View
219
Download
3
Category
Preview:
Citation preview
Copyright © 2005 Pearson Education, Inc.
Chapter 3
Radian Measure and
Circular Functions
Copyright © 2005 Pearson Education, Inc.
3.1
Radian Measure
Copyright © 2005 Pearson Education, Inc. Slide 3-3
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 3-4
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 3-5
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.
rad 12r
1r1r
2r
Copyright © 2005 Pearson Education, Inc. Slide 3-6
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 , there are radians in a complete rotation
Based on the reasoning just discussed:
r22
0360rad 2 rad 0180
rad 1
0180 03.57
0180rad
180
rad 01
Copyright © 2005 Pearson Education, Inc. Slide 3-7
Converting Between Degrees and Radians
From the preceding discussion these ratios both equal “1”:
To convert between degrees and radians: Multiply a degree measure by and simplify
to convert to radians.
Multiply a radian measure by and simplify to convert to degrees.
1rad
180
180
rad 0
0
0180
rad
rad
1800
Copyright © 2005 Pearson Education, Inc. Slide 3-8
Example: Degrees to Radians
Convert each degree measure to radians. a) 60
b) 221.7
00 6060 0180
rad rad
3
00 7.2217.221 0180
rad rad 896.3
Copyright © 2005 Pearson Education, Inc. Slide 3-9
Example: Radians to Degrees
Convert each radian measure to degrees.
a)
b) 3.25 rad
4
rad 11
4
rad 11
4
rad 11
rad
1800
0495
1
rad 3.25rad 25.3
rad
1800
02.186
Copyright © 2005 Pearson Education, Inc. Slide 3-10
Equivalent Angles in Degrees and Radians
6.2823601.0560
4.71270.7945
3.14180.5230
1.5790000
ApproximateExactApproximateExact
RadiansDegreesRadiansDegrees
6
4
3
2
3
2
Copyright © 2005 Pearson Education, Inc. Slide 3-11
Equivalent Angles in Degrees and Radians continued
Copyright © 2005 Pearson Education, Inc. Slide 3-12
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 3-13
Example: Finding Function Values of Angles in Radian Measure Find exact function value: a)
Convert radians to degrees.
b) 4
tan3
1804 4
tan tan3 3
tan 240
3
4sin
3
060tan
2
360sin
240sin3
4sin
0
0
Copyright © 2005 Pearson Education, Inc. Slide 3-14
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.
3.2
Applications of Radian Measure
Copyright © 2005 Pearson Education, Inc. Slide 3-16
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, and , with corresponding arc lengths and :
1 22s1s
2
2
1
1
ss
Copyright © 2005 Pearson Education, Inc. Slide 3-17
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:
Let one angle be with corresponding arc length and let the other central angle be a whole rotation, with arc length
2
2
1
1
ss
rs
rs
radians!in rad 2
2
rad
rs
srad
r2rad 2
Copyright © 2005 Pearson Education, Inc. Slide 3-18
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:
cm
54.6cm 21
1
.4cm8
23
8.8
s r
s
s
8
3
Copyright © 2005 Pearson Education, Inc. Slide 3-19
Example: Finding Arc Length continued
For the same circle with r = 18.2 cm and = 144, find the arc length
convert 144 to radians
144180
radi
144
an4
5s
cm
72.8cm 45
1
.7cm5
24
8.5
s r
s
s
Copyright © 2005 Pearson Education, Inc. Slide 3-20
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
ls
Copyright © 2005 Pearson Education, Inc. Slide 3-21
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.
180.8725 39.72 .6049 ft.
s r
s
Copyright © 2005 Pearson Education, Inc. Slide 3-22
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 3-23
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.
This same arc length will occur on the larger gear.
5225 225
180 4
5 12.52.5 cm.
2
4
5
84s r
Copyright © 2005 Pearson Education, Inc. Slide 3-24
Solution continued
An arc with this length on the larger gear corresponds to an angle measure , in radians where
Convert back to degrees.
4.8
125
19
8
2
25
s r
125
192
180117
radians)in measured (Angle
Copyright © 2005 Pearson Education, Inc. Slide 3-25
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, and , with corresponding sector areas and :
1 21A
2
2
1
1
AA
2A
Copyright © 2005 Pearson Education, Inc. Slide 3-26
Development of Formula for Area of Sector
Since this relationship is true for any two central angles and corresponding sectors in a circle of radius r:
Let one angle be with corresponding sector area and let the other central angle be a whole rotation, with sector area
2
2rA
2
2rA
radians!in rad 2rad
2
rA
rad 2rrad 2
2
2
1
1
AA
A
Copyright © 2005 Pearson Education, Inc. Slide 3-27
Area of a Sector
The area of a sector of a circle of radius r and central angle is given by
21, in radians.
2A r
2
2rA
Copyright © 2005 Pearson Education, Inc. Slide 3-28
Example: Area
Find the area of a sector with radius 12.7 cm and angle = 74.
Convert 74 to radians.
Use the formula to find the area of the sector of a circle.
74 radian74 s18
90
1.2 2
2 2 21.291 1
( ) 104.193 cm2
22
12.7A r
Copyright © 2005 Pearson Education, Inc. Slide 3-29
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.
3.3
The Unit Circle and Circular Functions
Copyright © 2005 Pearson Education, Inc. Slide 3-31
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 domain is a ratio of real numbers
The value of circular function of a real number “x” is the same as the corresponding trig function of “x radians”
Example: 23sin sin rad 23
2
1
6sin30sin 0 rad
84622.
Copyright © 2005 Pearson Education, Inc. Slide 3-32
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, andbeginning at (1, 0) mark offarc length s counterclockwiseif 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
Since s=r , and r = 1,
Define circular functions of s to be equal to trig functions of
0,1
yx, s
s
Copyright © 2005 Pearson Education, Inc. Slide 3-33
Circular Functions
0 tantan
1coscos
1sinsin
xx
ys
xx
r
xs
yy
r
ys
0 cotcot
0 1
secsec
0 1
csccsc
yy
xs
xxx
rs
yyy
rs
0,1
yx, s
Copyright © 2005 Pearson Education, Inc. Slide 3-34
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) 0,1
ss sin,coss
Copyright © 2005 Pearson Education, Inc. Slide 3-35
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 0,1
ss sin,coss
tt
s
ss
1cos
sintan t
1
stan
Copyright © 2005 Pearson Education, Inc. Slide 3-36
Unit Circle with Key Arc Lengths, Angles and Ordered Pairs Shown
1
2
1
2
6
5cos
2
3
0315sin2
2
3
2tan
3
21
23
7.13
2tan
6
5cos
87.
0315sin 71.
Copyright © 2005 Pearson Education, Inc. Slide 3-37
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:
Cotangent and Cosecant Functions:
| 2 12
s s n
|s s n
2 of multiple oddany bet can'
s
of multipleany bet can' s
Copyright © 2005 Pearson Education, Inc. Slide 3-38
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 3-39
Example: Finding Exact Circular Function Values Find the exact values of Evaluating a circular function of the real number
is equivalent to evaluating a trig function for radians.
Convert radian measure to degrees: What is the reference angle? Using our knowledge of relationships between
trig functions of angles and trig functions of reference angles:
7 7 7sin , cos , and tan .
4 4 4
7
4
7
4
045
0315180
4
7
2
245sin315sin
4
7sin oo
2
245cos315cos
4
7cos oo 145tan315tan
4
7tan oo
Copyright © 2005 Pearson Education, Inc. Slide 3-40
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 b) cos .6207 For the cotangent, secant, and cosecant
functions values, we must use the appropriate reciprocal functions.
c) cot 1.20711
cot1.2071 .3806tan1.2071
4252. 8135.
Copyright © 2005 Pearson Education, Inc. Slide 3-41
Finding an Approximate Number Given its Circular Function Value
Approximate the value of s in the interval given that:
With calculator set in radian mode use the inverse cosine key to get:
2,0
9685.cos s
9685.cos 1 2517.
574.1
2 and 0between sit' Yes,
specified? interval in the thisIs
Copyright © 2005 Pearson Education, Inc. Slide 3-42
Finding an Exact Number Given its Circular Function Value
Find the exact value of s in the interval given that:
What known reference angle has this exact tangent value?
Based on the interval specified, in what quadrant must the reference angle be placed?
The exact real number we seek for “s” is:
2
3,
1tan s
4450
III
4
s4
5
Copyright © 2005 Pearson Education, Inc. Slide 3-43
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.
3.4
Linear and Angular Speed
Copyright © 2005 Pearson Education, Inc. Slide 3-45
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 3-46
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)
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
distancespeed = or
time,s
vt
Copyright © 2005 Pearson Education, Inc. Slide 3-47
Formulas for Angular and Linear Speed
( in radians per unit time, in radians)
Linear SpeedAngular Speed
t
s
vtr
vt
v r
formulas! theseMemorize
Copyright © 2005 Pearson Education, Inc. Slide 3-48
Example: Using the Formulas
Suppose that point P is on a circle with radius 20 cm, and ray OP is rotating with angular speed 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.
18
PO
cmr 20
Copyright © 2005 Pearson Education, Inc. Slide 3-49
Solution: Find the angle generated by P in 6 seconds.
Which formula includes the unknown angle and other things that are known?
Substitute for to find
18
6 radians.
1
6
38
t
sec 6 rad/sec, 18
cm, 20 tr
t and t
Copyright © 2005 Pearson Education, Inc. Slide 3-50
Solution: Find the distance traveled by P in 6 seconds
The distance traveled is along an arc. What is the formula for calculating arc length?
The distance traveled by P along the circle is
20 cm20
33.
s r
rs
sec 6 rad/sec, 18
cm, 20 tr rad
3
Copyright © 2005 Pearson Education, Inc. Slide 3-51
Solution: Find the linear speed of P
There are three formulas for linear speed. You can use any one that is appropriate for the information that you know:
Linear speed:
20 20 1 106 cm per sec
3 3 9
2
6
6
0
3
sv
t
rvt
rv
t
sv
sec 6 rad/sec, 18
cm, 20 tr cm
3
20
srad
3
cm/sec 9
10
1820
:Better way
v
rv
Copyright © 2005 Pearson Education, Inc. Slide 3-52
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
speedlinear same at the moving be willredin point Every
r
v :different bemay speedsAngular
Copyright © 2005 Pearson Education, Inc. Slide 3-53
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
secondper radians 6
11
sec 60
rad 110
sec 60
min 1
rot 1
rad 2
min1
rot 55
Copyright © 2005 Pearson Education, Inc. Slide 3-54
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.
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.160
radians per sec6
8
30
6 16 50.3 cm pe s .3
r e8
c
v r
sec 60
min 1
rev 1
rad 2
min 1
rev 80
Copyright © 2005 Pearson Education, Inc. Slide 3-55
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
Recommended