Math Power 3

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533 533

6

The Circular Functions

and Their Graphs

I

n An August 2003,ugust 2003, the planet Mars the planet Mars passed closer to passed closer to Earth than it Earth than it had inhad in almost 60,000 years. Like Earth,

almost 60,000 years. Like Earth, Mars rotates on its axis and thus has daysMars rotates on its axis and thus has days and nights. The photos here were taken by

and nights. The photos here were taken by the Hubble telescope and showthe Hubble telescope and show two nearly opposite sides of Mars. (

two nearly opposite sides of Mars. ( Source:Source: www.hubblesite.org) In Exercise 92 of www.hubblesite.org) In Exercise 92 of Secti

Section 6.2,on 6.2, we exwe examine amine the lenthe length of agth of a Martian day.

Martian day.

Phenomena such as rotation of a planet Phenomena such as rotation of a planet on it

on its axiss axis,, high ahigh and lond low tidw tides,es, and chand chang- ang-ing of the seasons of the year are modeled ing of the seasons of the year are modeled by

by periodic functions. periodic functions. In thIn this chis chaptapterer,, wewe see how the trigonometric functions of the see how the trigonometric functions of the pre

previous chvious chapterapter,, introintroduced thduced there in theere in the context of ratios of the sides of a right context of ratios of the sides of a right tri-angle,

angle, can also be viecan also be viewed from the pewed from the perspec- rspec-tive of motion around a circle.

tive of motion around a circle.

6.1

6.1 Radian Measure

Radian Measure

6.2

6.2 The Unit Circle and Circular Functions

The Unit Circle and Circular Functions

6.3

6.3 Graphs of the Sine and Cosine

Graphs of the Sine and Cosine

Functions

6.4

6.4 Graphs of the Other Circular

Graphs of the Other Circular

Functions

Summary Exercises on Graphing Circular

Functions

6.5

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534

534 CHAPTER 6CHAPTER 6 The Circular Functions and Their Graphs The Circular Functions and Their Graphs

6.1

6.1 Radian Measure

Radian Measure

Radian Measure ss Converting Between Degrees and Converting Between Degrees and RadiansRadians ss Arc Length of a CircleArc Length of a Circle ss Area of a SectorArea of a Sector of a Circle of a Circle x x y y r r r r 0 0  _{= 1 radian}_{= 1 radian}   Figure 1 Figure 1 T

TEACHINGEACHINGTTIPIP Students may resistStudents may resist

the idea of using radian measure.

Explain that radians have wide

applications in angular motion

problems (seen in Section 6.2), as

well as engineering and science.

T

TEACHINGEACHINGTTIPIP Emphasize theEmphasize the

difference between an angle of

a

adegrees (writtendegrees (writtenaa°) and an°) and an

angle of

angle of aaradians (writtenradians (written

or

oraa ). Caution students that ). Caution students that

although radian measures are

often given exactly in terms of

often given exactly in terms of   _,_, they can also be approximated

they can also be approximated

using decimals. using decimals. a a r r ,, a a r r _,_,

Radian Measure

In most applications of trigonometryIn most applications of trigonometry,, angles are measuredangles are measured in degrees. In more advanced work in mathematics,

in degrees. In more advanced work in mathematics, radian measureradian measure of angles isof angles is preferred. Radian measure allows us to treat the trigonometric functions as preferred. Radian measure allows us to treat the trigonometric functions as func-tions with domains of

tions with domains of real numbers,real numbers, rather than angles.rather than angles. Figure

Figure 1 sho1 shows an ws an angle angle in standin standard positard position along ion along with with a circla circle of e of radius

radius r r . . The The vertex vertex of of is is at at the the center center of of the the circle. circle. Because Because angle angle interceptsintercepts an arc on the circle equal in length to the radius of the circl

an arc on the circle equal in length to the radius of the circle,e, we say that anglewe say that angle has a measure of 1 radian.

has a measure of 1 radian.

Radian

An angle with its vertex at the center of a circle that intercepts an arc on An angle with its vertex at the center of a circle that intercepts an arc on thethe circle equal in length to the radius of the circle has a measure of

circle equal in length to the radius of the circle has a measure of 1 radian.1 radian.

It follows that an angle of measure 2 radians intercepts an arc equal in It follows that an angle of measure 2 radians intercepts an arc equal in length

length to twto twice tice the radihe radius of us of the the circle,circle, an angle an angle of meof measure asure radian radian interceptsintercepts an arc

an arc equal iequal in length n length to half to half the radithe radius of us of the circthe circle,le, and so and so on. In on. In general,general, if if isis a central angle of a circle of

a central angle of a circle of radiusradius r r and and intercepts intercepts an an arc arc of of lengthlength ss,, ththen ten thehe radian

radian measure measure of of isis

Converting Between Degrees and Radians

The circumference of aThe circumference of a circle—

circle— the the distance distance around around the the circle—circle— is is given given by by wherewhere r r is the ra-is the ra-dius

dius of of the the circle. circle. The The formula formula shows shows that that the the radius radius can can be be laid laid off off times around a

times around a circle. Thercircle. Therefore,efore, an angle of an angle of 360°,360°, which corresponds which corresponds to a com-to a com-plete

plete circle,circle, intercepts intercepts an an arc arc equal equal in in length length to to times times the the radius radius of of the the circle.circle. Thus,

Thus, an an angle angle of of 360°360° has has a a measumeasure re of of radiradians:ans:

An angl

An angle of 180°e of 180° is half tis half the size of an anhe size of an angle of 360gle of 360°,°, so an angle oso an angle of 180°f 180° hashas half the radian measure of an angle of 360°.

half the radian measure of an angle of 360°.

W

We e can can use use the the relationsrelationship hip to to develop develop a a method method for for con- con-verting between degrees and radians

verting between degrees and radians as follows.as follows.

Divide by 180.

Divide by 180. oror Divide byDivide by  _._.

Converting Between Degrees and Radians

1.

1. Multiply Multiply a a degree degree measure measure by by radian radian and and simplify simplify to to convert convert to to radians.radians. 2.

2. Multiply Multiply a a radian radian measure measure by by 180180  and simplify and simplify to to convert convert to to degrees.degrees.

  180 180 1 radian 1 radian 180180   1 1   180 180radianradian 180 180    _radians_radians 180 180   _radians_radians Degree

Degree



radian relationshipradian relationship

180

180 11

2



22 



radiansradians   radiansradians

360 360  ₂₂  _radians._radians. 2 2  2 2  2 2  C C  ₂₂  _r_r C C ₂₂  _r_r ,, ss r r ..       1 1 2 2        

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6.1

6.1 Radian MeasureRadian Measure 535535

Some calculators (in radian Some calculators (in radian mode) have the capability to mode) have the capability to convert directly between convert directly between de-cimal degrees and radians. cimal degrees and radians. This screen shows the This screen shows the con-versions for Example 1. Note versions for Example 1. Note that when

that when exact exact values invol-values invol-ving

vingare required, such asare required, such as in part (a), calculator in part (a), calculator ap-proximations are not proximations are not accept-able. able. 4 4  

This screen shows how a This screen shows how a cal-culator converts the radian culator converts the radian measures in Example 2 to measures in Example 2 to de-gree measures. gree measures. x x y y 0 0 30 30 30 degrees 30 degrees x x y y 0 0 30 radians 30 radians Figure 2 Figure 2 EXAMPLE 1

EXAMPLE 1 Converting Degrees to RadiansConverting Degrees to Radians

Convert each degree measure to radians. Convert each degree measure to radians. (a)

(a) 4545°° (b)(b) 249.8249.8°°

Solution Solution

(a)

(a) Multiply Multiply by by radian.radian.

(b)

(b) Nearest thousandthNearest thousandth

Now try Exercises 1 and 13. Now try Exercises 1 and 13.

EXAMPLE 2

EXAMPLE 2 Converting Radians to DegreesConverting Radians to Degrees

Convert each radian measure to

Convert each radian measure to degrees.degrees.

((aa)) ((bb)) 4.25 (Give the answer in decimal degrees.)4.25 (Give the answer in decimal degrees.) Solution

Solution

((aa)) ((bb))

Multiply

Multiply by by Use Use a a calculator.calculator.

Now try Exercises 21 and 31. Now try Exercises 21 and 31. If no unit of angle

If no unit of angle measure is speciﬁed,measure is speciﬁed, then radian measure is undethen radian measure is understood.rstood.

C A U T I O N

C A U T I O N Figure 2 shows angles measuring 30 radians and 30Figure 2 shows angles measuring 30 radians and 30 °°. Be careful. Be careful not to confuse them.

not to confuse them.

The following table and Figure 3 on the next page give some equivalent The following table and Figure 3 on the next page give some equivalent angles measured in degrees and radians. Keep in mind that

angles measured in degrees and radians. Keep in mind that 180180    _radians._radians.

180 180   .. 4.25 4.25 _4.25_4.25



180180  





243.5243.5 9 9  4 4  9 9  4 4



180 180  



 405405 9 9  4 4 249.8 249.8  _249.8_249.8



  180

180radianradian





4.360 radians4.360 radians

  180 180 45 45  ₄₅₄₅



  180 180radianradian



   4 4 radianradian D

Deeggrreeees s RRaaddiiaanns s DDeeggrreeees s RRaaddiiaannss E

Exxaacct t AApppprrooxxiimmaatte e EExxaacct t AApppprrooxxiimmaattee 0 0°° 00 00 9900°° 1.571.57 30 30°° ..5522 118800°°   _3.14_3.14 45 45°° ..7799 227700°° 4.714.71 60 60°°   1..01055 336600°° 22  _6.28_6.28 3 3 3 3  2 2   4 4   6 6   2 2

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T

TEACHINGEACHINGTTIPIP To help students To help students

appreciate radian measure, show

them that in degrees the formula

for arc length is

(See Exercise 54.) (See Exercise 54.) s s   360 36022  r r    _r_r 180 180.. 536

Looking Ahead to Calculus Looking Ahead to Calculus

In calculus, radian measure is much In calculus, radian measure is much easier to work with than degree easier to work with than degree mea-sure. If

sure. If x x is measured in radians, thenis measured in radians, then the

the derivative derivative of of isis

However, if

However, if x x is measured in degrees,is measured in degrees, then

then the the derivative derivative of of isis

f f  x x   180 180coscos x x.. f f  x x_sin_sin x_x f f  x xcoscos x x.. f f  x x_sin_sin x_x Figure 3 Figure 3 W

We use radian measure to simplify certain formulas,e use radian measure to simplify certain formulas, two of which followtwo of which follow.. Each would be more

Each would be more complicated if expressed in degrees.complicated if expressed in degrees.

Arc Length of a Circle

We use theWe use the ﬁﬁrst for-rst for-mula to

mula to ﬁﬁnd the length of an arc of a circle. Thisnd the length of an arc of a circle. This formula is derived from the fact (proven in formula is derived from the fact (proven in geom-etry) that the length of an arc

etry) that the length of an arc is proportional to theis proportional to the measure of its central angle.

measure of its central angle. In Fi

In Figurgure 4,e 4, anganglele QOPQOPhas measure 1 radianhas measure 1 radian and intercepts an arc of length

and intercepts an arc of length r r on the circle.on the circle. Angle

Angle ROT ROT has measurehas measure   _{radians and intercepts}_{radians and intercepts}

an arc of length

an arc of length ss on the circle. Since the lengthson the circle. Since the lengths of the arcs are proportional to the measures of of the arcs are proportional to the measures of their central angles,

their central angles,

Multiplying both sides by

Multiplying both sides by r r gives the following result.gives the following result.

Arc Length

The length

The length ss of the arc intercepted on a circle of radiusof the arc intercepted on a circle of radius r r by a central angleby a central angle of measure

of measure   _{radians is given by the product of the radius and the radian}_{radians is given by the product of the radius and the radian}

measu

measure of the angre of the angle,le, oror



 _{in radians.}_{in radians.}

C A U T I O N

C A U T I O N When When applying applying the the formula formula the the value value of of   _{must be}_{must be}

expressed in radians. expressed in radians. ss _r_r  _,, s s _r_r  ,, s s r r    1 1 .. 0 0°°== 00 90 90°°== 30 30°°== 45 45°°== 60 60°°== 120 120°°== 135 135°°== 150 150°°== 180 180°°== 210 210°°== 225 225°°== 240 240°°== 330 330°°== 315 315°°== 300 300°°== 270 270°°==   2 2  3 3   4 4   6 6 11 11 6 6 7 7 4 4 5 5 3 3 5 5 4 4 7 7 6 6 5 5 6 6 3 3 4 4 2 2 3 3 4 4 3 3 33 2 2 y y r r r r OO P P  _radians_radians x x r r s s 1 radian 1 radian Q Q T T R R Figure 4 Figure 4

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6.1

r r= 18.2 cm= 18.2 cm s s 3 3  8 8 Equator Equator 34 34°° 40 40°° 6

6 _{Los Angeles}_{Los Angeles}

Reno Reno 6400 km 6400 km Figure 5 Figure 5 EXAMPLE 3

EXAMPLE 3 Finding Arc Length UsingFinding Arc Length Using

A circle has radius 18.2 cm. Find the length of the arc intercepted by a central A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having each of the

angle having each of the following measures.following measures.

((aa)) ((bb)) 144144°°

Solution Solution (a)

(a) As shown in theAs shown in the ﬁﬁgguurree,, aanndd

Arc length formula

Substitute for

Substitute forr r andand  _._.

(b)

(b) The The formula formula requires requires thatthat   _{be measured}_{be measured in radians.}_{in radians. First,}_{First, convert}_convert 

to radians by multiplying 144

to radians by multiplying 144 °° by by radian.radian.

Convert from degrees to radians.

The length

The length ss is given byis given by

Now try Exercises 49 and 51. Now try Exercises 49 and 51.

EXAMPLE 4

EXAMPLE 4 Using Latitudes to Find the Distance Between Two CitiesUsing Latitudes to Find the Distance Between Two Cities

Reno,

Reno, Nevada,Nevada, is approximately is approximately due north of Lodue north of Los Angels Angeles. The latitues. The latitude of Renode of Reno is 40

is 40°° N,N, while that of while that of Los Los Angeles is 34Angeles is 34°° N. (The N in 34N. (The N in 34°° N meansN means northnorth of theof the equator

equator.).) The radius of Earth is 6400 km. Find the north-souThe radius of Earth is 6400 km. Find the north-south distance betweenth distance between the two cities.

the two cities. Solution

Solution Latitude gives the measure of a central angle with vertex at EarthLatitude gives the measure of a central angle with vertex at Earth ’’ss center whose initial side goes through the equator

center whose initial side goes through the equator and whose terminal side goesand whose terminal side goes through the gi

through the given location. ven location. As shown in Figure As shown in Figure 5,5, the central anglthe central angle betweene between Reno

Reno and and Los Los Angeles Angeles is is The The distance distance between between the the two two citiescities can

can be be found found by by the the formula formula after after 66°° isis ﬁﬁrst converted to radians.rst converted to radians.

The distance between the two cities is The distance between the two cities is

Let and

Now try Exercise 55. Now try Exercise 55.

  ₃₀₃₀  _._. r r ₆₄₀₀₆₄₀₀ ss  _r_r   ₆₄₀₀₆₄₀₀



  30 30

21.4 cm21.4 cm ss _18.2_18.2



33   8 8



cmcm ss _r_r     33  8 8 .. r r  _{18.2 cm}_{18.2 cm} 3 3  8 8 radiansradians s s_r_r  s s

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538

2 2.. 5 5 cc m m 4 4 ..8 _8 c c m_m Figure 7 Figure 7 multiply multiply by

by ₂₄₂₄55 to to solve solve forfor  _._.

4.8 4.84848₁₀₁₀2424₅₅ _;_; r r The shaded The shaded region is a region is a sector of sector of the circle. the circle.   Figure 8 Figure 8 39.72 39.72°° .8725 ft .8725 ft Figure 6 Figure 6 EXAMPLE 5

EXAMPLE 5 Finding a Length UsingFinding a Length Using

A rope is being wound around a drum with radius .8725 ft. (See Figure 6.) How A rope is being wound around a drum with radius .8725 ft. (See Figure 6.) How much rope will be wound around the drum if the drum is rotated through an much rope will be wound around the drum if the drum is rotated through an angle of 39.72

angle of 39.72°°?? Solution

Solution The length of rope wound around the The length of rope wound around the drum is the arc length for a cir-drum is the arc length for a cir-cle of radius .8725 ft and a central angle of 39.72

cle of radius .8725 ft and a central angle of 39.72 °°. . Use Use the the formula formula withwith the angle converted to radian measure. The length of the rope wound around the the angle converted to radian measure. The length of the rope wound around the drum is approximately

drum is approximately

Now try Exercise 61(a). Now try Exercise 61(a).

EXAMPLE 6

EXAMPLE 6 Finding an Angle Measure UsingFinding an Angle Measure Using

T

Two gears are adjusted so that the smaller gear drives the larwo gears are adjusted so that the smaller gear drives the larger one,ger one, as shown inas shown in Figure 7. If the smaller gear rotates through 225

Figure 7. If the smaller gear rotates through 225 °°,, throuthrough how mgh how many deany degreesgrees will the larger gear rotate?

will the larger gear rotate? Solution

Solution FirstFirst fifind the radian meand the radian measure of the angle,sure of the angle, and thenand then fifind the arc lengthnd the arc length on the smaller gear that determines the motion of the larger gear. Since on the smaller gear that determines the motion of the larger gear. Since

for the smaller gear, for the smaller gear,

An arc with this length on the larger gear corresponds to an angle measure An arc with this length on the larger gear corresponds to an angle measure   _{,, in}_in

radia

radians,ns, wherewhere

Substitute for

Substitute forssand 4.8 forand 4.8 forr r ..

Converting

Converting  _{back to degrees shows that the larger gear}_{back to degrees shows that the larger gear rotates through}_{rotates through}

Convert

Convert to to degrees.degrees.

Area of a Sector of a Circle

AA sector of a circlesector of a circle is the portion of theis the portion of the interior of a circle intercepted by a central angle. Think of it as a

interior of a circle intercepted by a central angle. Think of it as a ““piece of pie.piece of pie.”” See Figure 8. A complete circle can be thought of as an angle with measure 2 See Figure 8. A complete circle can be thought of as an angle with measure 2 

radians. If a central angle for a sector has measure

radians. If a central angle for a sector has measure   _radians,_radia_{ns, then}_{then the}_{the secto}_sectorr

makes

makes up the up the fraction fraction of a of a complete complete circle. circle. The area The area of a of a complete complete circle circle withwith radius

radius r r is is Therefore,Therefore,

in radians. in radians.

 

area of the sector area of the sector  

2 2 



  r r 2 2



__ 11 2 2 r r 2 2___,, A A   _r_r22_..   2 2    125125₁₉₂₁₉₂  125 125  192 192



180 180  





117117.. 125 125  192 192



  .. 25 25  8 8 25 25  8 8  4.84.8  ss  _r_r  ss _r_r   _2.5_2.5



55   4 4



 12.5 12.5  4 4  25 25  8 8 cm.cm. 225 225  55  4 4 radians,radians, s s r r   ss  _r_r   _.8725_.8725



_39.72_39.72



  180 180





.6049 ft..6049 ft. ss _r_r  _,, s s r r  

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6.1

15 15°° 3 3 2 2 1 1 m m Figure 9 Figure 9

This discussion is summarized as follows. This discussion is summarized as follows.

Area of a Sector

The area of a sector of a circle of radius

The area of a sector of a circle of radius r r and central angleand central angle  _{is given by}_{is given by}



 _{in radians.}_{in radians.}

C A U T I O N

C A U T I O N As in the formulAs in the formula for arc lengtha for arc length,, the value of the value of   _{must be in radi-}_{must be in}

radi-ans

answhen using this formula for the area of a sector.when using this formula for the area of a sector.

EXAMPLE 7

EXAMPLE 7 Finding the Area of a Sector-Shaped FieldFinding the Area of a Sector-Shaped Field

Find the area of

Find the area of the sector-shapedthe sector-shaped ﬁﬁeld shown in Figure 9.eld shown in Figure 9. Solution

Solution FirstFirst,, conconvert 15vert 15°° to radians.to radians.

Now use the formula to

Now use the formula to ﬁﬁnd the area of a sector of a circle with radiusnd the area of a sector of a circle with radius

Now try Exercise 77. Now try Exercise 77. A A 11 2 2 r r 2 2__{ }_ 11 2 2



321321



2 2



  12 12





13,500 m13,500 m 2 2 r r  _321._321. 15 15  ₁₅₁₅



  180 180



   12 12radianradian A A  11 2 2 r r 2 2___,, Convert ea

Convert each degree ch degree measure to measure to radians. Learadians. Leave answers ve answers as multipas multiples of les of . See. See Example 1(a). Example 1(a). 1. 1. 6060 2.2. 9090 3.3. 150150 4.4. 270270 5. 5. 315315 6.6. 480480 7.7. 4545 8.8. 210210 Convert each degree measure to radians. See Example 1(b).

Convert each degree measure to radians. See Example 1(b).

9.

9. 3939 10.10. 7474 11.11. 1

122.. 1133.. 64.2964.29 14.14. 122.62122.62

Concept Check

Concept Check In In Exercises 15–Exercises 15– 18, 18, each each angle angle is is an an integer when integer when measured inmeasured in radians. Give the radian measure of the angle.

radians. Give the radian measure of the angle.

1 155.. 1166.. x x y y 0 0 θ θ θ θ 0 0 x x y y   174 1745050 139 1391010   1 1. . 22. . 33. . 44.. 5 5. . 66. . 77.. 8 8. . 99.. .68.68 10.10. 1.291.29 11. 11. 2.432.43 12.12. 3.053.05 13.13. 1.1221.122 14. 14. 2.1402.140 15.15. 11 16.16. 22  77  6 6    4 4 8 8  3 3 7 7  4 4 3 3  2 2 5 5  6 6   2 2   3 3

6.1

6.1 Exercises

Exercises

(8)

540

17. 17. 33 18.18. 11 21.21. 6060 22. 22. 480480 23.23. 315315 24.24. 120120 25. 25. 330330 26.26. 675675 27.27. 3030 28. 28. 6363 29.29. 114.6114.6 30. 30. 286.5286.5 31.31. 99.799.7 32.32. 19.619.6 33. 33. 564.2564.2 34.34. 198.9198.9 35. 35. 11 3366. . 3377.. 3 388. . 3399.. 11 40.40. 4 411. .



33 4422.. 11 2 2 1 1 2 2 1 1 2 2  



33 3 3



22

9

1 177.. 1188.. 19.

19. In your own words, explain how to convertIn your own words, explain how to convert (a)

(a) degree measure to radian measure;degree measure to radian measure; (b)(b) radian measure to degree measure.radian measure to degree measure. 20.

20. Explain the difference between degree measure Explain the difference between degree measure and radian measure.and radian measure.

Convert each radian measure to degrees. See Example 2(a). Convert each radian measure to degrees. See Example 2(a).

2

211.. 2222.. 2233.. 2244..

2

255.. 2266.. 2277.. 2288..

Convert each radian measure to degrees. Give answers using decimal degrees to the Convert each radian measure to degrees. Give answers using decimal degrees to the nearest tenth. See Example 2(b).

nearest tenth. See Example 2(b).

29. 29. 22 30.30. 55 31.31. 1.741.74 32. 32. .3417.3417 33.33. 9.847639.84763 34.34. 3.471893.47189

Relating Concepts

For individual or

For individual or collaborative investigationcollaborative investigation (Exercises 35

(Exercises 35––42)42)

In anticipation of the material in the next section, we show how to

In anticipation of the material in the next section, we show how to ﬁ ﬁnd the trigonomet-nd the trigonomet-ric function values of radian-measured angles. Suppose we want to

ric function values of radian-measured angles. Suppose we want to ﬁ ﬁnd nd . . One One wayway to

to do do this this is is to to convert convert radians radians toto 150150°° , , and then usand then use the mete the methods of Chahods of Chapter 5pter 5 to evaluate: to evaluate: .. (Section 5.3)(Section 5.3) S Siinne ie is s ppoossiittiivvee RReeffeerre ne ncce ae annggllee i in n qquuaaddrraannt t IIII.. ffoor r 115500

Use this technique to

Use this technique to ﬁ ﬁnd each function value. Give exact values.nd each function value. Give exact values.

sec sec  cos cos   3 3 cot cot22  3 3 csc csc   4 4 tan tan   4 4 sin sin55  6

6 sin 150sin 150sin 30sin 30 1 1 2 2 5 5  6 6 sin sin55₆₆   77  20 20    6 6 15 15  4 4 11 11  6 6 2 2  3 3 7 7  4 4 8 8  3 3   3 3 x x y y 0 0 θ θ x x y y 0 0 θ θ

9

(9)

Concept Check

Concept Check Find the exact length of each arc intercepted by the given centralFind the exact length of each arc intercepted by the given central angle.

angle.

4

433.. 4444..

Concept Check

Concept Check Find the radius of each circle.Find the radius of each circle.

4

455.. 4466..

Concept Check

Concept Check Find the measure of each central angle (in radians).Find the measure of each central angle (in radians).

4

477.. 4488..

Unless otherwise directed, give calculator approximations in your answers in the rest

of this exercise set.

Find the le

Find the length of each ngth of each arc intercepted arc intercepted by a central by a central angle angle in a circle in a circle of radius r. of radius r. SeeSee Example 3.

Example 3.

49.

49. cm, cm, radiansradians 50.50. cm, cm, radiansradians

51.

51. m,m, 52.52. cm,cm, 53.

53. Concept CheckConcept Check If the radius of a circle is doubled, how is the length of the arcIf the radius of a circle is doubled, how is the length of the arc intercepted by a

intercepted by a ﬁﬁxed central angle xed central angle changed?changed? 54.

54. Concept CheckConcept Check Radian measure simpliRadian measure simpliﬁﬁes many formulas, such as the formula fores many formulas, such as the formula for arc

arc length, length, . . Give Give the the corresponding corresponding formula formula when when is is measured measured in in degreesdegrees instead of radians.

instead of radians.

Distance Between Cities

Distance Between Cities Find the distance in kilometers between each pair of cities,Find the distance in kilometers between each pair of cities, assuming they lie on

assuming they lie on the same north-south line.the same north-south line. See Example 4.See Example 4.

55.

55. Panama City, Panama, 9Panama City, Panama, 9N, and Pittsburgh, Pennsylvania, 40N, and Pittsburgh, Pennsylvania, 40NN 56.

56. Farmersville, California, 36Farmersville, California, 36N, and Penticton, British Columbia, 49N, and Penticton, British Columbia, 49NN 57.

57. New York City, New York, 41New York City, New York, 41N, and Lima, Peru, 12N, and Lima, Peru, 12SS 58.

58. Halifax, Nova Scotia, 45Halifax, Nova Scotia, 45N, and Buenos Aires, N, and Buenos Aires, Argentina, 34Argentina, 34SS

  ss_r_r    ₁₃₅₁₃₅_ r r _71.9_71.9   ₆₀₆₀_ r r _4.82_4.82   1111  10 10 r r _.892_.892   22  3 3 r r _12.3_12.3   6 6 4 4   3 3 3 3   3 3    2 2 6 6  3 3  4 4 12 12   3 3 4 4   2 2 6.1

4 433. . 4444. . 4455.. 88 46.46. 66 47. 47. 11 48.48. 1.51.5 49.49. 25.8 cm25.8 cm 50. 50. 3.08 cm3.08 cm 51.51. 5.05 m5.05 m 52.

52. 169 cm169 cm 53.53. The length isThe length is

doubled. doubled. 54.54. 55. 55. 3500 km3500 km 56.56. 1500 km1500 km 57. 57. 5900 km5900 km 58.58. 8800 km8800 km s s   _r_r  180 180 4 4  2 2 

(10)

542

59.

59. 4444NN 60.60. 4343NN

61. (a)

61. (a)11.6 in.11.6 in. (b)(b)

62.

62. 12.7 cm12.7 cm 63.63. 38.538.5

64.

64. 18.7 cm18.7 cm 65.65. 146 in.146 in.

66. (a)

66. (a)39,616 rotations39,616 rotations

37

3755

59.

59. Latitude of Madison Latitude of Madison Madison, South Dakota, and Dallas, Texas, are 1200 kmMadison, South Dakota, and Dallas, Texas, are 1200 km apart and lie on the same north-south line.

apart and lie on the same north-south line. The latitude of Dallas is 33The latitude of Dallas is 33N. What isN. What is the latitude of Madison?

the latitude of Madison? 60.

60. Latitude of Toronto Latitude of Toronto Charleston, South Carolina, and Toronto, Canada, are 1100 kmCharleston, South Carolina, and Toronto, Canada, are 1100 km apart and lie on the same

apart and lie on the same north-south line. The latitude of Charleston is 33north-south line. The latitude of Charleston is 33N. WhatN. What is the latitude of Toronto?

is the latitude of Toronto?

Work each problem. See Examples 5 and 6. Work each problem. See Examples 5 and 6.

61.

61. Pulley Raising a Weight Pulley Raising a Weight

(a)

(a) How many inches will the weight in theHow many inches will the weight in the ﬁﬁgure rise if gure rise if the

the pulley pulley is is rotated rotated through through an an angle angle of of ?? (b)

(b) Through what angle, to the nearest minute, must theThrough what angle, to the nearest minute, must the pulley be rotated to raise the weight 6

pulley be rotated to raise the weight 6 in.?in.?

62.

62. Pulley Raising a Weight Pulley Raising a Weight Find the radius of the pulley inFind the radius of the pulley in the

the ﬁﬁgure if a rotation of 51.6gure if a rotation of 51.6raises the weight 11.4 cm.raises the weight 11.4 cm.

63.

63. Rotating Wheels Rotating Wheels The rotation of the smallerThe rotation of the smaller wheel in the

wheel in the ﬁﬁgure causes the larger wheel to rotate.gure causes the larger wheel to rotate. Through how many degrees will the larger wheel Through how many degrees will the larger wheel rotate if the smaller one rotates through 60.0 rotate if the smaller one rotates through 60.0??

64.

64. Rotating Wheels Rotating Wheels Find the radius of the largerFind the radius of the larger wheel in the

wheel in the ﬁﬁgure if the smaller wheel rotatesgure if the smaller wheel rotates 80.0

80.0when the larger wheel rotates 50.0when the larger wheel rotates 50.0..

65.

65. Bicycle Chain Drive Bicycle Chain Drive TheThe ﬁﬁguregure shows the chain drive of a bicycle. shows the chain drive of a bicycle. How far will the bicycle move if How far will the bicycle move if the pedals are rotated through the pedals are rotated through 180

180? Assume the radius of the? Assume the radius of the bicycle wheel is 13.6 in.

bicycle wheel is 13.6 in.

66.

66. Pickup Truck Speedometer Pickup Truck Speedometer The speedometer of TerryThe speedometer of Terry’’s small pickup truck iss small pickup truck is designed to be accurate with tires of radius 14 in.

designed to be accurate with tires of radius 14 in. (a)

(a) Find the number of rotations of a tire in 1 hr if Find the number of rotations of a tire in 1 hr if the truck is driven at 55 mph.the truck is driven at 55 mph. 71 715050 8 8.. 1 1 6 6 c c m m 5.23 5.23 cm cm 11.7 11.7 cm cm r r 1.38 in. 1.38 in. 4.72 in. 4.72 in. 9.27 in. 9.27 in. r r

(11)

(b)

(b) Suppose that oversize tires of radius 16 in. are placed on the truck. If the truck isSuppose that oversize tires of radius 16 in. are placed on the truck. If the truck is now driven for 1 hr

now driven for 1 hr with the speedometer reading 55 mph, how far has with the speedometer reading 55 mph, how far has the truck the truck gone?

gone? If the speed limit is 55 mph, doeIf the speed limit is 55 mph, does Terry deserve a speeding tickes Terry deserve a speeding ticket?t?

If a central angle is very small, there is little If a central angle is very small, there is little differ-ence in length between an arc and the inscribed ence in length between an arc and the inscribed chord. See the

chord. See the ﬁ ﬁgure. Approximate each of the fol-gure. Approximate each of the fol-lowing lengths by

lowing lengths by ﬁ ﬁnding the necessary arc length.nding the necessary arc length. (Note:

(Note: When a central When a central angle intercepts angle intercepts an arc, thean arc, the arc is said to

arc is said tosubtendsubtendthe angle.)the angle.)

67.

67. Length of a Train Length of a Train A railroad track in the desert is 3.5 km away. A train on theA railroad track in the desert is 3.5 km away. A train on the track

track subtends subtends (horizontally) (horizontally) an an angle angle of of .. Find Find the the length length of of the the train.train. 68.

68. Distance to a Boat Distance to a Boat The mast of Brent SimonThe mast of Brent Simon’’s boat is 32 ft high. If s boat is 32 ft high. If it subtends anit subtends an angle

angle of of , , how how far far away away is is it?it?

Concept Check

Concept Check Find the area of each sector.Find the area of each sector.

6

699.. 7700..

Concept Check

Concept Check Find the measure (in radians) of each central angle. The number in-Find the measure (in radians) of each central angle. The number in-side the sector is the area.

side the sector is the area.

7

711.. 7722..

Find the area of a

Find the area of a sector of a circle sector of a circle having radius r and central angle having radius r and central angle . See Example 7.. See Example 7.

73.

73. m, m, radiansradians 74.74. km, km, radiansradians

75.

75. ft, ft, radiansradians 76.76. yd, yd, radiansradians

77.

77. cm,cm, 78.78. m,m, 79.

79. mi,mi, 80.80. km,km,

Work each problem. Work each problem.

81.

81. Find Find the the measure measure (in radi(in radians) oans) of a f a central central angle angle of a of a sector sector of area of area 16 16 in a in a circlecircle of radius 3.0 in.

of radius 3.0 in. 82.

82. Find the radiFind the radius of a circle us of a circle in which a in which a central angle central angle of of radian determinradian determines a sector of es a sector of area area 64 64 mm22..   6 6 in. in.22   ₂₇₀₂₇₀_ r r _90.0_90.0   ₁₃₅₁₃₅_ r r _40.0_40.0   ₁₂₅₁₂₅_ r r _18.3_18.3   ₈₁₈₁_ r r _12.7_12.7   55  6 6 r r _90.0_90.0      2 2 r r _30.0_30.0   22  3 3 r r _59.8_59.8   55  6 6 r r _29.2_29.2   8 sq units 8 sq units 4 4 3 sq units 3 sq units 2 2 8 8 4 4    6 6 2 2    2 21010 3 32020 6.1

66. (b)

66. (b)62.9 mi; yes62.9 mi; yes

67. 67. .20 km.20 km 68.68. 850 ft850 ft 6 699. . 7700. . 7711.. 1.51.5 72. 72. 11 73.73. 1116.11116.1 74. 74. 3744.83744.8 75.75. 706.9706.9 76. 76. 10,602.910,602.9 77.77. 114.0114.0 78. 78. 365.3365.3 79.79. 1885.01885.0 80. 80. 19,085.219,085.2 81.81. 3.63.6 82. 82. 16 m16 m km km22 mi mi22 m m22 cm cm22 yd yd22 ft ft22 km km22 m m22 16 16  6 6  Arc length

Arc length≈≈length of inscribed chordlength of inscribed chord

Inscribed chord

Arc Arc

(12)

544

83. (a) 83. (a) ;; (b)(b)480 ft480 ft (c) (c) ftft (d) (d)approximately 672approximately 672 84. 84. 75.475.4 85. (a) 85. (a)140 ft140 ft (b)(b)102 ft102 ft (c) (c)622622 86. (a) 86. (a)550 m550 m (b)(b)1800 m1800 m 87. 87. 19001900ydyd22 88.88. 1.15 mi1.15 mi ft ft22 in. in.22 ft ft22 160 160 9 9



17.817.8 2 2  27 27 13 13 11 3 3  83.

83. Measures of a Structure Measures of a Structure TheThe ﬁﬁgure shows Medicine Wheel, a Native Americangure shows Medicine Wheel, a Native American structure in northern Wyomi

structure in northern Wyoming.ng. This circular structure is perhaps 250This circular structure is perhaps 2500 yr old. There0 yr old. There are 27 aboriginal spokes in the wheel, all equally spaced.

are 27 aboriginal spokes in the wheel, all equally spaced.

(a)

(a) Find the measure of each Find the measure of each central angle in degrees and in radians.central angle in degrees and in radians. (b)

(b) If the radius of the wheel is 76 ft,If the radius of the wheel is 76 ft, ﬁﬁnd the circumference.nd the circumference. (c)

(c) Find the length of each arc Find the length of each arc intercepted by consecutivintercepted by consecutive pairs of e pairs of spokes.spokes. (d)

(d) Find the area of Find the area of each sector formed by consecutive spokes.each sector formed by consecutive spokes. 84.

84. Area Cleaned by a Windshield Wiper Area Cleaned by a Windshield Wiper The FordThe Ford Model

Model A, built from 1928 to 193A, built from 1928 to 1931, had a single1, had a single windshield wiper on the driver

windshield wiper on the driver ’’s sides side.. The tThe totalotal arm and blade was 10 in. long and rotated back arm and blade was 10 in. long and rotated back and forth through an angle of 95

and forth through an angle of 95. The shaded. The shaded region in the

region in the ﬁﬁgure is the portion of the wind-gure is the portion of the wind-shield cleaned b

shield cleaned by the 7-in. wiper bladey the 7-in. wiper blade.. What isWhat is the area of the region cleaned?

the area of the region cleaned? 85.

85. Circular Railroad CurvesCircular Railroad Curves In the United States, circular railroad curves areIn the United States, circular railroad curves are designated by the

designated by the degree of curvature,degree of curvature,the central angle subtended by a chord of the central angle subtended by a chord of 100 ft.

100 ft. Suppose a portion of tracSuppose a portion of track has curvature 42k has curvature 42. . ((Source:Source: Hay, W.,Hay, W., Railroad Railroad Engineering,

Engineering,John WiJohn Wiley &ley & Sons, 19Sons, 1982.)82.) (a)

(a) What is the radius of What is the radius of the curve?the curve? (b)

(b) What is the length of the What is the length of the arc determined by the 100-ft chord?arc determined by the 100-ft chord? (c)

(c) What is the area of the portion of the circle bounded by the arc and the 100-ftWhat is the area of the portion of the circle bounded by the arc and the 100-ft chord?

chord? 86.

86. Land Required for a Solar-Power Plant Land Required for a Solar-Power Plant A 300-megawatt solar-power plantA 300-megawatt solar-power plant requires

requires approximately approximately 950,000 950,000 of of land land area area in in order order to to collect collect the the requiredrequired amount of energy from

amount of energy from sunlight.sunlight. (a)

(a) If this land area is circular, what is its radius?If this land area is circular, what is its radius? (b)

(b) If this land area is a 35If this land area is a 35sector of a circle, what is its radius?sector of a circle, what is its radius? 87.

87. Area of a Lot Area of a Lot A frequent problem in surveyingA frequent problem in surveying city lots and rural lands adjacent to curves of

city lots and rural lands adjacent to curves of highwayshighways and railways is that of finding the area when one or and railways is that of finding the area when one or more of the boundary lines is the arc of a circle. Find more of the boundary lines is the arc of a circle. Find the area of the lot shown in the figure. (

the area of the lot shown in the figure. (Source:Source:

Anderson, J. and E. Michael,

Anderson, J. and E. Michael, Introduction to Introduction to Surveying,

Surveying,McGraw-Hill, 1985.)McGraw-Hill, 1985.) 88.

88. Nautical Miles Nautical Miles Nautical miles Nautical milesare used byare used by ships and airplanes. They are different from ships and airplanes. They are different from

statute miles,

statute miles, which equal which equal 5280 ft.5280 ft. A nauticalA nautical mile is de

mile is deﬁﬁned to be the arc length along thened to be the arc length along the equator intercepted by a central angle

equator intercepted by a central angle AOB AOB of of 1 min, as illustrated in the

1 min, as illustrated in the ﬁﬁgure. If the equato-gure. If the equato-rial radius of Earth is 3963 mi, use the arc rial radius of Earth is 3963 mi, use the arc length formula to approximate the number of length formula to approximate the number of statute miles in 1 naut

statute miles in 1 nautical mile.ical mile. Round yourRound your answer to two decimal places.

answer to two decimal places. m m22 1 1 0 0ii n n.. 95 95°° 7 7 _i_i n n .. 60 60°° 30 yd 30 yd 40 yd 40 yd B B A A O O Nautical Nautical mile mile Not to scale Not to scale

(13)

89.

89. Circumference of EarthCircumference of Earth TheThe ﬁﬁrst accuraterst accurate estimate of the distance around Earth was estimate of the distance around Earth was done by the Greek astronomer Eratosthenes done by the Greek astronomer Eratosthenes (276

(276––195195BB..CC.), who noted that the noontime.), who noted that the noontime position of the sun at the summer solstice position of the sun at the summer solstice dif-fered

fered by by from from the the city city of of Syene Syene to to thethe city of Alexandria. (See the

city of Alexandria. (See the ﬁﬁgure.gure.)) The dThe dis- is-tance between these two cities is 496 mi. Use tance between these two cities is 496 mi. Use the arc length formula to estimate the

the arc length formula to estimate the radius of radius of Earth

Earth.. ThenThen ﬁﬁnd the circumference of Earth.nd the circumference of Earth. ((Source:Source: Zeilik, M.,Zeilik, M., Introductory Astronomy Introductory Astronomy and Astrophysics,

and Astrophysics, Third Edition, SaundersThird Edition, Saunders College Publishers, 1992.)

College Publishers, 1992.) 90.

90. Diameter of the Moon Diameter of the Moon The distance to the moon is approximately 238,900 mi. UseThe distance to the moon is approximately 238,900 mi. Use the arc length formula to estimate the

the arc length formula to estimate the diameterdiameterd d of of the the moon moon if if angle angle in in thethe ﬁﬁguregure is measured to be .517

is measured to be .517..

91.

91. Concept CheckConcept Check If the radius of a circle is doubled and the central angle of a sec-If the radius of a circle is doubled and the central angle of a sec-tor is unchanged, how is the area of

tor is unchanged, how is the area of the sector changed?the sector changed? 92.

92. Concept CheckConcept Check Give the corresponding formula for the area of a sector when theGive the corresponding formula for the area of a sector when the angle is measured in degrees.

angle is measured in degrees.

d d   Not to scale Not to scale   7 71212 6.2

6.2 The Unit Circle and Circular The Unit Circle and Circular FunctionsFunctions 545545

89.

89. radius: 3947 mi;radius: 3947 mi;

circumference: 24,800 mi

90.

90. approximately 2156 miapproximately 2156 mi

91.

91. The area is quadrupled.The area is quadrupled.

92. 92. A A   _r_r22  360 360 496 mi 496 mi Syene Syene Alexandria Alexandria Shadow Shadow 7 7°°1212' ' 7 7°°1212' ' Sun's rays at noon Sun's rays at noon

In Sec

In Section tion 5.2,5.2, we dewe deﬁﬁned the six trigonometric functions in such a way that thened the six trigonometric functions in such a way that the domain of each function was a set of

domain of each function was a set of anglesangles in standard position. These anglesin standard position. These angles can be measured in degrees or in ra

can be measured in degrees or in radians. In advanced courses,dians. In advanced courses, such as calculus,such as calculus, it is necessary to modify the trigonometric functions so that their domains it is necessary to modify the trigonometric functions so that their domains con-sist of

sist of real numbersreal numbers rather than angles. We do this by using the relationshiprather than angles. We do this by using the relationship between an angle

between an angle   _{and an arc of length}_{and an arc of length ss on a circle.}_{on a circle.}

Circular Functions

In In Figure Figure 10,10, we we start start at at the the point point and and measure measure anan arc of length

arc of length ss along along the the circle. circle. If If then then the the arc arc is is measured measured in in a a counter- counter-clockwise

clockwise direction,direction, and and if if then then the the direction direction is is clockwise. clockwise. (If (If thenthen no

no arc arc is is measured.) measured.) Let Let the the endpoint endpoint of of this this arc arc be be at at the the point point The The circlecircle in Figure 10 is a

in Figure 10 is a unit circleunit circle ——it has center at the origin and radius 1 unit (henceit has center at the origin and radius 1 unit (hence the name

the name unit circleunit circle). Recall from algebra that the equation of this circle is). Recall from algebra that the equation of this circle is

(Section 2.1) (Section 2.1) x x22   y y22__ _1._1.



x x,, y y



.. ss _0,_0, ss 0,0, ss 0,0,



1,1, 00



6.2

6.2 The Unit Circle and Circular

The Unit Circle and Circular

Functions

Circular Functions

Circular Functions ss Finding Values of Circular Finding Values of Circular FunctionsFunctions ss Determining a Number with a Determining a Number with a Given CircularGiven Circular Function Value

Function Value ss Angular and Linear SpeedAngular and Linear Speed

x x y y 0 0 x x= cos= cos s s y

y= sin= sin s s

( ( x x,, y y)) (1, 0) (1, 0) (0, 1) (0, 1) ((––1, 0)1, 0) (0, (0, ––1)1) Arc of length Arc of length s s Unit circle

Unit circle x x22++ y y22= 1= 1  

Figure 10 Figure 10

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546

T

TEACHINGEACHINGTTIPIP Draw a unit circleDraw a unit circle

on the chalkboard and trace a

point slowly counterclockwise,

starting

starting at at Students Students shouldshould

see that the

see that the y y -value (sine function)-value (sine function)

starts at 0, increases to 1 (at

radians), decreases to 0 (at

radians), decreases to 0 (at  _radi- radi-ans), decreases again to

ans), decreases again to1 (at1 (at

radians), and returns to 0 after

one full revolution (2

one full revolution (2  _radians)._radians). This type of analysis will help

This type of analysis will help

illus-trate the range associated with the

trate the range associated with the

sine and cosine functions.

3 3  2 2   2 2  1,1,00..

*The authors thank Professor Marvel Townsend o

*The authors thank Professor Marvel Townsend o f the University of Florida for her suggestion to includef the University of Florida for her suggestion to include this

this ﬁﬁgure.gure.

Looking Ahead to Calculus Looking Ahead to Calculus

If you plan to study calculus, you must If you plan to study calculus, you must become very familiar with radian become very familiar with radian mea-sure. In calculus, the trigonometric or sure. In calculus, the trigonometric or circular functions are always circular functions are always under-stood to have real number domains. stood to have real number domains.

We saw in the previous section that the radian measure of

We saw in the previous section that the radian measure of   _{is related to the}_{is related to the}

arc length

arc length ss. In . In fafactct,, foforr   _measu_measured_{red in}_{in radi}_radians,_{ans, we}_{we kno}_know_{w that}_that _Here,_Here,

so

so ss,, which is meawhich is measured in linsured in linear unitear units such as inches or ces such as inches or centimentimeters,ters, isis numerically equal to

numerically equal to   _{,, measu}_{measured in radia}_{red in radians. Thus,}_{ns. Thus, the trig}_{the trigonome}_{onometric fun}_{tric function}_ctionss

of angle

of angle   _in_{in radians}_{radians found}_{found by}_{by choosing}_{choosing a}_{a point}_point _on_{on the}_{the unit}_{unit circle}_{circle can}_{can be}_be

rewritten as functions of the arc length

rewritten as functions of the arc length ss,, a real numa real number. ber. When interpreWhen interpreted thisted this way

way,, they are callthey are calleded circular functions.circular functions.

Circular Functions

Since

Since x x represents the cosine of represents the cosine of ss andand y y represents the sine of represents the sine of ss,, and beand becaucausese of the discussion in Sect

of the discussion in Section 6.1 on conveion 6.1 on converting between degrees arting between degrees and radians,nd radians, wewe can summarize a gr

can summarize a great deal of informeat deal of information in a concise ation in a concise manner,manner, as seen inas seen in Fig

Figureure 11.11.**

Figure 11 Figure 11

N O T E

N O T E Since Since and and we we can can replacereplace x x andand y y in the equationin the equation and obtain the Pythagorean identity

and obtain the Pythagorean identity

The

The ordered ordered pair pair represents represents a a point point on on the the unit unit circle,circle, and and thereforetherefore

so

so 11 coscos ss 11 aanndd 11 sinsin ss1.1.



11 x x 11 aanndd 11  y y 1,1,



x x,, y y



Unit circle x x22₊₊_y_y22_{= 1}_{= 1} 3 3 2 2 5 5 3 3 7 7 4 4 4 4 3 3 5 5 4 4 7 7 6 6 5 5 6 6 3 3 4 4 2 2 3 3 11 11 6 6   3 3   2 2   4 4   6 6 0 0 cot cot s s  x x y

y (( y y0)0) sec sec s s 11 x x (( x x 0)0) csc csc s s  11 y

y (( y y 0)0)

tan tan s s  y y

x

x (( x x 0)0) cos

cos s s x x sin

sin s s  y y



x x,, y y



r

r  _1,_1,

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6.2

6.2 The Unit Circle and Circular The Unit Circle and Circular FunctionsFunctions 547547

x x y y (0, 1) (0, 1) (1, 0) (1, 0) ((––1, 0)1, 0) (0, (0, ––1)1) x x22++ y y22= 1= 1 s s== (cos

(cosss, sin, sinss) = ) = (( x x,, y y))

0 0   Figure 12 Figure 12

For any value of

For any value of ss,, botboth sh sinin ss and cosand cos ss exist,exist, so the domain so the domain of these functof these functions isions is the set of all real numbers. For tan

the set of all real numbers. For tan ss,, dedeﬁﬁned asned as x x must not equal 0. The onlymust not equal 0. The only way

way x x can equal 0 is when the arc lengthcan equal 0 is when the arc length ss is is and and so so on. on. ToTo avoid a 0 denominator

avoid a 0 denominator,, the domain of the tangent function must be restrithe domain of the tangent function must be restricted tocted to those values of

those values of ss satisfyingsatisfying

n

n any integer.any integer. The de

The deﬁﬁnition of secant also hasnition of secant also has x x in the denominatorin the denominator,, so the domain of so the domain of secant issecant is the same as the domain of tangent.

the same as the domain of tangent. Both cotangent and cosecant are deBoth cotangent and cosecant are de ﬁﬁned withned with a denominator of

a denominator of y y. . To To guarantee guarantee that that the the domain domain of of these these functions functions mustmust be the set of all values of

be the set of all values of ss satisfyingsatisfying

In summary

In summary,, the domains of the circular fthe domains of the circular functions are as follows.unctions are as follows.

Domains of the Circular Functions

Assume that

Assume that nn is any integer andis any integer and ss is a real number.is a real number. Sine and Cosine Functions:

Sine and Cosine Functions: T

Tangent and Secant angent and Secant Functions:Functions: Cotangent and Cosecant Functions: Cotangent and Cosecant Functions:

Finding Values of Circular Functions

The circular functions (functionsThe circular functions (functions of real numbers) are closely related to the trigonometric functions of angles of real numbers) are closely related to the trigonometric functions of angles measured in radians.

measured in radians. TTo see this,o see this, let us assume that alet us assume that anglengle   _{is in standard posi-}_{is in standard}

posi-tion,

tion, supersuperimposimposed on the unit cired on the unit circle,cle, as showas shown in Figure 12. Supn in Figure 12. Suppose furtpose furtherher that

that  _{is the}_{is the radian}_{radian measure of this angle. Using the arc length formula}_{measure of this angle. Using the arc length formula}

with

with we we have have Thus,Thus, the the length length of of the the intercepted intercepted arc arc is is the the realreal number that corresponds to the radian measure of

number that corresponds to the radian measure of   _{. Using the de}_{. Using the deﬁ}_{ﬁnitions of the}_{nitions of the}

trigonometric

trigonometric functions,functions, we hawe haveve

and so on. As sho

and so on. As shown here,wn here, the trigonometric functhe trigonometric functions and the circular functtions and the circular functionsions lead to the same function values,

lead to the same function values, provided we think of the angles as being in ra-provided we think of the angles as being in ra-dian measure. This leads

dian measure. This leads to the following important result to the following important result concerning evaluatioconcerning evaluationn of circular functions.

of circular functions.

Evaluating a Circular Function

Circular function values of real numbers are obtained in

Circular function values of real numbers are obtained in the same manner asthe same manner as trigonometric function values of angles measured in radians. This applies trigonometric function values of angles measured in radians. This applies both to methods of

both to methods of fifinding exact values (such as reference angle analysis)nding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when and to calculator approximations. Calculators must be in radian mode when fi

ﬁnding circular function values.nding circular function values. sin

sin   y y

r r 

y y 1

1  y y  sin ss,,sin aanndd ccooss   x x r r  x x 1 1  x x  coscos ss,, ss    _.. r r  _1,_1, ss  _r_r  {{ s s



s s_n_n  _}}



s s



s s₍₂_(2 n_n ₁₎₁₎   2 2



((_,,₎₎

ss  _n_n  _,, _n_{n any integer.}_{any integer.}

y y _0,_0, ss



₂_2n_n__₁₁



  2 2 ,,  3322  ,, 3 3  2 2 ,,    22,,   2 2,, y y x x,,

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548

x x y y (0, (0, ––1)1) (0, 1) (0, 1) ((––1, 0)1, 0) (1, 0) (1, 0)  ==33 2 2 0 0 Figure 13 Figure 13 T

TEACHINGEACHINGTTIPIP Point out the impor-Point out the

impor-tance of being able to sketch the

tance of being able to sketch the

unit circle as in Figure 13. This will

enable students to more easily

determine function values of

mul-tiples of

tiples of  

2

2..

EXAMPLE 1

EXAMPLE 1 Finding Exact Circular Function ValuesFinding Exact Circular Function Values

Find

Find the the exact exact values values of of andand Solution

Solution Evaluating Evaluating a a circular circular function function at at the the real real number number is equivalis equivalent ent toto evaluat

evaluating ing it it at at radians. radians. An An angle angle of of radians radians intersects intersects the the unit unit circle circle at at thethe point

point as as shown shown in in Figure Figure 13. 13. SinceSince

it follows that it follows that

EXAMPLE 2

EXAMPLE 2 Finding Exact Circular Function ValuesFinding Exact Circular Function Values

(a)

(a) Use Figure 11 toUse Figure 11 to ﬁﬁnd nd the the exact exact values values of of andand (b)

(b) Use Figure 11 toUse Figure 11 to ﬁﬁnd the exact value of nd the exact value of (c)

(c) Use reference angles Use reference angles and degree/radian conversion toand degree/radian conversion to ﬁﬁnd the exact value of nd the exact value of

Solution Solution (a)

(a) In In Figure Figure 11,11, we we see see that that the the terminal terminal side side of of radians intersects radians intersects the the unitunit circle

circle at at Thus,Thus,

(b)

(b) Angles Angles of of radians radians and and radians radians are are coterminal. coterminal. Their Their terminal terminal sidessides intersect

intersect the the unit unit circle circle at at soso

(c)

(c) An An angle angle of of radians radians corresponds corresponds to to an an angle angle of of 120120°°. In standard posi-. In standard posi-tio

tion,n, 120120°° lies in quadrant II with a reference angle of 60lies in quadrant II with a reference angle of 60 °°,, soso

Cosine is negative in quadrant II.

Reference angle

Reference angle(Section 5.3)(Section 5.3)

Now try Exercises 7, 17, and 21. Now try Exercises 7, 17, and 21.

N O T E

N O T E Examples 1 and 2 illustrate that there are several methods of Examples 1 and 2 illustrate that there are several methods of ﬁﬁndingnding exact circular function

exact circular function values.values. cos

cos22  3 3