AnglesAn angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side.

Aè

B

C

Terminal Side Initial Side

Vertex

Unit 4 Trigonometric Functions4.1 Angles and Their Measures (4.1)

Angles of the Rectangular Coordinate SystemAn angle is in standard position if• its vertex is at the origin of a rectangular coordinate system and• its initial side lies along the positive x-axis.

x

y

Terminal Side

Initial SideVertex

is positive

Positive angles rotate counterclockwise.

x

y

Terminal Side

Initial SideVertex

is negative

Negative angles rotate clockwise.

Complete Student CheckpointDraw each angle in standard position.a. 30° b. 210°

c. -120° d. 390°

30°

210°

-120° 390°

Measuring Angles Using DegreesThe figures below show angles classified by

their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º.

Acute angle0º < < 90º

90º

Right angle1/4 rotation

Obtuse angle90º < < 180º

180º

Straight angle1/2 rotation

Finding Complements and Supplements

For an xº angle, the complement is a 90º – xº angle. Thus, the complement’s measure is found by subtracting the angle’s measure from 90º.

For an xº angle, the supplement is a 180º – xº angle. Thus, the supplement’s measure is found by subtracting the angle’s measure from 180º.

In Navigation

In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north. For example, the line of travel in this figure has the bearing of 155º

Degree-Minute-Second (DMS) measure 42º24'36"

A degree, represented by the symbol º, is a unit of measure equal to 1/180th of a straight angle.

In the DMS system of angular measure, each degree is subdivided into:60 minutes, denoted by ꞌ,

1 60'

each minute is subdivided into:60 seconds, denoted by ꞌꞌ,

1' 60''

Convert 37.425 º to DMS.

0.425 60'

1

Convert 42º24'36" to degrees.

Need to convert the decimal part to minutes and seconds.First convert to minutes:

25.5'

Convert the decimal part to seconds:

0.5'60''

1'

30''

25' '37 30'

42 24'36'' 42 24' 361'

66 00' ''

1

60'

1''

42.41

Definition of a RadianOne radian is the measure of the central

angle of a circle that intercepts an arc equal in length to the radius of the circle.

Radian MeasureConsider an arc of length s on a circle or radius r. The measure of the central angle that intercepts the arc is = s/r radians

Or

s

r

A central angle, , in a circle of radius 12 feet intercepts an arc of length 42 feet. What is the radian measure of ?

s

r

42 ft

12 ft

3.5

Conversion between Degrees and Radians radians = 180º

To convert degrees to radians: (degrees)

To convert radians to degrees: (radians)

Convert each angle in degrees to radians:

40º -160º

75º

140

80

29

175

80

512

160180

89

radians

180

180

radians

Complete Student CheckpointConvert each angle in degrees to radians. a. 60º b. 270º c. -300

Convert each angle in radians to degrees. a. b. c. 6 radians

4

43

160

80

3

1270

80

32

300180

53

1 0

4

8

45o

804 1

3

240o

06

18

343.8o

Length of a Circular ArcLet r be the radius of a circle and the non-negative radian measure of a central angle of the circle. The length of the arcintercepted by the central angle is:

s = r

O

s

r

A circle has a radius of 7 inches. Find the length of the arc intercepted by a central angle of 2/3

Solution: s 7 inches 2

3

s

143

inches

Complete Student CheckpointFind the perimeter of a 45º slice of a medium (6 in. radius) pizza.

456 inches180

s

s 4.71 inches

must convert

degrees to radians

for the formulas = r

Converting to Nautical MilesA nautical mile, represented by naut mi, is the length of 1 minute of arc along the Earth’s equator (radius ≈3956 statute miles).

1' 1'60' 80

1

1

Applying the formula s = r :

radians10,800

(3956 stat m1 nau i)t mi10,800

1.15 stat mi

10,800 naut mi

391 sta

5t m

6 i

0.87 naut mi

The distance from Boston to San Francisco is 2698 stat mi. How many nautical miles is it from Boston to San Francisco?

2698 stat mi10,800

naut mi3956 stat mi

0.8or

7 n 269

aut mi8 sta

1 stat mi

t mi

2345 naut mi

Angles and Their Measure