The Beginnings of The Beginnings of Chapter 4Chapter 4

Section 4.1a: Angles, Section 4.1a: Angles, their measures,and their measures,and

arc length arc length

First, Some DefinitionsDegree – represented by the symbol , is a unit ofangular measure equal to 1/180th of a straight angle.

Note: Each degree is subdivided into 60 minutes(denoted by ’ ), and each minute is subdivided into60 seconds (denoted by ” ).

This is called the DMS form for angle measure.

Conversion ProblemsConvert each of the following from DMS to decimal form.

42 24 36 Each minute is 1/60th of a degree, and each second is 1/3600th

of a degree:24 36

42 24 36 4260 3600

42.41

60 5 56 5 56

60 5 56 6060 3600

60.099

Conversion ProblemsConvert the following from decimal to DMS form.

37.425We need to convert the fractional part to minutes and seconds.First, convert the 0.425 degrees to minutes:

600.425

1

25.5

Then, convert 0.5 minutes to seconds:

600.5

1

30

Final Answer: 37.425 37 25 30

RadiansSo, what’s the problem with degrees, anyway???

Degree units have no mathematical relationshipDegree units have no mathematical relationship whatsoever to linear units!!!whatsoever to linear units!!!

EX: There are 360 degrees in a circle with radius 1…

What relationship does the 360 have to the 1???

Is it 360 times as big???

Enter RADIANS to help solve these dilemmas…Enter RADIANS to help solve these dilemmas…

Definition: Radians

A central angle of a circle has measure 1 radianif it intercepts an arc with the same length as theradius.

a a

1 radian

Degree-Radian Conversion

To convert radians to degrees, use the conversionfactor:

πNote: radians and 180 both measure a straight angle!!!

We will use dimensional analysis to convert all angles.

180 radians

Guided Practice

rad90

180

How many radians are in 90 degrees?

radians90

180

radians

2

How many degrees are in radians?

180rad

3 rad

180

3

60

3

More DefinitionsIn navigation, the course or bearing of an object issometimes given as the angle of the line of travelmeasured clockwise from due north.Ex: Sketch a diagram of the path of a boat leaving a harborwith a bearing of 155

Harbor

Pathof boat

155

s r

If 0 is a central angle in a circle of radius r,and if 0 is measured in radians, then the lengths of the intercepted arc is given by

arc length (linear unit of measure)

= measure of angle of rotation in radians

= radius (linear unit of measure)

s

r

180

rs

If 0 is a central angle in a circle of radius r,and if 0 is measured in degrees, then the lengths of the intercepted arc is given by

sUse the appropriate arc length formula to find the missing information.

r 1 cm 70 rad70cm

7.5 ft 3 rad2.5ft

7 in 4 7rad4 in

5 m 182 m

7 60

180s

Find the perimeter of a 60 slice of a large (7 in. radius)pizza.

Delicious slice of pizza:

60

7 in

7 in s in

Perimeter:

7 in + 7 in + s in

Find s :

77.330

3

The perimeter is approximately 21.330 inchesThe perimeter is approximately 21.330 inches

The running lanes at a certain track are 1 meter wide. Theinside radius of lane 1 is 33 meters and the inside radius oflane 2 is 34 meters. How much longer is lane 2 than lane1 around one turn?

33 m

34 m

Lane 1Lane 2

Each lane is a semicircle withcentral angle andlength .s r r

Therefore, the difference intheir lengths is

34 33 Lane 2 is about 3.142 meters longer than lane 1Lane 2 is about 3.142 meters longer than lane 1..

5

It takes ten identical pieces to form a circular track for apair of toy racing cars. If the inside arc of each piece is3.4 inches shorter than the outside arc, what is the widthof the track?

What is the measure of each ofthese central angles?

Inside arc length for one pieceof track: 5ir Outside arc length for one pieceof track: 5or

It takes ten identical pieces to form a circular track for apair of toy racing cars. If the inside arc of each piece is3.4 inches shorter than the outside arc, what is the widthof the track?

5ir But we were given the difference between these arc lengths:

5or

5 5 3.4o ir r 3.4 5o ir r

5.411 inches

The concentric circles on an archery target are 6 inchesapart. The inner circle (red) has perimeter of 37.7 inches.What is the perimeter of the next-largest (yellow) circle?

6 6d

37.7d Perimeter of inner circle:

6 6d Perimeter of outer circle:

12d 37.7 12

75.399 inches

Whiteboard Conversion Problems

Convert each of the following from DMS to decimal form.

34 30 18 30 18

34 30 18 3460 3600

34.505

119 15 37 15 37

119 15 37 11960 3600

119.260

Whiteboard Conversion ProblemsConvert each of the following from decimal to DMS form.

10.98Convert the 0.98 degreesto minutes: 60

0.98 58.81

10 58 48 Convert the 0.8 minutesto seconds:

600.8 48

1

25.29

Convert the 0.29 degreesto minutes:

600.29 17.4

1

25 17 24 Convert the 0.4 minutesto seconds:

600.4 24

1

Whiteboard Practice

150

Convert each of the following from DMS to radians.

(a)5

rad180 6

(b) 75 30 75 30 60 75.5

75.5 1.318rad180

Whiteboard Practice

7

10

Convert each of the following from radians to degrees.

(a) 180126

(b) 1.3180

74.485

Convert from DMS to radians

11.83180

0.206 rad

Convert from radians to degrees

13

20

180

117