
55 rot 2 rad 1 min 110 rad 11




radians per second
1 min 1 rot 60 sec
60 sec
6
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Slide 3-53
Example: A belt runs a pulley of radius
6 cm at 80 revolutions per min.

a) Find the angular speed
of the pulley in radians
per second.
80 rev 2 rad 1 min



1 min 1 rev 60 sec
160 8


radians per sec
60
3


b) Find the linear speed
of the belt in centimeters
per second.
The linear speed of the
belt will be the same as
that of a point on the
circumference of the
pulley.
v  r
 8
 6
 3
Copyright © 2005 Pearson Education, Inc.

  16  50.3 cm per sec.

Slide 3-54
Homework

3.4 Page 119
All: 3 – 43

MyMathLab Assignment 3.4 for practice

MyMathLab Homework Quiz 3.4 will be due for a
grade on the date of our next class meeting

Copyright © 2005 Pearson Education, Inc.
Slide 3-55