Section5.1Angles and Their Measure

Angles

A ray is a part of a line that has only one endpoint and extends forever in the opposite direction. An angleis formed by two rays that have a common endpoint. One ray is called the initial side and the other the terminal side.A rotating ray is often a useful way to think about angles. The endpoint of an angle's initial side and terminal side isthe vertex of the angle. An angle is in standard position if1. its vertex is at the origin of a rectangular coordinate system2. its initial side lies along the positive x-axis

Positive angles are generated by counterclockwise rotation. Thus, angle is positive. Negative angles are generatedby clockwise rotation as you see angle in the diagram. An angle is called quadran

tal if its terminal side lies on the x-axis or the y-axis. If a standard angle has a terminal side that lies in a quadrant then we say that the angle lies in that quadrant. Angle lies in quadrant II. Angle liesin quadrant III.

Measuring Angles Using Degrees

Names of Angles

Degree-Minute-Second• 1 Degree = 60 Minutes• 1 Minute = 60 seconds

• Thus»1° = 60’»1’ = 60”»1° = 3600”

• Convert 50°6’21” to a decimal in degrees

• Convert 40°10’25” to a decimal in degrees

• Convert 73°40’40” to a decimal in degrees

• Convert 21.256° to degree-minute-second

• Convert 18.255° to degree-minute-second

• Convert 29.411° to degree-minute-second

5 Min Challenge• Convert to a decimal

• Convert to D-M-S

"10'1530

"9'20123"18'432

"40'3122

42.30 37.5118.12773.31

Measuring Angles Using Radians

Example

What is the radian measure of for an arc of length 20 inches and a radius of 5 inches.

20 inches

5 inches

Relationship between Degrees and Radians

Example

0

0

0

0

0

Convert each angle in degrees to radians.

a. 135

b. -120

c. -150

d. 90

e. 180

ExampleConvert each angle in radians to degrees.

a. 2

b.

c. 3

5d. 6

2e. 3

Top 10 #1-5

• Convert the following to degree measure

2

2

93

47

54

3

• Convert the following to radian measure

• 220°• 315°• -90°• 900°• -270°

Top 10 #6-10

Drawing Angles in Standard Position

Angles Formed by Revolution of Terminal Sides

Example

Draw and label each angle in standard position.3a. 2

b. =2

7c. =4

Degree and Angle Measures of Selected Positive and Negative Angles

Coterminal Angles

Example

0

0

0

0

Assume the following angles are in standard position.

Find a positive angle less than 360 that is coterminalwith each of the following.

a. 390

b. 405

c. -135

ExampleAssume the following angles are in standard position. Find a positive angle less than 2 that is coterminalwith each of the following.

5a. 2

11b. 4

c. -6

Example

0

0

Find a positive angle less than 2 or 360 that is coterminalwith each of the following.

a. 765

22b. 6

19c. -6

Find, and sketch the coterminal angle with…

2

6

135

3

Complementary & Supplementary Angles• Complementary angles

– Sum of the two angles = 90°; Or…

• Supplementary angles– Sum of the two angles = 180° °; Or…

Find the complement & Supplement of…

2

6

52

54

• Find the complement & supplement of…

4

72

83

The Length of a Circular Arc

Example

0

A circle has a radius of 7 inches. Find the length

of the arc intercepted by a central angle of 120 .

Example

0

A circle has a radius of 5 inches. Find the length

of the arc intercepted by a central angle of 150 .

Linear and Angular Speed

ExampleA windmill in Holland is used to generate electricity. Its blades are 12 feet in length. The blades rotate at eight revolutions per minute. Find the linear speed, in feet per minute of the tops of the blades.

Exit SlipConvert from degrees to radians1) 15 2) 120 3) 315

Convert from radians to degrees4) 5)

Draw each angle in standard position6) 7)

8) 9)

Find a positive or negative co-terminal angle with the following10) 11)

35

57

32

4

5

4

38

32

413