Front Matter1 Triangles and Circles2 The Trigonometric Ratios3 Laws of Sines and Cosines4 Trigonometric Functions5 Equations and Identities6 Radians7 Circular Functions8 More Functions and Identities9 Vectors10 Polar Coordinates and Complex Numbers
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Imagine that you are riding on a Ferris wheel of radius 100 feet, and each rotation takes eight minutes. We can use angles in standard position to describe your location as you travel around the wheel. The figure at right shows the locations indicated by ( heta = 0degree,~ 90degree,~ 180degree,) and (270degree ext.) But degrees are not the only way to specify location on a circle.

You are watching: How many radians in one revolution


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We could use percent of one complete rotation and label the same locations by (p = 0,~ p = 25,~ p = 50,~ extand~ p = 75 ext.) Or we could use the time elapsed, so that for this example we would have (t = 0,~ t = 2,~t = 4,~ extand~ t = 6) minutes.

Another useful method uses distance traveled, or arclength, along the circle. How far have you traveled around the Ferris wheel at each of the locations shown?

Subarea Arclength

Recall that the circumference of a circle is proportional to its radius,


eginequation*lertC = 2 pi rendequation*

If we walk around the entire circumference of a circle, the distance we travel is (2pi) times the length of the radius, or about 6.28 times the radius. If we walk only part of the way around the circle, then the distance we travel depends also on the angle of displacement.


For example, an angle of (45degree) is (dfrac18) of a complete revolution, so the arclength, (s ext,) from point (A) to point (B) in the figure at right is (dfrac18) of the circumference. Thus


*

eginequation*s = dfrac18(2pi r) = dfracpi4 rendequation*

Similarly, the angle of displacement from point (A) to point (C) is (dfrac34) of a complete revolution, so the arclength along the circle from (A) to (C ext,) shown at right, is


*

eginequation*s = dfrac34(2pi r) = dfrac3pi2 rendequation*

In general, for a given circle the length of the arc spanned by an angle is proportional to the size of the angle.

Arclength on a Circle.
eginequation*lert extbfArclength~ = ~ lert( extbffraction of one revolution) cdot (2pi r)endequation*

The Ferris wheel in the introduction has circumference


eginequation*C = 2pi (100) = 628~ extfeetendequation*

so in half a revolution you travel 314 feet around the edge, and in one-quarter revolution you travel 157 feet.


To indicate the same four locations on the wheel by distance traveled, we would use


eginequation*s = 0,~ s = 157,~ s = 314,~ extand~ s = 471 ext,endequation*

as shown at right.


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Example 6.1.

What length of arc is spanned by an angle of (120degree) on a circle of radius 12 centimeters?


Solution.

Because (dfrac120360 = dfrac13 ext,) an angle of (120degree) is (dfrac13) of a complete revolution, as shown at right.


Using the formula above with (r = 12 ext,) we find that


eginequation*s = dfrac13(2pi cdot 12) = dfrac2 pi3 cdot 12 = 8pi ~ extcmendequation*

or about 25.1 cm.


Checkpoint 6.2.

How far have you traveled around the edge of a Ferris wheel of radius 100 feet when you have turned through an angle of (150degree ext?)


Answer.

(261.8) ft


Subarea Measuring Angles in Radians

If you think about measuring arclength, you will see that the degree measure of the spanning angle is not as important as the fraction of one revolution it covers. This observation suggests a new unit of measurement for angles, one that is better suited to calculations involving arclength. We"ll make one change in our formula for arclength, from


eginequation* extbfArclength~ = ~ ( extbffraction of one revolution) cdot (2pi r)endequation*

to


eginequation*lert extbfArclength~ = ~ lert( extbffraction of one revolution imes 2pi) cdot rendequation*

We"ll call the quantity in parentheses, (fraction of one revolution ( imes 2pi)), the radian measure of the angle that spans the arc.

Radians.

The radian measure of an angle is given by


eginequation*lert( extbffraction of one revolution imes 2pi)endequation*

For example, one complete revolution, or (360degree ext,) is equal to (2pi) radians, and one-quarter revolution, or (90degree ext,) is equal to (dfrac14(2pi)) or (dfracpi2) radians. The figure below shows the radian measure of the quadrantal angles.


Example 6.3.

What is the radian measure of an angle of (120degree ext?)


Solution.

An angle of (120degree) is (dfrac13) of a complete revolution, as we saw in the previous example. Thus, an angle of (120degree) has a radian measure of(dfrac13(2pi) ext,) or (dfrac2pi3 ext.)


Checkpoint 6.4.

What fraction of a revolution is (pi) radians? How many degrees is that?


Answer.

Half a revolution, (180degree)


Radian measure does not have to be expressed in multiples of (pi ext.) Remember that (pi approx 3.14 ext,) so one complete revolution is about 6.28 radians, and one-quarter revolution is (dfrac14(2pi) = dfracpi2 ext,) or about 1.57 radians. The figure below shows decimal approximations for the quadrantal angles.


DegreesRadians:Exact ValuesRadians: DecimalApproximations
(0degree)(0)(0)
(90degree)(dfracpi2)(1.57)
(180degree)(pi)(3.14)
(270degree)(dfrac3pi2)(4.71)
(360degree)(2pi)(6.28)

Note 6.5.

You should memorize both the exact values of these angles in radians and their approximations!

Example 6.6.

In which quadrant would you find an angle of 2 radians? An angle of 5 radians?


Solution.

Look at the figure above. The second quadrant includes angles between (dfracpi2) and (pi ext,) or 1.57 and 3.14 radians, so 2 radians lies in the second quadrant. An angle of 5 radians is between 4.71 and 6.28, or between (dfrac3pi2) and (2pi) radians, so it lies in the fourth quadrant.


Checkpoint 6.7.

Draw a circle centered at the origin and sketch (in standard position) angles of approximately 3 radians, 4 radians, and 6 radians.


Answer.

Subsection Converting Between Degrees and Radians

It is not difficult to convert the measure of an angle in degrees to its measure in radians, or vice versa. One complete revolution is equal to 2 radians or to (360degree ext,) so


eginequation*2pi ~ extradians = 360degreeendequation*

Dividing both sides of this equation by 2 gives us a conversion factor:

Unit Conversion for Angles.
eginequation*lertdfrac180degreepi~ extradians = 1endequation*
Note 6.8.

To convert from radians to degrees we multiply the radian measure by (dfrac180degreepi ext.)

To convert from degrees to radians we multiply the degree measure by (dfracpi180 ext.)

Example 6.9.

Convert 3 radians to degrees.

Convert 3 degrees to radians.


Solution.

(displaystyle (3 ~ extradians) imes left(dfrac180degreepi ight) = dfrac540degreepi approx 171.9degree)

(displaystyle (3degree) imes left(dfracpi180degree ight) = dfracpi60approx 0.05~ extradians.)


Checkpoint 6.10.

Convert (60degree) to radians. Give both an exact answer and an approximation to three decimal places.

Convert (dfrac3pi4) radians to degrees.


Answer.

(displaystyle dfracpi3 approx 1.047)

(displaystyle 135degree)


From our conversion factor we also learn that


eginequation*lert 1~ extradian = dfrac180degreepi approx 57.3degreeendequation*

So while (1degree) is a relatively small angle, 1 radian is much larger — nearly (60degree ext,) in fact.


But this is reasonable, because there are only a little more than 6 radians in an entire revolution. An angle of 1 radian is shown above.

We"ll soon see that, for many applications, it is easier to work entirely in radians. For reference, the figure below shows a radian protractor.


Subarea Arclength Formula

Measuring angles in radians has the following advantage: To calculate an arclength we need only multiply the radius of the circle by the radian measure of the spanning angle, ( heta ext.) Look again at our formula for arclength:


eginequation*lert extbfArclength~ = ~ lert( extbffraction of one revolution imes 2pi) cdot rendequation*

The quantity in parentheses, fraction of one revolution ( imes 2pi ext,) is just the measure of the spanning angle in radians. Thus, if ( heta) is measured in radians, we have the following formula for arclength, (s ext.)

Arclength Formula.

On a circle of radius (r ext,) the length (s) of an arc spanned by an angle ( heta) in radians is


eginequation*lerts = r hetaendequation*

In particular, if ( heta = 1) we have (s = r ext.) We see that an angle of one radian spans an arc whose length is the radius of the circle. This is true for a circle of any size, as illustrated at right: an arclength equal to one radius determines a central angle of one radian, or about (57.3degree ext.)


In the next example, we use the arclength formula to compute a change in latitude on the Earth"s surface. Latitude is measured in degrees north or south of the equator.

Example 6.11.

The radius of the Earth is about 3960 miles. If you travel 500 miles due north, how many degrees of latitude will you traverse?


Solution.

We think of the distance 500 miles as an arclength on the surface of the Earth, as shown at right. Substituting (s = 500) and (r = 3960) into the arclength formula gives


eginalign*500 amp = 3960 heta\ heta amp = dfrac5003960 = 0.1263~ extradiansendalign*

To convert the angle measure to degrees, we multiply by (dfrac180degreepi) to get


eginequation*0.1263left(dfrac180degreepi ight) = 7.23degreeendequation*

Your latitude has changed by about (7.23degree ext.)


Checkpoint 6.12.

The distance around the face of a large clock from 2 to 3 is five feet. What is the radius of the clock?


Answer.

(9.55) ft


Subarea Unit Circle

On a unit circle, (r = 1 ext,) so the arclength formula becomes (s = heta ext.) Thus, on a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.

Example 6.13.

You have walked 4 miles around a circular pond of radius one mile. What is your position relative to your starting point?


Solution.

The pond is a unit circle, so you have traversed an angle in radians equal to the arc length traveled, 4 miles. An angle of 4 radians is in the middle of the third quadrant relative to your starting point, more than halfway but less than three-quarters around the pond.


Checkpoint 6.14.

An ant walks around the rim of a circular birdbath of radius 1 foot. How far has the ant walked when it has turned through an angle of (210degree ext?)


Answer.

(3.67) ft


Review the following skills you will need for this section.

Algebra Refresher 6.1.

Use the appropriate conversion factor to convert units.


(dfrac1~ extmile1.609~ extkilometers = 1)

10 miles = km

50 km = miles

(dfrac1~ extacre0.405~ exthectare = 1)

40 acres = hectares

5 hectares = acres

(dfrac1~ exthorsepower746~ extwatts = 1)

250 horsepower = watts

1000 watts = horsepower

(dfrac1~ exttroy ounce480~ extgrains = 1)

0.5 troy oz = grains

100 grains = troy oz


(underlineqquadqquadqquadqquad)


Algebra Refresher Answers


a.(16.09) km b. (31.08) mi

a. (16.2) hectares b.(12.35) acres

a. (186,500) watts b. (1.34) horsepower

a. (240) grains b. (0.21) troy oz


Subsection Section 6.1 Summary

Subsubarea Vocabulary

Arclength

Radian

Conversion factor

Latitude

Unit circle

Subsubarea Concepts

The distance we travel around a circle of radius is proportional to the angle of displacement.
eginequation* extbfArclength~ = ~ ( extbffraction of one revolution) cdot (2pi r)endequation*

We measure angles in radians when we work with arclength.

Radians.

The radian measure of an angle is given by


eginequation*( extbffraction of one revolution imes 2pi)endequation*

An arclength equal to one radius determines a central angle of one radian.

Radian measure can be expressed as multiples of (pi) or as decimals.


Degrees(dfrac extRadians: extExact Values)(dfrac extRadians: Decimal extApproximations)
(0degree)(0)(0)
(90degree)(dfracpi2)(1.57)
(180degree)(pi)(3.14)
(270degree)(dfrac3pi2)(4.71)
(360degree)(2pi)(6.28)

We multiply by the appropriate conversion factor to convert between degrees and radians.

Unit Conversion for Angles.
eginequation*dfrac180degreepi~ extradians = 1endequation*

To convert from radians to degrees we multiply the radian measure by (dfrac180degreepi ext.)

To convert from degrees to radians we multiply the degree measure by (dfracpi180 ext.)

Arclength Formula.On a circle of radius (r ext,) the length (s) of an arc spanned by an angle ( heta) in radians is


eginequation*s = r hetaendequation*

On a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.

Subsubarea Study Questions

The length of a circular arc depends on what two variables?

Define the radian measure of an angle.

What is the conversion factor from radians to degrees?

On a unit circle, the length of an arc is equal to what other quantity?

Subsubsection Skills

Express angles in degrees and radians #1–8, 25–32

Sketch angles given in radians #1 and 2, 11 and 12

Estimate angles in radians #9–10, 13–24

Use the arclength formula #33–46

Find coordinates of a point on a unit circle #47–52

Calculate angular velocity and area of a sector #55–60

Exercises Homework 6.1

1.
Radians(0)(dfracpi4)(dfracpi2)(dfrac3pi4)(pi)(dfrac5pi4)(dfrac3pi2)(dfrac7pi4)(2 pi)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)

Convert each angle to degrees.

Sketch each angle on a circle like this one, and label in radians.


2.
Radians(0)(dfracpi6)(dfracpi3)(dfracpi2)(dfrac2pi3)(dfrac5pi6)(pi)(dfrac7pi6)(dfrac4pi3)(dfrac3pi2)(dfrac5pi3)(dfrac11pi6)(2 pi)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)(hphantom0000)

Convert each angle to degrees.

Sketch each angle on a circle like this one, and label in radians.


Exercise Group.

For Problems 3–6, express each fraction of one complete rotation in degrees and in radians.


3.

(displaystyle dfrac13)

(displaystyle dfrac23)

(displaystyle dfrac43)

(displaystyle dfrac53)


4.

(displaystyle dfrac15)

(displaystyle dfrac25)

(displaystyle dfrac35)

(displaystyle dfrac45)


5.

(displaystyle dfrac18)

(displaystyle dfrac38)

(displaystyle dfrac58)

(displaystyle dfrac78)


6.

(displaystyle dfrac112)

(displaystyle dfrac16)

(displaystyle dfrac512)

(displaystyle dfrac56)


Exercise Group.

For Problems 7–8, label each angle in standard position with radian measure.


7.

Rotate counter-clockwise from 0.


8.

Rotate clockwise from 0.


Exercise Group.

For Problems 9–10, give a decimal approximation to hundredths for each angle in radians.


9.

(displaystyle dfracpi6)

(displaystyle dfrac5pi6)

(displaystyle dfrac7pi6)

(displaystyle dfrac11pi6)

10.

(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


11.

Locate and label each angle from Problem 9 on the unit circle below. (The circle is marked off in tenths of a radian.)


12.

Locate and label each angle from Problem 10 on the unit circle below. (The circle is marked off in tenths of a radian.)


Exercise Group.

From the list below, choose the best decimal approximation for each angle in radians in Problems 13–20. Do not use a calculator; use the fact that (pi) is a little greater than 3.


eginequation*0.52,~~ 0.79,~~ 2.09,~~ 2.36,~~ 2.62,~~ 3.67,~~ 5.24,~~ 5.50 endequation*
13.

(dfrac2pi3)

14.

(dfracpi4)

15.

(dfrac5pi6)

16.

(dfrac5pi3)

17.

(dfracpi6)

18.

(dfrac7pi4)

19.

(dfrac3pi4)

20.

(dfrac7pi6)


Exercise Group.

For Problems 21–24, say in which quadrant each angle lies.


21.

(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)

22.

(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)

23.

(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)

24.

(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


Exercise Group.

For Problems 25–28, complete the table.


25.
Radians(dfracpi6)(dfracpi4)(dfracpi3)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)

26.
Radians(dfrac2pi3)(dfrac3pi4)(dfrac5pi6)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)

27.
Radians(dfrac7pi6)(dfrac5pi4)(dfrac4pi3)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)

28.
Radians(dfrac5pi3)(dfrac7pi4)(dfrac11pi6)
Degrees(hphantom0000)(hphantom0000)(hphantom0000)

Exercise Group.

For Problems 29–30, convert to radians. Round to hundredths.


29.

(displaystyle 75degree)

(displaystyle 236degree)

(displaystyle 327degree)

30.

(displaystyle 138degree)

(displaystyle 194degree)

(displaystyle 342degree)


Exercise Group.

For Problems 31–32, convert to degrees. Round to tenths.


31.

(displaystyle 0.8)

(displaystyle 3.5)

(displaystyle 5.1)

32.

(displaystyle 1.1)

(displaystyle 2.6)

(displaystyle 4.6)


Exercise Group.

For Problems 33–37, use the arclength formula to answer the questions. Round answers to hundredths


33.

Find the arclength spanned by an angle of (80degree) on a circle of radius 4 inches.

34.

Find the arclength spanned by an angle of (200degree) on a circle of radius 18 feet.

35.

Find the radius of a cricle if an angle of (250degree) spans an arclength of 18 meters.

36.

Find the radius of a cricle if an angle of (20degree) spans an arclength of 0.5 kilometers.

37.

Find the angle subtended by an arclength of 28 centimeters on a circle of diameter 20 centimeters.

38.

Find the angle subtended by an arclength of 1.6 yards on a circle of diameter 2 yards.


Exercise Group.

For Problems 39–46, use the arclength formula to answer the questions.


39.

Through how many radians does the minute hand of a clock sweep between 9:05 pm and 9:30 pm?

The dial of Big Ben"s clock in London is 23 feet in diameter. How long is the arc traced by the minute hand between 9:05 pm and 9:30 pm?

40.

The largest clock ever constructed was the Floral Clock in the garden of the 1904 World"s Fair in St. Louis. The hour hand was 50 feet long, the minute hand was 75 feet long, and the radius of the clockface was 112 feet.

If you started at the 12 and walked 500 feet clockwise around the clockface, through how many radians would you walk?

If you started your walk at noon, how long would it take the minute hand to reach your position? How far did the tip of the minute hand move in its arc?

41.

In 1851 Jean-Bernard Foucault demonstrated the rotation of the earth with a pendulum installed in the Pantheon in Paris. Foucault"s pendulum consisted of a cannonball suspended on a 67 meter wire, and it swept out an arc of 8 meters on each swing. Through what angle did the pendulum swing? Give your answer in radians and then in degrees, rounded to the nearest hundredth.

42.

A wheel with radius 40 centimeters is rolled a distance of 1000 centimeters on a flat surface. Through what angle has the wheel rotated? Give your answer in radians and then in degrees, rounded to one decimal place.

43.

Clothes dryers draw 3.5 times as much power as washing machines, so newer machines have been engineered for greater efficiency. A vigorous spin cycle reduces the time needed for drying, and some front-loading models spin at a rate of 1500 rotations per minute.

If the radius of the drum is 11 inches, how far do your socks travel in one minute?

How fast are your socks traveling during the spin cycle?

44.

The Hubble telescope is in orbit around the earth at an altitude of 600 kilometers, and completes one orbit in 97 minutes.

How far does the telescope travel in one hour? (The radius of the earth is 6400 kilometers.)

What is the speed of the Hubble telescope?

45.

See more: How Many Yards Are 50 Feet Is How Many Yards, 50 Feet To Yards Conversion Calculator

The first large windmill used to generate electricity was built in Cleveland, Ohio in 1888. Its sails were 17 meters in diameter, and moved at 10 rotations per minute. How fast did the ends of the sails travel?

46.

The largest windmill operating today has wings 54 meters in length. To be most efficient, the tips of the wings must travel at 50 meters per second. How fast must the wings rotate?


For Problems 47–52, find two points on the unit circle with the given coordinate.Sketch the approximate location of the points on the circle. (Hint: what is the equation for the unit circle?)