Lecture Two – Trigonometric
Section 2.1 – Degrees, Radians, Angles and Triangles
Basic Terminology
Two distinct points determine line AB.
Line segment AB: portion of the line between A and B.
Ray AB: portion of the line AB starts at A and continues through B, and past B.
A
B
Angles in General
B
An angle is formed by 2 rays with the same end point.
The two rays are the sides of the angle.
Terminal side
O
Angle  = AOB

Initial side
O is the common endpoint and it is called vertex of the angle
An angle is in a Counterclockwise (CCW) direction: positive angle
An angle is in a Clockwise (CW) direction: negative angle
Type of Angles: Degree
1
A
Complementary angles:  +  = 90
Supplementary angles:  +  = 180
Example
Give the complement and the supplement of each angle: 40 110
Solution
a. 40
Complement: 90 - 40 = 50
Supplement:180 - 40 = 140
b. 110
Complement: 90 - 110 = -20
Supplement:180 - 110 = 70
Degrees, Minutes, Seconds
1: 1 degree
1  60
1 : 1 minute
1  60
1 : 1 second
1  3600
1 full Rotation or Revolution = 360
1  60  3600


1  1  1
60
3600
  
Example
Change 27.25 to degrees and minutes
Solution
27.25 = 27 + .25
= 27 + .25(60)
= 27 + 15
= 27 15
Example
Example
Add 48 49 and 72 26
Subtract 24 14 and 90
Solution
Solution
49
26
120 75
90
89 60
 24 14   24 14
65 46
48
 72
120 75 = 120 60+15
= 121 15
2

Angles in Standard Position
An angle is said to be in standard position if its initial side is along the positive x-axis and its vertex is at
the origin.
If angle  is in standard position and the terminal side of  lies in quadrant I, then we say  lies in QI
  QI
If the terminal side of an angle in standard position lies along one of the axes (x-axis or y-axis), such as
angles with measures 90, 180, 270, then that called a quadrantal angle.
Two angles in standard position with the same terminal side are called coterminal angles.
Q II
90
QI
( , +)
(+ , +)
180
0
360
-90
Q III
( , )
270
Q IV
(+ , )
Example
Find all angles that are coterminal with 120.
Solution:
120 + 360k
Example
Find the angle of least possible positive measure coterminal with an
angle of 908.
Solution
908  2.360  188
An angle of 908 is coterminal with an angle of 188
3
Example
CD players always spin at the same speed. Suppose a Constant Angular Velocity player makes 480
revolutions per minute. What degrees will a point on the edge of a CD spins for 2 seconds?
Solution
The player revolves 480 times in one minute  480  480  8 times per sec.
1
60
In 2 sec, the CD will spin: 2.8 = 16 times
Therefore; CD will revolve 16.360  5760
Triangles
Equilateral – All angles always equal to
Isosceles: 2 sides and angles are equals
60& all sides are equals
Scalene: No equal sides or angles
Right: Has a right angle 90.
Obtuse: Has an angle more than 90.
Acute: All angles are less than 90.
4
Radians
Degrees - Radians
 measures
  2
one full rotation
The measure of 
in radians is 2
1  1 rad
1 = 1 degree
If no unit of angle measure is specified, then the angle is to be measured in radians.
Full Rotation : 360  2 rad
180   rad
Converting from Degrees to Radians
180   rad
180 180
 1   rad
180
Multiply a degree measure by  rad and simplify to convert to radians.
180
Example
Convert 45 to radians
Solution
 
45  45  rad
180
  rad
4
5
Example
Convert -450 to radians
Solution
 
450  450  rad
180
  5 rad
2
Example
Convert 249.8 to radians
Solution
 
249.8  249.8  rad
180
 4.360 rad
Converting from Radians to Degrees
Multiply a radian measure by 180 radian and simplify to convert to degrees.

180   rad


 180 

 1 rad
Example
Convert 1 to degrees
Solution
   


1 rad  1 180  1 180  57.3
3.14

Example
Example
Convert 4 to degrees
Convert 4.5 to degrees
3
Solution
Solution
 
 
4  4 180   240
3
3 
4.5  4.5 180
6

 257.8
Exercises
Section 2.1– Degrees, Radians, Angles and Triangles
1.
Indicate the angle if it is an acute or obtuse. Then give the complement and the supplement of each
angle.
a) 10
b) 52
c) 90
d) 120
e) 150
2.
Change 10 45 to decimal degrees.
3.
Convert 34 51 35 to decimal degrees.
4.
Convert 274 18 59 to decimal degrees.
5.
Change 74 8 14 to decimal degrees to the nearest thousandth.
6.
Convert 89.9004 to degrees, minutes, and seconds.
7.
Convert 34.817 to degrees, minutes, and seconds.
8.
Convert 122.6853 to degrees, minutes, and seconds.
9.
Convert 178.5994 to degrees, minutes, and seconds.
10.
Perform each calculation
a) 51 29  32 46
b) 90  7312
c) 90  36 18 47
d) 75 15  83 32
11.
Find the angle of least possible positive measure coterminal with an angle of -75.
12.
Find the angle of least possible positive measure coterminal with an angle of -800.
13.
Find the angle of least possible positive measure coterminal with an angle of 270.
14.
A vertical rise of the Forest Double chair lift 1,170 feet and the length of the chair lift as 5,570
feet. To the nearest foot, find the horizontal distance covered by a person riding this lift.
15.
A tire is rotating 600 times per minute. Through how many degrees does a point of the edge of
the tire move in 1 second?
2
16.
A windmill makes 90 revolutions per minute. How many revolutions does it make per second?
17.
Use a calculator to convert 256 20 to radians to the nearest hundredth of a radian.
18.
Convert 78.4 to radians
19.
Convert 11 to degrees
6
20.
Convert  5 to degrees
3
21.
Convert  to degrees
6
22.
Use the calculator to convert 2.4 to degree measure to the nearest tenth of a degree.
7
Solution
Section 2.1– Degrees, Radians, Angles and Triangles
Exercise
Indicate the angle if it is an acute or obtuse. Then give the complement and the supplement of each angle.
a) 10
b) 52
c) 90
d) 120
e) 150
Solution
a) Acute;
Complement is 90  10 = 80;
Supplement is 180  10 = 170.
b) Acute;
Complement is 90  52 = 38;
Supplement is 180  52 = 128.
c) Neither (right angle); Complement is 90  90 = 0;
Supplement is 180  90 = 90.
d) Obtuse;
Complement is 90  120 = 30; Supplement is 180  120 = 60.
e) Obtuse;
Complement is 90  150 = 60; Supplement is 180  150 = 30.
Exercise
Change 10 45 to decimal degrees
Solution
10 45 = 10 + 45
 10  45 1 
60
 10  0.75
 10.75
Exercise
Convert 34 51 35 to decimal degrees.
Solution
34 51 35  34  51  35
 34  51  1  35  1
60
3600
 34  0.85  0.00972
 34.85972
1
Exercise
Convert 274 18 59 to decimal degrees.
Solution
274 18 59  274  18  59
 274  18  1  59  1
60
3600
 274  0.3  0.016389
 274.316389
Exercise
Change 74 8 14 to decimal degrees to the nearest thousandth
Solution
74 8 14  74  8  14
60 3600
 74  0.1333  0.0039
 74.137
Exercise
Convert 89.9004 to degrees, minutes, and seconds.
Solution
89.9004  89  0.9004
 89  0.9004   60
 89 54.024
 89 54  0.024
 89 54 0.024   60
 89 54 1.44
Exercise
Convert 34.817 to degrees, minutes, and seconds
Solution
34.817  34  0.817
 34  0.817  60
 34  49.02
 34  49  .02  60
2
 34  49  1.2
 34 49 1.2
Exercise
Convert 34.817 to degrees, minutes, and seconds.
Solution
34.817  34  0.817
 34  0.817   60
 34 49.02
 34 49  0.02
 34 49 0.02   60
 34 49 1.2
Exercise
Convert 122.6853 to degrees, minutes, and seconds.
Solution
122.6853  122  .6853
 122  0.6853   60
 122 41.118
 122 41  0.118
 122 41 0.118   60
 122 41 7.1
Exercise
Convert 178.5994 to degrees, minutes, and seconds.
Solution
178.5994  178  .5994
 178  .5994   60
 178 35.964
 178 35  .964
 178 35 0.964   60
 178 35 57.84
3
Exercise
Perform each calculation
a) 51 29  32 46
c) 90  36 18 47
b) 90  7312
d) 75 15  83 32
Solution
a) 5129  3246
51 29
 32 46
83 75
83 75  115
84 15
b) 90  7312
89 60
 73 12
16 48
c) 90  36 18 47
90
89 59 60
 36 18 47   36 18 47
53 41 13
d) 75 15  83 32
75 15
83 32
158 47
Exercise
Find the angle of least possible positive measure coterminal with
an angle of -75.
Solution
360  75  285
4
Exercise
Find the angle of least possible positive measure coterminal with an angle of -800.
Solution
3  360  800  280
Exercise
Find the angle of least possible positive measure coterminal with an angle of 270.
Solution
360  270  630
Exercise
A vertical rise of the Forest Double chair lift 1,170 feet and the length of the chair lift as 5,570 feet. To
the nearest foot, find the horizontal distance covered by a person riding this lift.
Solution
B
x2  11702  55702
5570
x2  55702  11702
1170
x  55702  11702
A
x  5,445.73 ft
x
C
Exercise
A tire is rotating 600 times per minute. Through how many degrees does a point of the edge of the tire
move in 1 second?
2
Solution
1 600 rev  1min  360  1800 deg/ sec
min 60sec 1rev
2
Exercise
A windmill makes 90 revolutions per minute. How many revolutions does it make per second?
Solution
90 rev  1min  1.5 rev / sec
min 60sec
5
Exercise
Use a calculator to convert 256 20 to radians to the nearest hundredth of a radian.
Solution
256 20  256  20
60
 256  2
6
 1538
6
1538   4.47 rad
6 180
Exercise
Convert -78.4 to radians
Solution
 
78.4  78.4  rad
180
 1.37 rad
Exercise
Convert 11 to degrees
6
Solution
11
11 180
rad 
•
6
6

 330
Exercise
Convert  5 to degrees
3
Solution

5
5 180
rad  
•
3
3

 300
6
Exercise
Convert  to degrees
6
Solution
 
 (rad )   180 
6
6 
 30
Exercise
Use the calculator to convert 2.4 to degree measure to the nearest tenth of a degree.
Solution
2.4 rad  2.4 •

180

432

 137.5
7