Section5.1
Angles and Their Measure
Angles
A ray is a part of a line that has only one endpoint
and extends forever in the opposite dire ction. A n angle
is form ed by tw o rays that have a com m on endpoint. O ne
ray is called the initial side and the other the term inal side.
A rotating ray is often a useful w ay to think about angles.
T he endpoint of an angle's initial side and term inal side is
the vertex of the angle.
A n angle is in standard pos ition if
1. its vertex is at the origin of a rectangular coordinate system
2. its initial side lies along the positive x-axis
P ositive angles are generated by counterclockw ise rotation.
T hus, angle  is positive. N egative angles are generated
by clockw ise rotation as you see angle  in the diagram .
A n angle is called quadran tal if its term inal side lies on the
x-axis or the y-axis. If a standard ang le has a term inal side
that lies in a quadrant then w e say that the angle lies in
that quadrant. A ngle  lies in quadrant II. A ngle  lies
in 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

30 15 '10 "

2 43 '18 "

22 31 '40 "

123 20 '9 "
• Convert to D-M-S
30 . 42
31 . 73


51 . 37

127 . 18

Measuring Angles Using
Radians
Example
W hat is the radian m easure of  for an ar c of
20 inches
length 20 inches and a radius of 5 inche s.
5 inches
Relationship between Degrees
and Radians
Example
C onvert each angle in degrees to radians .
a. 135
0
b. -120
0
c. -150
0
d. 90
0
e. 180
0
Example
C onvert each angle in radians to degrees.
a.

2
b. 
c. 

3
d.
5
6
e.
2
3
Top 10 #1-5
• Convert the following to degree measure


9
4
5
3
2
2
3
7
4
Top 10 #6-10
• Convert the following to radian measure
•
•
•
•
•
220°
315°
-90°
900°
-270°
Drawing Angles in Standard
Position
Angles Formed by Revolution of Terminal Sides
Example
D raw and label each angle in standard po sition.
a.  
3
2
b.  = 2 
c.  =
7
4
Degree and Angle Measures of
Selected Positive and Negative Angles
Coterminal Angles
Example
A ssum e the follow ing angles are in stand ard position.
0
F ind a positive angle less than 360 that is coterm inal
w ith each of the follow ing.
a. 390
b. 405
0
0
c. -135
0
Example
A ssum e the follow ing angles are in stand ard position.
F ind a positive angle less than 2  that i s coterm inal
w ith each of the follow ing.
5
a.
2
b.
1 1
4
c. -

6
Example
Find a positive angle less than 2  or 36 0 that is coterm inal
0
w ith each of the follow ing.
a. 765
b.
0
22 
6
c. -
19 
6
Find, and sketch the coterminal angle with…


13 
3
2
6
5
Complementary & Supplementary Angles
• Complementary angles

– Sum of the two angles = 90°; Or… 2
• Supplementary angles
– Sum of the two angles = 180° °; Or… 
Find the complement & Supplement of…

2
4
6
5
5
• Find the complement & supplement of…

3
2
4
8
7
The Length of a Circular Arc
Example
A circle has a radius of 7 inches. Find the length
0
of the arc intercepted by a central angle of 120 .
Example
A circle has a radius of 5 inches. Find the length
0
of the arc intercepted by a central angle of 150 .
Linear and Angular Speed
Example
A w indm ill in H olland is used to generate electricity.
Its blades are 12 feet in length. T he blades rotate
at eight revolutions per m inute. Find th e linear
speed, in feet per m inute of the tops of the blades.
Exit Slip
Convert from degrees to radians
1) 15
2) 120
3) 315
Convert from radians to degrees
4) 5
5) 7 
3
5
Draw each angle in standard position
6)  2 
7) 5
4
3
8)
8
3
9)

4
Find a positive or negative co-terminal angle with the following
10) 2 
11) 13 
3
4