Front Matter1 Triangles and also Circles2 The Trigonometric Ratios3 laws of Sines and Cosines4 Trigonometric Functions5 Equations and Identities6 Radians7 circular Functions8 an ext Functions and also Identities9 Vectors10 Polar coordinates and complicated Numbers
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Imagine that you room riding top top a Ferris wheel the radius 100 feet, and each rotation take away eight minutes. We have the right to use angles in standard position to describe your place as you travel approximately the wheel. The figure at right reflects the locations indicated by ( heta = 0degree,~ 90degree,~ 180degree,) and (270degree ext.) however degrees are not the only means to specify location on a circle.

You are watching: One revolution is how many radians


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

Another useful technique uses distance traveled, or arclength, follow me the circle. How far have you traveled approximately the Ferris wheel at every of the locations shown?

Subsection Arclength

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

eginequation*lertC = 2 pi rendequation*

If us walk approximately the whole circumference of a circle, the distance we take trip is (2pi) times the size of the radius, or around 6.28 times the radius. If we walk only part of the means around the circle, then the distance we take trip depends also on the edge of displacement.

For example, an angle of (45degree) is (dfrac18) of a finish revolution, for this reason the arclength, (s ext,) from allude (A) to point (B) in the figure at ideal is (dfrac18) the the circumference. Thus


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

Similarly, the edge of displacement from suggest (A) to point (C) is (dfrac34) of a complete revolution, for this reason the arclength follow me the circle from (A) to (C ext,) displayed at right, is


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

In general, because that a provided circle the size of the arc spanned by an edge is proportional to the size of the angle.

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

The Ferris wheel in the development has circumference

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

so in fifty percent a change you take trip 314 feet roughly the edge, and also in one-quarter revolution you travel 157 feet.

To suggest the same 4 locations top top the wheel by distance traveled, we would certainly use

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

as displayed at right.


Example 6.1.

What length of arc is extended by an angle of (120degree) top top a one of radius 12 centimeters?


Because (dfrac120360 = dfrac13 ext,) an edge of (120degree) is (dfrac13) the a finish revolution, as presented 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 around 25.1 cm.

Checkpoint 6.2.

How far have friend traveled approximately the edge of a Ferris wheel that radius 100 feet as soon as you have turned v an edge of (150degree ext?)


(261.8) ft

Subsection Measuring angles in Radians

If you think around measuring arclength, you will check out that the level measure the the covering angle is not as important as the portion of one revolution it covers. This observation suggests a brand-new unit of measurement because that angles, one the is far better suited come calculations including arclength. We"ll make one adjust in ours formula for arclength, from

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


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

We"ll contact the amount in parentheses, (fraction of one change ( imes 2pi)), the radian measure of the angle the spans the arc.


The radian measure up of an angle is given by

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

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

Example 6.3.

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


An angle of (120degree) is (dfrac13) the a finish revolution, together we witnessed in the vault example. Thus, an edge 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?


Half a revolution, (180degree)

Radian measure does not have to be express in multiples the (pi ext.) Remember the (pi approx 3.14 ext,) for this reason one complete transformation is around 6.28 radians, and also one-quarter transformation is (dfrac14(2pi) = dfracpi2 ext,) or about 1.57 radians. The figure listed below shows decimal approximations for the quadrantal angles.

DegreesRadians:Exact ValuesRadians: DecimalApproximations

Note 6.5.

You should memorize both the precise values the these angle in radians and also their approximations!

Example 6.6.

In i beg your pardon quadrant would you find an angle of 2 radians? An edge of 5 radians?


Look in ~ the number above. The second quadrant includes angles in between (dfracpi2) and also (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 also 6.28, or in between (dfrac3pi2) and also (2pi) radians, so that lies in the fourth quadrant.

Checkpoint 6.7.

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


Subsection Converting between Degrees and also Radians

It is not an overwhelming to transform the measure of an angle in levels to its measure up in radians, or evil versa. One complete transformation is same to 2 radians or come (360degree ext,) so

eginequation*2pi ~ extradians = 360degreeendequation*

Dividing both political parties of this equation by 2 offers 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 up by (dfrac180degreepi ext.)

To convert from levels to radians we multiply the level measure by (dfracpi180 ext.)

Example 6.9.

Convert 3 radians to degrees.

Convert 3 levels to radians.


(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) come radians. Give both specific answer and also an approximation to three decimal places.

Convert (dfrac3pi4) radians come degrees.


(displaystyle dfracpi3 approx 1.047)

(displaystyle 135degree)

From our conversion variable we likewise learn that

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

So if (1degree) is a reasonably small angle, 1 radian is much larger — practically (60degree ext,) in fact.

however this is reasonable, since there are only a little an ext than 6 radians in whole revolution. An angle of 1 radian is displayed above.

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

Subsection Arclength Formula

Measuring angle in radians has the following advantage: To calculate an arclength we need only main point the radius of the circle by the radian measure up of the covering angle, ( heta ext.) watch again at our formula for arclength:

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

The amount in parentheses, portion of one transformation ( imes 2pi ext,) is just the measure of the extending angle in radians. Thus, if ( heta) is measure up in radians, we have actually the complying with formula for arclength, (s ext.)

Arclength Formula.

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

eginequation*lerts = r hetaendequation*

In particular, if ( heta = 1) we have (s = r ext.) We view that an edge of one radian spans one arc whose size is the radius of the circle. This is true because that a one of any kind of size, as depicted at right: an arclength equal to one radius identify a main angle the one radian, or about (57.3degree ext.)

In the following example, we usage the arclength formula come compute a adjust in latitude ~ above the Earth"s surface. Latitude is measure up in levels north or southern of the equator.

Example 6.11.

The radius that the earth is about 3960 miles. If you take trip 500 miles due north, how countless degrees of latitude will certainly you traverse?


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

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

To transform the angle measure to degrees, us multiply through (dfrac180degreepi) to get

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

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

Checkpoint 6.12.

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


(9.55) ft

Subsection Unit Circle

On a unit circle, (r = 1 ext,) for this reason the arclength formula becomes (s = heta ext.) Thus, on a unit circle, the measure up of a (positive) edge in radians is same to the size of the arc it spans.

Example 6.13.

You have actually walked 4 miles about a one pond that radius one mile. What is your position relative to your starting point?


The pond is a unit circle, therefore you have traversed an edge in radians equal to the arc length traveled, 4 miles. An angle of 4 radians is in the center of the 3rd quadrant loved one to your starting point, an ext than halfway however less 보다 three-quarters about the pond.

Checkpoint 6.14.

An ant walks roughly the in salt of a one birdbath the radius 1 foot. How much has the ant walked as soon as it has actually turned through an angle of (210degree ext?)


(3.67) ft

Review the following an abilities you will require for this section.

Algebra Refresher 6.1.

Use the proper conversion aspect to transform 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 speech = watts

1000 watts = horsepower

(dfrac1~ exttroy ounce480~ extgrains = 1)

0.5 trojan oz = grains

100 seed = troy oz


Algebra Refresher Answers

a.(16.09) kilometres 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

Subsubsection Vocabulary



Conversion factor


Unit circle

Subsubsection Concepts

The distance we travel approximately a one of radius is proportional come the edge of displacement.
eginequation* extbfArclength~ = ~ ( extbffraction of one revolution) cdot (2pi r)endequation*

We measure angle in radians when we job-related with arclength.


The radian measure of an edge is offered by

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

An arclength equal to one radius identify a main angle that one radian.

Radian measure have the right to be expressed as multiples of (pi) or as decimals.

Degrees(dfrac extRadians: extExact Values)(dfrac extRadians: Decimal extApproximations)

We main point by the suitable conversion variable to convert in between degrees and radians.

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

To transform from radians to degrees we main point the radian measure by (dfrac180degreepi ext.)

To convert from degrees to radians us multiply the degree measure through (dfracpi180 ext.)

Arclength Formula.On a circle of radius (r ext,) the size (s) of one arc covered by an edge ( heta) in radians is

eginequation*s = r hetaendequation*

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

Subsubsection research Questions

The length of a one arc counts on what two variables?

Define the radian measure up of one angle.

What is the conversion aspect from radians come degrees?

On a unit circle, the size of an arc is same to what other quantity?

Subsubsection Skills

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

Sketch angles offered in radians #1 and also 2, 11 and also 12

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

Use the arclength formula #33–46

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

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

Exercises Homework 6.1

Radians(0)(dfracpi4)(dfracpi2)(dfrac3pi4)(pi)(dfrac5pi4)(dfrac3pi2)(dfrac7pi4)(2 pi)

Convert each angle to degrees.

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

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

Convert every angle to degrees.

Sketch every angle ~ above a circle prefer this one, and also label in radians.

Exercise Group.

For difficulties 3–6, to express each portion of one finish rotation in degrees and also in radians.


(displaystyle dfrac13)

(displaystyle dfrac23)

(displaystyle dfrac43)

(displaystyle dfrac53)


(displaystyle dfrac15)

(displaystyle dfrac25)

(displaystyle dfrac35)

(displaystyle dfrac45)


(displaystyle dfrac18)

(displaystyle dfrac38)

(displaystyle dfrac58)

(displaystyle dfrac78)


(displaystyle dfrac112)

(displaystyle dfrac16)

(displaystyle dfrac512)

(displaystyle dfrac56)

Exercise Group.

For troubles 7–8, brand each angle in standard place with radian measure.


Rotate counter-clockwise indigenous 0.


Rotate clockwise from 0.

Exercise Group.

For problems 9–10, provide a decimal approximation to hundredths for each edge in radians.


(displaystyle dfracpi6)

(displaystyle dfrac5pi6)

(displaystyle dfrac7pi6)

(displaystyle dfrac11pi6)


(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


Locate and label each angle from trouble 9 top top the unit one below. (The circle is significant off in one per 10 of a radian.)


Locate and also label each angle from difficulty 10 ~ above the unit one below. (The circle is marked off in one per 10 of a radian.)

Exercise Group.

From the perform below, pick the ideal decimal approximation for each edge in radians in troubles 13–20. Perform not use a calculator; usage the reality that (pi) is a tiny greater than 3.

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
















Exercise Group.

For troubles 21–24, to speak in i m sorry quadrant every angle lies.


(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)


(displaystyle dfracpi4)

(displaystyle dfracpi4)

(displaystyle dfrac5pi4)

(displaystyle dfrac7pi4)

Exercise Group.

For difficulties 25–28, finish the table.





Exercise Group.

For difficulties 29–30, transform to radians. Round to hundredths.


(displaystyle 75degree)

(displaystyle 236degree)

(displaystyle 327degree)


(displaystyle 138degree)

(displaystyle 194degree)

(displaystyle 342degree)

Exercise Group.

For problems 31–32, transform to degrees. Round to tenths.


(displaystyle 0.8)

(displaystyle 3.5)

(displaystyle 5.1)


(displaystyle 1.1)

(displaystyle 2.6)

(displaystyle 4.6)

Exercise Group.

For problems 33–37, usage the arclength formula to answer the questions. Ring answers come hundredths


Find the arclength extended by an edge of (80degree) on a one of radius 4 inches.


Find the arclength extended by an edge of (200degree) on a one of radius 18 feet.


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


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


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


Find the edge subtended by one arclength the 1.6 yards top top a circle of diameter 2 yards.

Exercise Group.

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


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

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


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

If you began at the 12 and walked 500 feet clockwise roughly the clockface, with how countless radians would certainly you walk?

If you started your walk at noon, just how long would it take it the minute hand come reach her position? How much did the reminder of the minute hand move in its arc?


In 1851 Jean-Bernard Foucault prove the rotation of the earth with a pendulum installed in the Pantheon in Paris. Foucault"s pendulum consisted of a cannonball exposed on a 67 meter wire, and also it brushed up out an arc that 8 meters on each swing. Through what edge did the pendulum swing? offer your price in radians and also then in degrees, rounded come the nearest hundredth.


A wheel with radius 40 centimeters is rolling a street of 1000 centimeters on a flat surface. Through what angle has actually the wheel rotated? provide your price in radians and then in degrees, rounded come one decimal place.


Clothes dryers attract 3.5 time as lot power as washing machines, so newer machines have been engineered for better efficiency. A vigorous spin cycle reduce the time essential for drying, and some front-loading models spin at a rate of 1500 rotations per minute.

If the radius that the north is 11 inches, how much do her socks travel in one minute?

How rapid are your socks traveling throughout the rotate cycle?


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

How much does the telescope take trip in one hour? (The radius that the planet is 6400 kilometers.)

What is the rate of the Hubble telescope?


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The very first large windmill provided to generate power was built in Cleveland, Ohio in 1888. Its sails were 17 meter in diameter, and moved at 10 rotations per minute. How quick did the end of the sails travel?


The biggest windmill operation today has actually wings 54 meters in length. To be many efficient, the advice of the wings have to travel at 50 meters every second. How fast must the wings rotate?

For difficulties 47–52, find two clues on the unit circle v the given coordinate.Sketch the approximate place of the points on the circle. (Hint: what is the equation because that the unit circle?)