Radian measure. - University of Wisconsin–Eau Claire€¦ · Radian measure Today’s goal is to...

Radian measure.

Radian measure. 1 / 12

Measuring angles

When someone talks about an angle of 1◦ how do you know what theymean?

You start by remembering that 360◦ corresponds to a full revolution.

So 1◦ should be one 360’th of a full revolution.

This is 1◦.

Zoomed in

Does this seem arbitrary to you? Why is 360◦ a full revolution?


Measuring angles

When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.


This is 1◦.

Zoomed in



Measuring angles

When someone talks about an angle of 1◦ how do you know what theymean?You start by remembering that 360◦ corresponds to a full revolution.


This is 1◦. Zoomed in



Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.

Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans

Distance=s.s radians

Definition

An arc of length s on a circle of radius r covers an angle of sr radians.

Example: What are the radian measures of these angles?

2 cm

2 cm

1.5 cm

6 cm


Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.

You have now covered an angle of s radi-ans


Definition



2 cm

2 cm

1.5 cm

6 cm


Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.

You have now covered an angle of s radi-ans

Distance=s.

s radians

Definition



2 cm

2 cm

1.5 cm

6 cm


Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans

Distance=s.

s radians

Definition



2 cm

2 cm

1.5 cm

6 cm


Radian measureToday’s goal is to develop a measure of angles which is less arbitrary.Imagine you are standing on a circle ofradius 1 foot.Start walking counter-clockwise. Stopwhen you have covered s feet.You have now covered an angle of s radi-ans


Definition



2 cm

2 cm

1.5 cm

6 cm


Converting between radians and degrees

Definition

An arc of length s on a circle of radius r covers an angle ofs

rradians.

If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.

Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .

This angle measures2πr

r=

2π

radians.

360◦ is the same as 2π radians.



Definition


rradians.

If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.

If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .


r=

2π

radians.




Definition


rradians.

If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?

s = 2πr .


r=

2π

radians.




Definition


rradians.

If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .


r=

2π

radians.




Definition


rradians.

If you know the degree measure of an angle, can you find its radianmeasure?Let’s start with 360◦.Drawing a 360◦ arc on a circle of radius r is the same asdrawing a whole circle.If the radius is r , then what is the length of this arc? In otherwords, What is the circumference of the circle of radius r?s = 2πr .


r= 2π radians.




So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?

How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.

So the radian measure of a 1◦ arc isarc length

radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦

A d◦ angle is made of d angles of measure 1◦. These will total d · π

180radians.Concept check: Use this to compute radian measure of a 30◦ angle.



So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.

360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.


radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦





So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.

So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.


radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦





So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?

2πr/360 = πr/180.


radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦





So far: Since a 360◦ arc on a radius r circle has length = 2πr , 360◦ isequal to 2π radians.What about a 1◦ arc?How many of these does it take to fill a 360◦ arc? a silly question.360 of them.So 360 · (The arc length of a 1◦ arc) = 2πr .What is the arc length of a 1◦ arc?2πr/360 = πr/180.


radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦






So the radian measure of a 1◦ arc is

arc length

radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦







radius=

πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦







radius=πr/180

r=

π

180.

1◦ =π

180radians.

What about d◦





Theorem

To convert from degrees to radians, multiply by π180 .

To convert from radians to degrees multiply by 180π .

Here is a right angle, which measures 90◦.What is its radian measure?


Arc length and sector area

Suppose you have a blueberry pie or radius rand you cut out a piece whose angle is π/4radians.Let’s find out how much area this piece of piehas.

How many of these pieces are in the whole ofthe pie? How many times does π

4 fit into2π? 2π

π/4 =

8

There are eight pieces like this in the wholepie, whose total area is:

πr2

.

The area of this piece iswhole area

8=

πr2

8

Next slide: What if we cut a different angle? Then how much area do wehave?




How many of these pieces are in the whole ofthe pie?

How many times does π4 fit into

2π? 2ππ/4 =

8


πr2

.


8=

πr2

8






4 fit into2π?

2ππ/4 =

8


πr2

.


8=

πr2

8






4 fit into2π? 2π

π/4 =

8


πr2

.


8=

πr2

8






4 fit into2π? 2π

π/4 = 8


πr2

.


8=

πr2

8






4 fit into2π? 2π

π/4 = 8

There are eight pieces like this in the wholepie, whose total area is: πr2.


8=

πr2

8






4 fit into2π? 2π

π/4 = 8

There are eight pieces like this in the wholepie, whose total area is: πr2.


8= πr2

8



Arc length and sector areaStart with a circle of radius r . Start at the center and draw an angle of θradians. How much area is in this sector?

We know that the area of the whole circle is 2π.

How many times can you fit this sector into the whole circle?

number of radians in a full rotation

θ=

2π

θ

The area of one copy of the sector is

area of circle

number of sectors in the circle=

πr2

2π/θ=

r2θ

2

Theorem

If you start with a circle of radius r and draw a sector of angle θ radiansthen the area of that sector is r2θ

2 .


sector area and radian measure

Theorem

If you start with a circle of radius r and draw a sector of angle θ then thearea of that sector is r2θ

2 .

Exercises :

1 If you make an angle of 1 radian in a circle or radius 1 then howmuch area is in its sector? r2·θ

2 =

1·12 =

12

2 Draw a right angle in a circle of radius 2. How much area is in itssector? r2·θ

2 =

4·π/42 =

π/2

3 If you make an angle of 1◦ in a circle of radius 20 then how mucharea is in its sector? (You need to convert to radians)r2·θ2 =

400·1· π180

2 =

10π9



Theorem


2 .

Exercises :


2 = 1·12 =

12


2 =

4·π/42 =

π/2


400·1· π180

2 =

10π9



Theorem


2 .

Exercises :


2 = 1·12 = 1

2


2 =

4·π/42 =

π/2


400·1· π180

2 =

10π9



Theorem


2 .

Exercises :


2 = 1·12 = 1

2


2 = 4·π/42 =

π/2


400·1· π180

2 =

10π9



Theorem


2 .

Exercises :


2 = 1·12 = 1

2


2 = 4·π/42 = π/2


400·1· π180

2 =

10π9



Theorem


2 .

Exercises :


2 = 1·12 = 1

2


2 = 4·π/42 = π/2


400·1· π180

2 = 10π9


Angular speed and linear speed

Definition

Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a second

The linear speed is given byv = how fast a point on the edge of the wheel is moving

How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1

3 = 2/3.In one second the point has travelled 2/3 feet.

The linear speed is 2/3.

Theorem

The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per time.



Definition

Suppose that you have a wheel rotating about its axis.The angular speed of the wheel rotation (in radians per second) is givenby ω = number of radians the wheel rotates through in a secondThe linear speed is given byv = how fast a point on the edge of the wheel is moving




Theorem




Definition


How are these related?

Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1



Theorem




Definition


How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.

In one second a point on the circle traces a 2 radian arcof radius 1/3.The length is ` = θ · r = 2 · 1



Theorem




Definition


How are these related?Suppose a gear of radius 1/3 feet is spinning at 2 radiansper second. Find its linear speed.In one second a point on the circle traces a 2 radian arcof radius 1/3.

The length is ` = θ · r = 2 · 13 = 2/3.

In one second the point has travelled 2/3 feet.


Theorem




Definition



3 = 2/3.

In one second the point has travelled 2/3 feet.


Theorem




Definition



3 = 2/3.In one second the point has travelled 2/3 feet.The linear speed is 2/3.

Theorem




Theorem

The linear and angular speed are related by v = ω · r where r is the radiusand ω is measured in radians per second or radians per minute or radiansper hour.

If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?

Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is

ω = 6 revolutions per minute= 6 · 2π radians per minute= 12π radians per minute

Step 2: Get linear velocity:

v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute = 1 foot/second



Theorem


If a gear of radius 5 inches is spinning at 6 revolutions per minute. Whatis its linear speed?Step 1: Compute angular velocity in radians per minute. One revolution is2π radians: The angular velocity is

ω = 6 revolutions per minute=

6 · 2π radians per minute= 12π radians per minute





Theorem



ω = 6 revolutions per minute= 6 · 2π radians per minute=

12π radians per minute





Theorem








Theorem





v = (radius) · ω =

5 · 12 feet/minute = 60 feet/minute = 1 foot/second



Theorem





v = (radius) · ω = 5 · 12 feet/minute =

60 feet/minute = 1 foot/second



Theorem





v = (radius) · ω = 5 · 12 feet/minute = 60 feet/minute =

1 foot/second



Theorem







exit quiz:

A gear of radius 2 is spinning 30 degrees per minute.

1 What is its angular speed in radians per minute?

2 What is its linear velocity?

3 This gear is connected to another gear of radius 1 so that is has thesame linear velocity. What is its angular velocity, in radians perminute and in degrees per minute.


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