Copyright © 2005 Pearson Education, Inc. Chapter 3 Radian Measure and Circular Functions.

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

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


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)


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


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


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


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


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


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


Equivalent Angles in Degrees and Radians continued


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”


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


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


3.2

Applications of Radian Measure


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


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


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


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


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


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


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


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


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


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


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


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


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


Homework

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




3.3

The Unit Circle and Circular Functions


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.


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


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


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


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


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.


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


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.


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


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.


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


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


Homework

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




3.4

Linear and Angular Speed


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)


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


Formulas for Angular and Linear Speed

( in radians per unit time, in radians)

Linear SpeedAngular Speed

t

s

vtr

vt

v r

formulas! theseMemorize


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


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


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


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


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


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


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


Homework

3.4 Page 119 All: 3 – 43



Copyright © 2005 Pearson Education, Inc. Chapter 3 Radian Measure and Circular Functions.

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Transcript of Copyright © 2005 Pearson Education, Inc. Chapter 3 Radian Measure and Circular Functions.