Front Matter1 Triangles and Circles2 The Trigonometric Ratios3 legislations of Sines and also Cosines4 Trigonometric Functions5 Equations and also Identities6 Radians7 circular Functions8 more Functions and Identities9 Vectors10 Polar works with and complex Numbers
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Imagine the you are riding ~ above a Ferris wheel the radius 100 feet, and also each rotation takes eight minutes. We have the right to use angle in standard position to describe your location as you travel roughly the wheel. The number at right reflects the locations shown by \\(\\theta = 0\\degree,~ 90\\degree,~ 180\\degree,\\) and \\(270\\degree\\text.\\) yet degrees room not the only means to specify ar on a circle.

You are watching: How many radians in one revolution


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

Another useful an approach uses street traveled, or arclength, follow me the circle. How far have you traveled around the Ferris wheel at each of the locations shown?

Subsection Arclength

Recall that the one of a one is proportional come its radius,

\\beginequation*\\blertC = 2 \\pi r\\endequation*

If us walk roughly the whole circumference the a circle, the distance we take trip is \\(2\\pi\\) times the size of the radius, or around 6.28 times the radius. If us walk only part of the way around the circle, climate the distance we travel depends likewise on the edge of displacement.

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


\\beginequation*s = \\dfrac18(2\\pi r) = \\dfrac\\pi4 r\\endequation*

Similarly, the edge of displacement from point \\(A\\) to point \\(C\\) is \\(\\dfrac34\\) of a complete revolution, for this reason the arclength along the circle from \\(A\\) come \\(C\\text,\\) shown at right, is


\\beginequation*s = \\dfrac34(2\\pi r) = \\dfrac3\\pi2 r\\endequation*

In general, for a given circle the length of the arc spanned by an edge is proportional come the dimension of the angle.

Arclength top top a Circle.
\\beginequation*\\blert\\textbfArclength~ = ~ \\blert(\\textbffraction of one revolution) \\cdot (2\\pi r)\\endequation*

The Ferris wheel in the advent has circumference

\\beginequation*C = 2\\pi (100) = 628~ \\textfeet\\endequation*

so in fifty percent a transformation you take trip 314 feet approximately the edge, and also in one-quarter transformation you take trip 157 feet.

To show the same 4 locations on the wheel by street traveled, we would certainly use

\\beginequation*s = 0,~ s = 157,~ s = 314,~ \\textand~ s = 471\\text,\\endequation*

as presented at right.


Example 6.1.

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


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

Using the formula above with \\(r = 12\\text,\\) we uncover that

\\beginequation*s = \\dfrac13(2\\pi \\cdot 12) = \\dfrac2 \\pi3 \\cdot 12 = 8\\pi ~ \\textcm\\endequation*

or around 25.1 cm.

Checkpoint 6.2.

How much have girlfriend traveled about the sheet of a Ferris wheel of radius 100 feet when you have turned with an edge of \\(150\\degree\\text?\\)


\\(261.8\\) ft

Subsection Measuring angles in Radians

If you think around measuring arclength, friend will see that the degree measure that the covering angle is no as crucial as the portion of one change it covers. This observation suggests a new unit of measurement because that angles, one the is better suited come calculations involving arclength. We\"ll do one adjust in our formula for arclength, from

\\beginequation*\\textbfArclength~ = ~ (\\textbffraction that one revolution) \\cdot (2\\pi r)\\endequation*


\\beginequation*\\blert\\textbfArclength~ = ~ \\blert(\\textbffraction of one revolution\\times 2\\pi) \\cdot r\\endequation*

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


The radian measure of an angle is offered by

\\beginequation*\\blert(\\textbffraction that one revolution\\times 2\\pi)\\endequation*

For example, one finish revolution, or \\(360\\degree\\text,\\) is same to \\(2\\pi\\) radians, and also one-quarter revolution, or \\(90\\degree\\text,\\) is same to \\(\\dfrac14(2\\pi)\\) or \\(\\dfrac\\pi2\\) radians. The figure listed below shows the radian measure of the quadrantal angles.

Example 6.3.

What is the radian measure of an edge of \\(120\\degree\\text?\\)


An angle of \\(120\\degree\\) is \\(\\dfrac13\\) that a finish revolution, as we experienced in the previous example. Thus, an angle of \\(120\\degree\\) has actually a radian measure up of\\(\\dfrac13(2\\pi)\\text,\\) or \\(\\dfrac2\\pi3\\text.\\)

Checkpoint 6.4.

What fraction of a change is \\(\\pi\\) radians? How plenty of degrees is that?


Half a revolution, \\(180\\degree\\)

Radian measure up does not need to be expressed in multiples of \\(\\pi\\text.\\) Remember that \\(\\pi \\approx 3.14\\text,\\) therefore one complete revolution is about 6.28 radians, and one-quarter revolution is \\(\\dfrac14(2\\pi) = \\dfrac\\pi2\\text,\\) or around 1.57 radians. The figure below shows decimal approximations for the quadrantal angles.

DegreesRadians:Exact ValuesRadians: DecimalApproximations

Note 6.5.

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

Example 6.6.

In i m sorry quadrant would you uncover an edge of 2 radians? An angle of 5 radians?


Look in ~ the figure above. The second quadrant has angles between \\(\\dfrac\\pi2\\) and also \\(\\pi\\text,\\) or 1.57 and also 3.14 radians, for this reason 2 radians lies in the 2nd quadrant. An edge of 5 radians is in between 4.71 and 6.28, or between \\(\\dfrac3\\pi2\\) and also \\(2\\pi\\) radians, so that lies in the 4th quadrant.

Checkpoint 6.7.

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


Subsection Converting in between Degrees and also Radians

It is not complicated to convert the measure up of an edge in degrees to its measure up in radians, or evil versa. One complete change is equal to 2 radians or come \\(360\\degree\\text,\\) so

\\beginequation*2\\pi ~\\textradians = 360\\degree\\endequation*

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

Unit Conversion because that Angles.
\\beginequation*\\blert\\dfrac180\\degree\\pi~\\textradians = 1\\endequation*
Note 6.8.

To transform from radians to degrees we main point the radian measure up by \\(\\dfrac180\\degree\\pi\\text.\\)

To transform from degrees to radians we multiply the level measure by \\(\\dfrac\\pi180\\text.\\)

Example 6.9.

Convert 3 radians to degrees.

Convert 3 degrees to radians.


\\(\\displaystyle (3 ~\\textradians) \\times \\left(\\dfrac180\\degree\\pi\\right) = \\dfrac540\\degree\\pi \\approx 171.9\\degree\\)

\\(\\displaystyle (3\\degree) \\times \\left(\\dfrac\\pi180\\degree\\right) = \\dfrac\\pi60\\approx 0.05~ \\textradians.\\)

Checkpoint 6.10.

Convert \\(60\\degree\\) to radians. Provide both specific answer and also an approximation to 3 decimal places.

Convert \\(\\dfrac3\\pi4\\) radians come degrees.


\\(\\displaystyle \\dfrac\\pi3 \\approx 1.047\\)

\\(\\displaystyle 135\\degree\\)

From ours conversion aspect we likewise learn that

\\beginequation*\\blert 1~\\textradian = \\dfrac180\\degree\\pi \\approx 57.3\\degree\\endequation*

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

however this is reasonable, due to the fact that there are just a little more than 6 radians in whole revolution. An angle of 1 radian is shown above.

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

Subsection Arclength Formula

Measuring angle in radians has the complying with advantage: To calculation an arclength we need only multiply the radius the the one by the radian measure up of the extending angle, \\(\\theta\\text.\\) look at again at our formula because that arclength:

\\beginequation*\\blert\\textbfArclength~ = ~ \\blert(\\textbffraction that one revolution\\times 2\\pi) \\cdot r\\endequation*

The quantity in parentheses, portion of one change \\(\\times 2\\pi\\text,\\) is simply the measure of the spanning angle in radians. Thus, if \\(\\theta\\) is measure in radians, we have the adhering to formula for arclength, \\(s\\text.\\)

Arclength Formula.

On a circle of radius \\(r\\text,\\) the size \\(s\\) of an arc spanned by an angle \\(\\theta\\) in radians is

\\beginequation*\\blerts = r\\theta\\endequation*

In particular, if \\(\\theta = 1\\) we have \\(s = r\\text.\\) We watch that an angle of one radian spans one arc whose length is the radius of the circle. This is true because that a circle of any type of size, as depicted at right: an arclength equal to one radius identify a central angle of one radian, or around \\(57.3\\degree\\text.\\)

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

Example 6.11.

The radius the the planet is around 3960 miles. If you take trip 500 miles due north, how numerous degrees of latitude will certainly you traverse?


We think that the street 500 miles as an arclength top top the surface of the Earth, as displayed at right. Substituting \\(s = 500\\) and also \\(r = 3960\\) right into the arclength formula gives

\\beginalign*500 \\amp = 3960 \\theta\\\\\\theta \\amp = \\dfrac5003960 = 0.1263~ \\textradians\\endalign*

To transform the angle measure to degrees, we multiply by \\(\\dfrac180\\degree\\pi\\) to get

\\beginequation*0.1263\\left(\\dfrac180\\degree\\pi\\right) = 7.23\\degree\\endequation*

Your latitude has changed by about \\(7.23\\degree\\text.\\)

Checkpoint 6.12.

The distance around the confront of a big clock from 2 to 3 is 5 feet. What is the radius that the clock?


\\(9.55\\) ft

Subsection Unit Circle

On a unit circle, \\(r = 1\\text,\\) therefore the arclength formula i do not care \\(s = \\theta\\text.\\) Thus, on a unit circle, the measure up of a (positive) angle in radians is equal to the size of the arc it spans.

Example 6.13.

You have walked 4 miles around a circular pond that radius one mile. What is your place relative to your beginning point?


The pond is a unit circle, so you have actually traversed an angle in radians equal to the arc size traveled, 4 miles. An edge of 4 radians is in the middle of the 3rd quadrant family member to your beginning point, much more than halfway but less 보다 three-quarters about the pond.

Checkpoint 6.14.

An ant walks roughly the in salt of a circular birdbath that radius 1 foot. How much has the ant walked as soon as it has actually turned with an angle of \\(210\\degree\\text?\\)


\\(3.67\\) ft

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

Algebra Refresher 6.1.

Use the ideal conversion element to convert units.

\\(\\dfrac1~ \\textmile1.609~\\textkilometers = 1\\)

10 miles = km

50 kilometres = miles

\\(\\dfrac1~ \\textacre0.405~\\texthectare = 1\\)

40 acres = hectares

5 hectares = acres

\\(\\dfrac1~ \\texthorsepower746~\\textwatts = 1\\)

250 speech = watts

1000 watts = horsepower

\\(\\dfrac1~ \\texttroy ounce480~\\textgrains = 1\\)

0.5 trojan oz = grains

100 grains = 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\\) seed b. \\(0.21\\) trojan oz

Subsection ar 6.1 Summary

Subsubsection Vocabulary



Conversion factor


Unit circle

Subsubsection Concepts

The street we travel about a circle of radius is proportional to the edge of displacement.
\\beginequation*\\textbfArclength~ = ~ (\\textbffraction that one revolution) \\cdot (2\\pi r)\\endequation*

We measure angles in radians once we occupational with arclength.


The radian measure up of an angle is provided by

\\beginequation*(\\textbffraction of one revolution\\times 2\\pi)\\endequation*

An arclength equal to one radius determines a main angle the one radian.

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

Degrees\\(\\dfrac\\textRadians:\\textExact Values\\)\\(\\dfrac\\textRadians: Decimal\\textApproximations\\)

We main point by the ideal conversion factor to convert in between degrees and radians.

Unit Conversion for Angles.
\\beginequation*\\dfrac180\\degree\\pi~\\textradians = 1\\endequation*

To transform from radians to degrees we main point the radian measure by \\(\\dfrac180\\degree\\pi\\text.\\)

To convert from levels to radians we multiply the level measure through \\(\\dfrac\\pi180\\text.\\)

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

\\beginequation*s = r\\theta\\endequation*

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

Subsubsection study Questions

The size of a circular arc counts on what 2 variables?

Define the radian measure of one angle.

What is the conversion factor from radians to degrees?

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

Subsubsection Skills

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

Sketch angles provided 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 works with of a allude on a unit one #47–52

Calculate angular velocity and also area of a ar #55–60

Exercises Homework 6.1

Radians\\(0\\)\\(\\dfrac\\pi4\\)\\(\\dfrac\\pi2\\)\\(\\dfrac3\\pi4\\)\\(\\pi\\)\\(\\dfrac5\\pi4\\)\\(\\dfrac3\\pi2\\)\\(\\dfrac7\\pi4\\)\\(2 \\pi\\)

Convert every angle come degrees.

Sketch every angle top top a circle choose this one, and also label in radians.

Radians\\(0\\)\\(\\dfrac\\pi6\\)\\(\\dfrac\\pi3\\)\\(\\dfrac\\pi2\\)\\(\\dfrac2\\pi3\\)\\(\\dfrac5\\pi6\\)\\(\\pi\\)\\(\\dfrac7\\pi6\\)\\(\\dfrac4\\pi3\\)\\(\\dfrac3\\pi2\\)\\(\\dfrac5\\pi3\\)\\(\\dfrac11\\pi6\\)\\(2 \\pi\\)

Convert every angle come degrees.

Sketch each angle top top a circle choose this one, and also label in radians.

Exercise Group.

For troubles 3–6, refer each portion of one finish rotation in degrees and 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 difficulties 7–8, label each edge in standard place with radian measure.


Rotate counter-clockwise indigenous 0.


Rotate clockwise indigenous 0.

Exercise Group.

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


\\(\\displaystyle \\dfrac\\pi6\\)

\\(\\displaystyle \\dfrac5\\pi6\\)

\\(\\displaystyle \\dfrac7\\pi6\\)

\\(\\displaystyle \\dfrac11\\pi6\\)


\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac5\\pi4\\)

\\(\\displaystyle \\dfrac7\\pi4\\)


Locate and also label every angle from problem 9 top top the unit circle below. (The circle is marked off in tenths of a radian.)


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

Exercise Group.

From the list below, pick the ideal decimal approximation for each angle in radians in troubles 13–20. Do not usage a calculator; use the fact that \\(\\pi\\) is a small greater than 3.

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
















Exercise Group.

For difficulties 21–24, to speak in which quadrant every angle lies.


\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac5\\pi4\\)

\\(\\displaystyle \\dfrac7\\pi4\\)


\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac5\\pi4\\)

\\(\\displaystyle \\dfrac7\\pi4\\)


\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac5\\pi4\\)

\\(\\displaystyle \\dfrac7\\pi4\\)


\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac\\pi4\\)

\\(\\displaystyle \\dfrac5\\pi4\\)

\\(\\displaystyle \\dfrac7\\pi4\\)

Exercise Group.

For troubles 25–28, complete the table.





Exercise Group.

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


\\(\\displaystyle 75\\degree\\)

\\(\\displaystyle 236\\degree\\)

\\(\\displaystyle 327\\degree\\)


\\(\\displaystyle 138\\degree\\)

\\(\\displaystyle 194\\degree\\)

\\(\\displaystyle 342\\degree\\)

Exercise Group.

For troubles 31–32, convert to degrees. Ring to tenths.


\\(\\displaystyle 0.8\\)

\\(\\displaystyle 3.5\\)

\\(\\displaystyle 5.1\\)


\\(\\displaystyle 1.1\\)

\\(\\displaystyle 2.6\\)

\\(\\displaystyle 4.6\\)

Exercise Group.

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


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


Find the arclength extended by an edge of \\(200\\degree\\) ~ above a circle of radius 18 feet.


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


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


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


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

Exercise Group.

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


Through how numerous radians walk the minute hand that 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. How long is the arc traced through the minute hand in between 9:05 pm and 9:30 pm?


The largest clock ever created 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 the the clockface to be 112 feet.

If you started at the 12 and also walked 500 feet clockwise roughly the clockface, v how countless radians would you walk?

If you started your walk in ~ noon, just how long would certainly it take the minute hand to reach your position? How far did the reminder of the minute hand relocate in its arc?


In 1851 Jean-Bernard Foucault demonstrated the rotation the the earth with a pendulum mounted in the Pantheon in Paris. Foucault\"s pendulum consisted of a cannonball suspended on a 67 meter wire, and it swept out one arc that 8 meter on each swing. With what angle did the pendulum swing? give your price in radians and also then in degrees, rounded come the nearest hundredth.


A wheel through radius 40 centimeters is rolled a street of 1000 centimeters top top a level surface. V what angle has the wheel rotated? offer your answer in radians and then in degrees, rounded to one decimal place.


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

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

How quick are her socks traveling throughout the spin cycle?


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

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

What is the rate of the Hubble telescope?


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The very first large windmill offered to generate electrical power was built in Cleveland, Ohio in 1888. The sails to be 17 meter in diameter, and also moved in ~ 10 rotations every minute. How rapid did the ends of the sails travel?


The largest windmill operation today has wings 54 meters in length. Come be many efficient, the advice of the wings must travel at 50 meters every second. How quick must the wings rotate?

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