Chapter 3bethanyshultz.weebly.com/uploads/2/2/6/1/22614328/... · Copyright © 2005 Pearson...

Copyright © 2005 Pearson Education, Inc.

Chapter 3

Radian Measure and Circular Functions


3.1

Radian Measure

Copyright © 2005 Pearson Education, Inc. Slide 3-3

Measuring Angles

n  Thus far we have measured angles in degrees n  For most practical applications of trigonometry

this the preferred measure n  For advanced mathematics courses it is more

common to measure angles in units called “radians”

n  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


Radian Measure

n  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)


Comments on Radian Measure

n  A radian is an amount of rotation that is independent of the radius chosen for rotation

n  For example, all of these give a rotation of 1 radian: q  radius of 2 rotated along an arc length of 2 q  radius of 1 rotated along an arc length of 1 q  radius of 5 rotated along an arc length of 5, etc.

rad 12r

1r1r

2r


More Comments on Radian Measure

n  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

n  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 n  Based on the reasoning just discussed:

rπ2π2

0360rad 2 =π=rad π 0180=rad 1 ≈

π

0180 03.57

0180rad =π

180rad π 01=


Converting Between Degrees and Radians

n  From the preceding discussion these ratios both equal “1”:

n  To convert between degrees and radians: n  Multiply a degree measure by and simplify

to convert to radians.

n  Multiply a radian measure by and simplify to convert to degrees.

1rad

180180

rad 0

0 ==π

π

0180rad π

rad 1800

π


Example: Degrees to Radians

n  Convert each degree measure to radians. n  a) 60°

n  b) 221.7°

⋅= 00 6060 =0180rad π rad

3π

⋅= 00 7.2217.221 ≈0180rad π rad 896.3


Example: Radians to Degrees

n  Convert each radian measure to degrees.

n  a)

n  b) 3.25 rad

4rad 11π

⋅=4rad 11

4rad 11 ππ

=rad

1800

π0495

⋅=1

rad 3.25rad 25.3 ≈rad

1800

π02.186


Equivalent Angles in Degrees and Radians

6.28 2π 360°

1.05 60°

4.71 270°

.79 45°

3.14 π 180°

.52 30°

1.57 90°

0 0 0°

Approximate Exact Approximate Exact

Radians Degrees Radians Degrees

6π

4π

3π

2π

32π


Equivalent Angles in Degrees and Radians continued


Finding Trigonometric Function Values of Angles Measured in Radians

n  All previous definitions of trig functions still apply n  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

n  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!

n  When using a calculator to find trig functions of angles measured in radians, be sure to first set the calculator to “radian mode”


Example: Finding Function Values of Angles in Radian Measure n  Find exact function value: n  a)

n  Convert radians to degrees.

n  b) 4tan3π

1804 4tan tan3 3

tan 240

3

π ⎛ ⎞= ⋅⎜ ⎟

⎝ ⎠

=

=

o

o

4sin3π

== 060tan2360sin

240sin34sin

0

0

−=−=

=π


Homework

n  3.1 Page 97 n  All: 1 – 4, 7 – 14, 25 – 32, 35 – 42, 47 – 52,

61 – 72

n  MyMathLab Assignment 3.1 for practice

n  MyMathLab Homework Quiz 3.1 will be due for a grade on the date of our next class meeting


3.2

Applications of Radian Measure


Arc Lengths and Central Angles of a Circle

n  Given a circle of radius “r”, any angle with vertex at the center of the circle is called a “central angle”

n  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”

n  From geometry it is know that in a specific circle the length of an arc is proportional to the measure of its central angle

n  For any two central angles, and , with corresponding arc lengths and :

1θ 2θ2s1s

2

2

1

1

θθss

=


Development of Formula for Arc Length

n  Since this relationship is true for any two central angles and corresponding arc lengths in a circle of radius r:

n  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

θθss

=

rs=

θθrs =

radians!in θrad 22

rad ππ

θrs

=

srad θ

rπ2rad 2π


Example: Finding Arc Length

n  A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having the following measure:

cm

54.6 cm 21

1

.4cm8

2 38.8

s r

s

s

θ

π

π

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

= ≈

83π

θ =


Example: Finding Arc Length continued

n  For the same circle with r = 18.2 cm and θ = 144°, find the arc length

n  convert 144° to radians

144180

radi

144

an45

s

π

π

⎛ ⎞= ⎜ ⎟

⎝ ⎠

=

o

cm

72.8 cm 45

1

.7cm5

2 48.5

s r

s

s

θ

π

π

=

⎛ ⎞= ⎜ ⎟

⎝ ⎠

= ≈


Note Concerning Application Problems Involving Movement Along an Arc

n  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 =


Example: Finding a Length

n  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°?

n  Convert 39.72° to radian measure.

180.8725 39.72 .6049 ft.

s r

s π

θ=

⎡ ⎤⎛ ⎞= ≈⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦


Example: Finding an Angle Measure

n  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?

n  The motion of the small gear will generate an arc length on the small gear and an equal movement on the large gear


Solution

n  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.

n  This same arc length will occur on the larger gear.

5225 225180 45 12.52.5 cm.24

584

s r

π π

π πθ

π

⎛ ⎞= =⎜ ⎟

⎝ ⎠

⎛ ⎞= = = =⎜ ⎟

⎝ ⎠

o


Solution continued

n  An arc with this length on the larger gear corresponds to an angle measure θ, in radians where

n  Convert back to degrees.

4.8

12519

8

2

25s rθ

θ

π

π

θ

=

=

=

125192

180 117π

π ⎛ ⎞≈⎜ ⎟

⎝ ⎠

oo

radians)in measured (Angle


Sectors and Central Angles of a Circle

n  The pie shaped portion of the interior of circle intercepted by the central angle is called a “sector”

n  From geometry it is know that in a specific circle the area of a sector is proportional to the measure of its central angle

n  For any two central angles, and , with corresponding sector areas and :

1θ 2θ1A

2

2

1

1

θθAA

=2A


Development of Formula for Area of Sector

n  Since this relationship is true for any two central angles and corresponding sectors in a circle of radius r:

n  Let one angle be with corresponding sector area and let the other central angle be a whole rotation, with sector area

2

2rA=

θ 2

2θrA =radians!in θrad 2rad

2

ππ

θrA

=

rad θ2rπrad 2π

2

2

1

1

θθAA

=

A


Area of a Sector

n  The area of a sector of a circle of radius r and central angle θ is given by

21 , in radians.2

A r θ θ=

2

2θrA =


Example: Area

n  Find the area of a sector with radius 12.7 cm and angle θ = 74°.

n  Convert 74° to radians.

n  Use the formula to find the area of the sector of a circle.

74 radian74 s18

90

1.2 2π⎛ ⎞= =⎜ ⎟

⎝ ⎠o

2 2 21.291 1 ( ) 104.193 cm2

22

12.7A r θ= = ≈


Homework

n  3.2 Page 103 n  All: 1 – 10, 17 – 23, 27 – 42




3.3

The Unit Circle and Circular Functions


Circular Functions Compared with Trigonometric Functions

n  “Circular Functions” are named the same as trig functions (sine, cosine, tangent, etc.)

n  The domain of trig functions is a set of angles measured either in degrees or radians

n  The domain of circular functions is a set of real numbers

n  The value of a trig function of a specific angle in its domain is a ratio of real numbers

n  The value of circular function of a real number “x” is the same as the corresponding trig function of “x radians”

n  Example: =23sin sin rad 23

⎟⎠

⎞⎜⎝

⎛ ==21

6sin30sin 0 radπ

84622.−≈


Circular Functions Defined

n  The definition of circular functions begins with a unit circle, a circle of radius 1 with center at the origin

n  Choose a real number s, and beginning at (1, 0) mark off arc length s counterclockwise if s is positive (clockwise if negative)

n  Let (x, y) be the point on the unit circle at the endpoint of the arc

n  Let be the central angle for the arc measured in radians

n  Since s=r , and r = 1,

n  Define circular functions of s to be equal to trig functions of

( )0,1

( )yx, s

θ

θ

θ

θ=s

θ


Circular Functions

( )0 tantan

1coscos

1sinsin

≠==

====

====

xxys

xxrxs

yyrys

θ

θ

θ ( )

( )

( )0 cotcot

0 1secsec

0 1csccsc

≠==

≠===

≠===

yyxs

xxx

rs

yyy

rs

θ

θ

θ

( )0,1

( )yx, s

θ


Observations About Circular Functions

n  If a real number s is represented “in standard position” as an arc length on a unit circle,

n  the ordered pair at the endpoint of the arc is:

n  (cos s, sin s) ( )0,1

( )ss sin,coss


Further Observations About Circular Functions

n  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

n  The two triangles formed are similar ( )0,1

( )ss sin,coss

ttsss ===1cos

sintan t

1

stan


Unit Circle with Key Arc Lengths, Angles and Ordered Pairs Shown

1

2

1−

2−

=65cos π

23

−

=0315sin22

−

=32tan π 3

2123

−=−

7.1−≈32tan π

≈65cos π

87.−

≈0315sin 71.−


Domains of the Circular Functions

n  Assume that n is any integer and s is a real number.

n  Sine and Cosine Functions: (-∞, ∞)

n  Tangent and Secant Functions:

n  Cotangent and Cosecant Functions:

| 2 12

s s n π⎧ ⎫⎛ ⎞≠ +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

{ }|s s nπ≠

⎟⎠

⎞⎜⎝

⎛2

of multiple oddany bet can' πs

( )π of multipleany bet can' s

( )


Evaluating a Circular Function

n  Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians.

n  This applies both methods of finding exact values (such as reference angle analysis) and to calculator approximations.

n  Calculators must be in radian mode when finding circular function values.


Example: Finding Exact Circular Function Values n  Find the exact values of n  Evaluating a circular function of the real number

is equivalent to evaluating a trig function for radians.

n  Convert radian measure to degrees: n  What is the reference angle? n  Using our knowledge of relationships between

trig functions of angles and trig functions of reference angles:

7 7 7sin , cos , and tan .4 4 4π π π

74π

74π

045

031518047

=⋅π

π

2245sin315sin

47sin −=−== ooπ

2245cos315cos

47cos === ooπ 145tan315tan

47tan −=−== ooπ


Example: Approximating Circular Function Values with a Calculator

n  Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode.)

n  a) cos 2.01 ≈ b) cos .6207 ≈ n  For the cotangent, secant, and cosecant

functions values, we must use the appropriate reciprocal functions.

n  c) cot 1.2071 1cot1.2071 .3806

tan1.2071= ≈

4252.− 8135.


Finding an Approximate Number Given its Circular Function Value

n  Approximate the value of s in the interval given that:

n  With calculator set in radian mode use the inverse cosine key to get:

⎥⎦

⎤⎢⎣

⎡2,0 π

9685.cos =s

≈− 9685.cos 1 2517.

⎟⎠

⎞⎜⎝

⎛ ≈ 574.12

and 0between sit' Yes, π

( )specified? interval in the thisIs


Finding an Exact Number Given its Circular Function Value

n  Find the exact value of s in the interval given that:

n  What known reference angle has this exact tangent value?

n  Based on the interval specified, in what quadrant must the reference angle be placed?

n  The exact real number we seek for “s” is:

⎥⎦

⎤⎢⎣

⎡23, ππ

1tan =s

4450 π

=

III

=+=4π

πs45π


Homework

n  3.3 Page 113 n  All: 3 – 6, 11 – 18, 23 – 32, 49 – 60




3.4

Linear and Angular Speed


Circular Motion

n  When an object is traveling in a circular path, there are two ways of describing the speed observed:

n  We can describe the actual speed of the object in terms of the distance it travels per unit of time (linear speed)

n  We can also describe how much the central angle changes per unit of time (angular speed)


Linear and Angular Speed

n  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)

n  Angular Speed: the amount of rotation per unit of time, where θ is the angle of rotation measured in radians and t is the time.

tθ

ω =

distancespeed = or time

,=svt


Formulas for Angular and Linear Speed

( in radians per unit time, θ in radians)

Linear Speed Angular Speed

tθ

ω =svtrvt

v r

θ

ω

=

=

=

ω

formulas! theseMemorize


Example: Using the Formulas

n  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.

18π

PO

cmr 20=


Solution: Find the angle generated by P in 6 seconds.

n  Which formula includes the unknown angle and other things that are known?

n  Substitute for to find

18

6 radians.1

6

38

θω

π θ

πθ

π

=

=

= =

t

sec 6 rad/sec, 18

cm, 20 === tr πω

t and ω θtθ

ω =


Solution: Find the distance traveled by P in 6 seconds

n  The distance traveled is along an arc. What is the formula for calculating arc length?

n  The distance traveled by P along the circle is

20 cm2033

.π π

θ=

⎛ ⎞= =⎜ ⎟

⎝ ⎠

s r

θrs =

sec 6 rad/sec, 18

cm, 20 === tr πω rad

3 πθ =


Solution: Find the linear speed of P

n  There are three formulas for linear speed. You can use any one that is appropriate for the information that you know:

n  Linear speed:

20 20 1 106 cm per sec3 3 9

2

6

6

03

π π π

π

= =

= ÷ = ⋅ =

svt

ω

θ

rvtrv

tsv

=

=

=

sec 6 rad/sec, 18

cm, 20 === tr πω cm

320 π

=srad 3

πθ =

cm/sec 9

1018

20

:Better way

ππ

ω

=⋅=

=

v

rv


Observations About Combinations of Objects Moving in Circular Paths

n  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

n  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

rv

=ω :different bemay speedsAngular


Observations About Angular Speed

n  Angular speed is sometimes expressed in units such as revolutions per unit time or rotations per unit time

n  In these situations you should convert to the units of radians per unit time by normal unit conversion methods before using the formulas

n  Example: Express 55 rotations per minute in terms of angular speed units of radians per second

secondper radians

611

sec 60rad 110

sec 60min 1

rot 1rad 2

min1rot 55 πππ

==⋅⋅


Example: A belt runs a pulley of radius 6 cm at 80 revolutions per min. n  a) Find the angular speed

of the pulley in radians per second.

n  b) Find the linear speed of the belt in centimeters per second.

n  The linear speed of the belt will be the same as that of a point on the circumference of the pulley. 160 radians per sec

6830ππ

ω = =

6 16 50.3 cm pe s .3

r e8 c

v rω

ππ

=

⎛ ⎞= = ≈⎜ ⎟

⎝ ⎠

sec 60min 1

rev 1rad 2

min 1rev 80

⋅⋅=π

ω


Homework

n  3.4 Page 119 n  All: 3 – 43



Date post:	23-Jul-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Chapter 3bethanyshultz.weebly.com/uploads/2/2/6/1/22614328/... · Copyright © 2005 Pearson...

Documents