Front Matter1 Triangles and also Circles2 The Trigonometric Ratios3 laws of Sines and Cosines4 Trigonometric Functions5 Equations and Identities6 Radians7 one Functions8 more Functions and also Identities9 Vectors10 Polar works with and facility Numbers
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Imagine the you space riding top top a Ferris wheel of radius 100 feet, and also each rotation bring away eight minutes. We deserve to use angles in standard place to define your ar as girlfriend travel about the wheel. The number at right mirrors the locations shown by \(\theta = 0\degree,~ 90\degree,~ 180\degree,\) and \(270\degree\text.\) yet degrees are not the only method to specify place on a circle.

You are watching: How many radians are in a revolution


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

Another useful technique uses street traveled, or arclength, along the circle. How much have you traveled around the Ferris wheel at each of the locations shown?

Subsection Arclength

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

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

If us walk around the whole circumference of a circle, the distance we take trip is \(2\pi\) times the length of the radius, or around 6.28 time the radius. If us walk only component of the means around the circle, climate the distance we travel depends also on the edge of displacement.

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


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

Similarly, the edge of displacement from suggest \(A\) to suggest \(C\) is \(\dfrac34\) the a finish revolution, so the arclength along the circle indigenous \(A\) come \(C\text,\) presented at right, is


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

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

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

The Ferris wheel in the arrival has circumference

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

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

To indicate the same 4 locations top top the wheel by distance 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 size of arc is extended by an edge of \(120\degree\) top top a circle of radius 12 centimeters?


Because \(\dfrac120360 = \dfrac13\text,\) an edge of \(120\degree\) is \(\dfrac13\) of a finish revolution, as presented at right.

Using the formula over with \(r = 12\text,\) we discover 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 edge of a Ferris wheel the radius 100 feet as soon as you have actually turned v an edge of \(150\degree\text?\)


\(261.8\) ft

Subsection Measuring angles in Radians

If girlfriend think about measuring arclength, friend will check out that the level measure that the spanning angle is not as essential as the portion of one revolution it covers. This observation suggests a new unit of measurement for angles, one the is far better suited to calculations involving arclength. We"ll make one readjust 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 speak to the amount in parentheses, (fraction of one transformation \(\times 2\pi\)), the radian measure of the angle the spans the arc.


The radian measure of an edge is offered by

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

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

Example 6.3.

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


An angle of \(120\degree\) is \(\dfrac13\) that a finish revolution, as we witnessed in the vault 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 revolution is \(\pi\) radians? How plenty of degrees is that?


Half a revolution, \(180\degree\)

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

DegreesRadians:Exact ValuesRadians: DecimalApproximations

Note 6.5.

You have to memorize both the specific values of these angle in radians and 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 at the number above. The second quadrant has angles between \(\dfrac\pi2\) and also \(\pi\text,\) or 1.57 and 3.14 radians, so 2 radians lies in the second quadrant. An angle of 5 radians is in between 4.71 and 6.28, or in between \(\dfrac3\pi2\) and also \(2\pi\) radians, so it lies in the 4th quadrant.

Checkpoint 6.7.

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


Subsection Converting in between Degrees and Radians

It is not challenging to transform the measure of an edge in levels to its measure in radians, or vice versa. One complete transformation is same to 2 radians or come \(360\degree\text,\) so

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

Dividing both political parties of this equation by 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 convert from levels to radians we multiply the level measure through \(\dfrac\pi180\text.\)

Example 6.9.

Convert 3 radians come degrees.

Convert 3 levels 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 \(60\degree\) come radians. Give both precise answer and also an approximation to three decimal places.

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


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

\(\displaystyle 135\degree\)

From our conversion variable we additionally learn that

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

So when \(1\degree\) is a reasonably small angle, 1 radian is much larger — virtually \(60\degree\text,\) in fact.

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

We"ll soon see that, for many applications, that is simpler to work completely in radians. Because that reference, the figure listed below shows a radian protractor.

Subsection Arclength Formula

Measuring angles in radians has the following advantage: To calculate an arclength we need only main point the radius of the one by the radian measure of the spanning angle, \(\theta\text.\) look at again in ~ our formula for arclength:

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

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

Arclength Formula.

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

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

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

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

Example 6.11.

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


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

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

To convert the angle measure up to degrees, we multiply through \(\dfrac180\degree\pi\) to get

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

Your latitude has readjusted by around \(7.23\degree\text.\)

Checkpoint 6.12.

The distance around the face of a big 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, \(r = 1\text,\) so the arclength formula i do not care \(s = \theta\text.\) Thus, on a unit circle, the measure of a (positive) angle in radians is same to the size of the arc the spans.

Example 6.13.

You have walked 4 miles approximately a one pond the radius one mile. What is your position relative to your beginning point?


The pond is a unit circle, for this reason you have actually traversed an edge in radians equal to the arc size traveled, 4 miles. An edge of 4 radians is in the middle of the 3rd quadrant loved one to your starting point, more than halfway yet less 보다 three-quarters around the pond.

Checkpoint 6.14.

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


\(3.67\) ft

Review the following skills you will need for this section.

Algebra Refresher 6.1.

Use the ideal conversion aspect to transform units.

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

10 mile = km

50 km = miles

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

40 acre = hectares

5 hectares = acres

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

250 speech = watts

1000 watt = horsepower

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

0.5 troy oz = grains

100 seed = trojan oz


Algebra Refresher Answers

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

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

a. \(186,500\) watt 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 approximately a one of radius is proportional come the angle of displacement.
\beginequation*\textbfArclength~ = ~ (\textbffraction the one revolution) \cdot (2\pi r)\endequation*

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


The radian measure of an angle is offered by

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

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

Radian measure deserve to be expressed as multiples that \(\pi\) or together decimals.

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

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

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

To convert from radians to degrees we multiply the radian measure up by \(\dfrac180\degree\pi\text.\)

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

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

\beginequation*s = r\theta\endequation*

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

Subsubsection examine Questions

The length of a circular arc depends on what 2 variables?

Define the radian measure of an angle.

What is the conversion variable from radians come degrees?

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

Subsubsection Skills

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

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

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

Use the arclength formula #33–46

Find collaborates of a suggest on a unit circle #47–52

Calculate angular velocity and also area that 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 each angle to degrees.

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

Exercise Group.

For problems 3–6, refer each fraction 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 difficulties 7–8, label each edge in standard position with radian measure.


Rotate counter-clockwise from 0.


Rotate clockwise indigenous 0.

Exercise Group.

For difficulties 9–10, offer a decimal approximation come hundredths because that 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 ~ above the unit one below. (The circle is significant off in tenths of a radian.)


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

Exercise Group.

From the list below, select the ideal decimal approximation because that each edge in radians in troubles 13–20. Execute not use a calculator; usage the truth 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 problems 21–24, speak in i beg your pardon quadrant each 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 difficulties 25–28, complete the table.





Exercise Group.

For difficulties 29–30, convert to radians. Ring 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 difficulties 33–37, use the arclength formula to answer the questions. Round answers to hundredths


Find the arclength covered by an edge of \(80\degree\) ~ above a one of radius 4 inches.


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


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


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


Find the edge subtended by an arclength that 28 centimeters top top a one of diameter 20 centimeters.


Find the edge subtended by one arclength that 1.6 yards ~ above a one of diameter 2 yards.

Exercise Group.

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


Through how plenty of radians go the minute hand that a clock sweep in between 9:05 pm and 9:30 pm?

The dial of huge 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?


The biggest clock ever built was the Floral Clock in the garden the the 1904 World"s same 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 about the clockface, with how numerous radians would certainly you walk?

If you began your walk in ~ noon, how long would it take the minute hand to reach your position? How far did the pointer of the minute hand move in that arc?


In 1851 Jean-Bernard Foucault prove the rotation that the planet with a pendulum set up in the Pantheon in Paris. Foucault"s pendulum contained a cannonball rely on a 67 meter wire, and also it brushed up out an arc of 8 meters on each swing. V what angle did the pendulum swing? give your price in radians and then in degrees, rounded come the nearest hundredth.


A wheel with radius 40 centimeters is rolled a distance of 1000 centimeters on a level surface. With what angle has the wheel rotated? give your prize in radians and then in degrees, rounded come one decimal place.


Clothes dryers draw 3.5 times as lot power as washing machines, so newer machines have actually been engineered for greater efficiency. A vigorous spin cycle reduce the time necessary for drying, and some front-loading models spin in ~ a rate of 1500 rotations per minute.

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

How rapid are her socks traveling during the turn cycle?


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

How much does the telescope take trip in one hour? (The radius the 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 electrical power was constructed in Cleveland, Ohio in 1888. The sails to be 17 meters in diameter, and moved at 10 rotations per minute. How quick did the end of the sails travel?


The biggest windmill operating today has actually wings 54 meter in length. Come be most efficient, the advice of the wings need to travel in ~ 50 meters per second. How quick must the wings rotate?

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