4.1 radian and degree measure changing degrees to radians linear speed angular speed

4.1 Radian and 4.1 Radian and Degree measure Degree measure

Changing Degrees to RadiansChanging Degrees to Radians

Linear speedLinear speed

Angular speedAngular speed

Definition of an angle

An angle is made from two rays with a common initial point.

In standard position the initial side is on the x axis

side

Initial

side

alTer min

Positive angle vs. Negative angle

Positive angles are Counter clockwise C.C.W.

Negative angles are Clockwise C.W.

Angles with the same initial side and terminal side are coterminal.

The measure of an angle is from initial side to terminal side

Vertex at the origin (Center)

r

r

Angle

Central

Definition of a Radian

Radian is the measure of the arc of a unit circle.

Unit circle is a circle with a radius of 1.

The quadrants in terms of Radians

What is the circumference of a circle with radius 1?

The quadrants in terms of Radians

What is the circumference of a circle with radius 1? 2

1

20

The quadrants in terms of Radians

The circumference can be cut into parts.

1

20

2

2

3

The quadrants in terms of Radians

The circumference can be cut into parts.

1

20

2

2

3

I

20

II

2

III

2

3

IV

2

2

3

Find the Coterminal Angle

Since equals 0. it can be added or subtracted from any angle to find a coterminal angle.

Given

2

4

3

4

52

4

3

4

112

4

3

Complementary Angles – two angles are complementary

if their sum is 90 degrees or .2

Supplementary Angles have a sum of 180 degrees or .

Find the complementary and supplementary angles for

5

2

5

2

2

1010

4

10

5

5

25

3

5

2

5

5

Radian vs. Degree measurements

360º =

180º =

So or

2

rad180

1

Radian vs. Degree measurements

360º =

180º =

So or

To convert Degrees into Radians multiply by

To convert Radians into Degrees multiply by

2

rad180

1

180

1 rad

180

180

Conversions: Radians Degrees

To convert degrees to radians, multiply by 180

rad

To convert radians to degrees, multiply by

rad

180

Converting an angle from to decimal form. "'SMD

"29'15152

3600

29

60

15152

25806.152

Change 140º to RadiansChange to degrees

Use degree to rads.

Use rads to degrees

3

7

180

443460953.29

7

180

140

180*140

180

4203

1260180*

3

7

How to use radian to find Arc length

The geometry way was to find the circumference of the circle and multiply by the fraction. Central angle

360º

In degrees Are length called S would be

rS 2

360

How to use radian to find Arc length

In degrees Are length called S would be

In radian the equation is

rS 2

360

rS

r = 9, θ = 215º Changing to rads

Are length S

r

36

43

180215

936

43

S

772.334

43

S

Linear speed and Angular speed

Linear speed is

Angular speed is

Assuming “constant speed”

t

S

time

lengtharc

ttime

angleCentral

Linear and Angular Speeds

Consider a particle moving at a constant speed along a circular arc ofradius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is

arc length

times

tLinear speed v

Moreover, if

is the angle (in radian measure) corresponding to the arc

length s, then the angular speed

(the lowercase Greek letter omega)

of the particle is

Angular speed

central angle

time

t

A relationship between linear speed and angular speed is

v r

Area of a Sector of a Circle

A1

2r2

where is measured in radians

A sprinkler on a golf course fairway is set to spray water over a distanceof 70 feet and rotates through an angle of 120 degrees. Find the area of the fairway watered by the sprinkler.

12070 ft

120o = how many radians?

120 23radians

A1

270 2 2

3

49003

5131 ft 2

79-99 odd, 107

Finding Linear Speed

The second hand of a clock is 10.2 cm long. Find the linear speed of theTip of the second hand as it passes around the clock face.

arc length

times

tLinear speed v

Arc length s

2r

2 (10.2)20.4 cm

,v s

t20.4 cm60 sec

=1.068 cm/sec