LESSON 4 COTERMINAL ANGLES
Topics in this lesson:
1.
THE DEFINITION AND EXAMPLES OF COTERMINAL ANGLES
2.
FINDING COTERMINAL ANGLES
3.
TRIGONOMETRIC FUNCTIONS OF COTERMINAL ANGLES
1.
THE DEFINITION AND EXAMPLES OF COTERMINAL ANGLES
Definition Two angles are said to be coterminal if their terminal sides are the
same.
Examples Here are some examples of coterminal angles.
1.
The two angles of
140 and
220 are coterminal angles.
Animation of the making of these two coterminal angles.
2.
The two angles of
7
and
6
31
are coterminal angles.
6
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Animation of the making of these two coterminal angles.
3.
The two angles of
1845 and
1395 are coterminal angles.
Animation of the making of these two coterminal angles.
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
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2.
FINDING COTERMINAL ANGLES
Theorem The difference between two coterminal angles is a multiple (positive or
negative) of 2 or 360 .
Examples Find three positive and three negative angles that are coterminal with
the following angles.
660
1.
1
660
2
1020
NOTE:
360
2
360
1380
1380
1020
3
660
360
4
300
360
NOTE:
5
4
60
NOTE:
6
NOTE:
1020
60
360
5
420
420
360
6
780
360
( 660
360 )
360
660
2 ( 360 )
300
60
300
360
( 660
360 )
360
660
2( 360 )
420
60
360
[660
2 (360 )]
360
660
3( 360 )
3 (360 )]
360
660
4( 360 )
780
420
360
[660
Copyrighted by James D. Anderson, The University of Toledo
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NOTE: There are other answers for this problem. If the difference between
660 and another angle is a multiple of 360 , then this
the angle of
second angle is an answer to problem.
87
5
2.
1
87
5
2
87
5
10
5
77
5
2
77
5
2
77
5
10
5
67
5
NOTE:
3
67
5
NOTE:
4
3
3
5
NOTE:
6
10
5
2
9 (2 )
87
5
9
3
5
10
5
13
5
2
3
5
13
5
2
87
5
2 (2 )
87
5
2
2 (2 )
57
5
67
5
2
87
5
2
57
5
5
13
5
77
5
67
5
2
87
5
5
67
5
2
10
5
87
5
90
5
87
5
2
3 (2 )
3
5
13
5
2
87
5
10
5
23
5
9 (2 )
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2
87
5
10 (2 )
NOTE:
6
23
5
13
5
87
5
2
10 (2 )
2
87
5
11 (2 )
NOTE: There are also other answers for this problem. If the difference
87
between the angle of
and another angle is a multiple of 2 , then
5
this second angle is an answer to problem.
87
is made. We will need the following property of
5
arithmetic, which comes from the check for long division.
Let’s see how the angle
b
q
a
a
qb
r
r
Dividing both sides of the equation a q b r by b, we obtain the equation
a
r
q
. For the work that we will do in order to find one particular coterminal
b
b
angle of a given angle in radians, we will want the number q above to be an even
number.
Now, consider the fraction of
87
in the angle
5
87
5
87
5
given above.
17
5 87
5
37
35
2
87
2
17
. The number 17 is an odd number. However, we may write 17
5
5
as 17 16 1 , where the number 16 is an even number. Thus, we have that
Thus,
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87
5
17
2
5
87
Thus,
5
16
7
. Now, multiply both sides of this equation by
5
(16
87
5
1)
16
2
5
7
5
16
1
16
2
5
7
5
16
8( 2 )
5
5
2
5
16
7
5
to obtain that
7
5
87
. In order to make the angle
5
7
87
, it will take eight complete revolutions and an additional rotation of
5
5
87
going in the counterclockwise direction in order to make the angle
.
5
This equation tells us how to make the angle
Thus, the two angles
7
5
87
5
7
and
5
are coterminal angles.
2 .
Copyrighted by James D. Anderson, The University of Toledo
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Notice that
Animation of the making of these two coterminal angles.
Another animation of the making of these two coterminal angles.
Examples Find the angle between 0 and 2
is coterminal with the following angles.
1.
or the angle between
2
and 0 that
85
6
Consider the fraction of
85
.
6
85
in the angle
6
14
6 85
6
85
6
25
24
1
85
1
14
. The 14 is an even number. So, multiply both sides of
6
6
this equation by to obtain that
Thus,
85
6
1
6
14
In order to make the angle
and an additional rotation of
Thus, the angle of
6
14
6
7(2 )
6
85
, it will take seven complete revolutions
6
6
going in the counterclockwise direction.
is the coterminal angle that we are looking for.
Copyrighted by James D. Anderson, The University of Toledo
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Animation of the making of these two coterminal angles. Another animation
of the making of these two coterminal angles.
Answer:
6
91
4
2.
91
.
4
91
Consider the fraction of
in the angle
4
22
4 91
8
91
4
11
8
3
91
3
22
. The 22 is an even number. Now, multiply both sides of
4
4
this equation by
to obtain that
Thus,
91
4
22
3
4
22
3
4
11 ( 2 )
3
4
91
, it will take eleven complete
4
3
revolutions and an additional rotation of
going in the clockwise
4
3
direction. Thus, the angle of
is the coterminal angle that we are
4
looking for. Animation of the making of these two coterminal angles.
In order to make the angle
Copyrighted by James D. Anderson, The University of Toledo
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Answer:
3
4
112
3
3.
112
3
112
Consider the fraction of
in the angle
3
.
37
3 112
112
3
9
22
21
1
112
1
37
Thus,
. The 37 is an odd number. So, we may write 37 as
3
3
37 36 1 , where the number 36 is an even number. Thus, we have that
112
3
37
112
3
obtain that
Thus,
1
3
( 36
36
112
3
1)
1
3
36
1
1
3
36
3
3
1
3
4
3
36
4
. Now, multiply both sides of this equation by
3
36
4
3
36
4
3
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18 ( 2 )
4
3
to
112
3
In order to make the angle
, it will take eighteen complete
revolutions and an additional rotation of
4
3
direction. Thus, the angle of
4
3
going in the clockwise
is the coterminal angle that we are
looking for.
Answer:
4
3
59
2
4.
59
.
2
59
Consider the fraction of
in the angle
2
29
2 59
59
2
4
19
18
1
59
1
29
Thus,
. The 29 is an odd number. So, we may write 29 as
2
2
29 28 1 , where the number 28 is an even number. Thus, we have that
59
2
29
1
2
( 28
Thus,
59
2
28
3
. Now, multiply both sides of this equation by
2
1)
1
2
28
1
1
2
28
that
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2
2
1
2
28
3
2
to obtain
59
2
3
2
28
3
2
28
14 ( 2 )
3
2
59
, it will take fourteen complete
2
3
revolutions and an additional rotation of
going in the counterclockwise
2
3
direction. Thus, the angle of
is the coterminal angle that we are looking
2
for. Animation of the making of these two coterminal angles.
In order to make the angle
Answer:
3
2
17
2
5.
Consider the fraction of
17
in the angle
2
17
.
2
8
2 17
17
2
16
1
17
1
8
. The 8 is an even number. Now, multiply both sides of
2
2
this equation by
to obtain that
Thus,
17
2
8
1
2
8
2
4( 2 )
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2
17
, it will take four complete revolutions
2
In order to make the angle
and an additional rotation of
2
going in the clockwise direction. Thus, the
angle of
is the coterminal angle that we are looking for. Animation of
2
the making of these two coterminal angles.
Answer:
6.
2
82
41( 2 ) . In order to make
The number 82 is an even number. Thus, 82
82 , it will take forty-one complete revolutions. Thus, the
the angle
angle of 0 is the coterminal angle that we are looking for.
Answer: 0
7.
21
The 21 is an odd number. So, we may write 21 as 21 20 1, where the
number 20 is an even number. Now, multiply both sides of this equation by
to obtain that
21
( 20 1)
20
10 ( 2 )
21 , it will take ten complete revolutions
In order to make the angle
and an additional rotation of
going in the clockwise direction. Thus, the
angle of
is the coterminal angle that we are looking for.
Answer:
8.
4110
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Use your calculator to find the whole number of times that 360 will divide
into 4110 . Since 4110 360 11.41666 .... and 360 times 11 equals
3960, then we have that
11
360 4110
3960
150
Thus, 4110 11( 360 ) 150 . When working in degrees, the whole number of
11, in this case, does not have to be even. Thus,
4110 11( 360 )
150
4110
11( 360 )
150
4110 , it will take eleven complete
In order to make the angle
revolutions and an additional rotation of 150 going in the counterclockwise
direction. Thus, the angle of 150 is the coterminal angle that we are
looking for.
Answer: 150
9.
8865
Use your calculator to find the whole number of times that 360 will divide
into 8865 . Since 8865 360 24.625 and 360 times 24 equals 8640, then
we have that
24
360 8865
8640
225
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Thus, 8865 24 ( 360 ) 225 . Multiplying both sides of this equation by
we obtain that 8865 24 ( 360 ) 225 . Thus,
8865
24 ( 360 )
225
8865
24 ( 360 )
1,
225
8865 , it will take twenty-four complete
In order to make the angle
revolutions and an additional rotation of 225 going in the clockwise
direction. Thus, the angle of 225 is the coterminal angle that we are
looking for.
Answer:
225
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3.
TRIGONOMETRIC FUNCTIONS OF COTERMINAL ANGLES
Theorem Let
and
be two coterminal angles. Then
1.
cos
cos
4.
sec
sec
2.
sin
sin
5.
csc
csc
3.
tan
tan
6.
cot
cot
We can use this theorem to find any one of the six trigonometric functions of an
angle that is numerically bigger than 2 or 360 . When given an angle that is
numerically bigger than 2
or 360 , we will want to find the angle that is
coterminal to it and is numerically smaller than 2 or 360 .
Examples Use a coterminal angle to find the exact value of the six trigonometric
functions of the following angles.
1.
11
3
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Doing one subtraction of 2
11
3
Thus, the two angles of
, we obtain that
2
=
11
3
11
3
6
3
5
3
5
5
and
are coterminal. The angle
is
3
3
in the IV quadrant and has a reference angle of
3
. Thus, by the theorem
above, we have that
2.
cos
11
3
cos
5
3
sin
11
3
sin
5
3
tan
11
3
tan
5
3
cos
1
2
3
sin
tan
3
3
2
3
3
sec
11
3
csc
11
3
2
cot
11
3
1
19
4
Doing one addition of 2
19
4
, we obtain that
2
Doing a second addition of 2
11
4
2
=
19
4
8
4
11
4
8
4
3
4
, we obtain that
=
11
4
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2
3
3
19
4
Thus, the two angles of
and
3
are coterminal. The angle
4
3
is in the III quadrant and has a reference angle of . Thus, by the
4
4
theorem above, we have that
3.
cos
19
4
cos
3
4
sin
19
4
sin
3
4
tan
19
4
tan
3
4
cos
4
2
2
sin
tan
2
4
2
1
4
149
6
Consider the fraction of
149
in the angle
6
149
6
.
24
6 149
12
149
6
29
24
5
Thus,
149
6
24
5
. This implies that
6
149
6
24
5
6
12 ( 2 )
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5
6
sec
19
4
2
csc
19
4
2
cot
19
4
1
149
6
, it will take twelve complete
revolutions and an additional rotation of
5
going in the counterclockwise
6
In order to make the angle
direction. Thus, the angle of
5
is the coterminal angle that we are looking
6
5
is in the II quadrant and has a reference angle of .
6
6
Thus, by the theorem above, we have that
for. The angle
4.
cos
149
6
cos
5
6
sin
149
6
sin
5
6
tan
149
6
tan
5
6
cos
sin
3
6
1
2
6
tan
2
1
6
3
58
3
58
Consider the fraction of
in the angle
3
58
3
58
.
3
19
3 58
3
28
27
1
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sec
149
6
csc
149
6
cot
149
6
2
3
2
3
Thus,
58
3
19
1
3
4
. This implies that
3
18
58
3
4
3
18
4
3
58
, it will take nine complete revolutions
3
In order to make the angle
and an additional rotation of
Thus, the angle of
9( 2 )
4
going in the counterclockwise direction.
3
4
is the coterminal angle that we are looking for. The
3
4
is in the III quadrant and has a reference angle of . Thus, by the
3
3
theorem above, we have that
angle
5.
cos
58
3
cos
4
3
cos
sin
58
3
sin
4
3
sin
tan
58
3
tan
4
3
tan
3
1
2
3
3
3
2
3
sec
58
3
csc
58
3
cot
58
3
169
4
169
Consider the fraction of
in the angle
4
169
4
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.
2
2
3
1
3
169
4
42
4 169
16
9
8
1
Thus,
169
4
1
. This implies that
4
42
169
4
42
21( 2 )
4
169
4
In order to make the angle
revolutions and an additional rotation of
Thus, the angle of
4
4
, it will take twenty-one complete
4
going in the clockwise direction.
is the coterminal angle that we are looking for. The
angle
is in the IV quadrant and has a reference angle of . Thus, by the
4
4
theorem above, we have that
cos
169
4
cos
sin
169
4
sin
tan
169
4
tan
4
4
4
cos
2
4
sin
tan
2
2
4
4
2
1
Copyrighted by James D. Anderson, The University of Toledo
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sec
169
4
csc
169
4
cot
169
4
2
2
1
527
6
6.
Consider the fraction of
527
6
527
in the angle
6
.
87
6 527
527
6
48
47
42
5
Thus,
527
6
5
6
87
527
6
86
11
. This implies that
6
86
In order to make the angle
11
6
43 ( 2 )
527
6
, it will take forty-three complete
11
6
revolutions and an additional rotation of
direction. Thus, the angle of
looking for. The angle
of
6
cos
11
6
11
6
11
6
going in the clockwise
is the coterminal angle that we are
is in the I quadrant and has a reference angle
. Thus, by the theorem above, we have that
527
6
cos
11
6
cos
3
6
2
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sec
527
6
2
3
sin
527
6
sin
11
6
sin
tan
527
6
tan
11
6
tan
6
1
2
1
6
3
csc
527
6
cot
527
6
2
3
17
2
7.
Consider the fraction of
17
17
. Since
2
2
17
in the angle
2
8
1
,
2
then
17
2
8
4( 2 )
2
17
, it will take four complete revolutions
2
In order to make the angle
and an additional rotation of
2
2
going in the clockwise direction. Thus, the
angle of
is the coterminal angle that we are looking for. The angle
2
lies on the negative y-axis. Thus, by the theorem above, we have that
cos
17
2
cos
sin
17
2
sin
tan
17
2
tan
2
2
2
0
sec
17
2
= undefined
1
csc
17
2
1
1
= undefined
0
cot
17
2
0
1
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2
0
8.
65
64
32 ( 2 )
Since 65
, then in order to make the angle
65 , it will take thirty-two complete revolutions and an additional
rotation of going in the counterclockwise direction. Thus, the angle of
is the coterminal angle that we are looking for. The angle
lies on the
negative x-axis. Thus, by the theorem above, we have that
9.
cos 65
cos
sin 65
sin
tan 65
tan
1
sec 65
0
0
1
0
1
csc 65
= undefined
cot 65
= undefined
30
30
15 ( 2 ) , then in order to make the angle
30 , it
Since
will take fifteen complete revolutions going in the clockwise direction. Thus,
the angle of 0 is the coterminal angle that we are looking for. The angle 0
lies on the positive x-axis. Thus, by the theorem above, we have that
10.
cos ( 30 )
cos 0
1
sin ( 30 )
sin 0
0
tan ( 30 )
tan 0
0
1
sec ( 30 )
1
csc ( 30 ) = undefined
0
cot ( 30 ) = undefined
9840
Use your calculator to find the whole number of times that 360 will divide
into 9840 . Since 9840 360 27.3333 .... and 360 times 27 equals 9720,
then we have that
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27
360 9840
9720
120
Thus, 9840
27 ( 360 )
9840
120 . Thus,
27 ( 360 )
120
9840
27 ( 360 )
120
9840 , it will take twenty-seven complete
In order to make the angle
revolutions and an additional rotation of 120 going in the counterclockwise
direction. Thus, the angle of 120 is the coterminal angle that we are
looking for. The angle 120 is in the II quadrant and has a reference angle
of 60 . Thus, by the theorem above, we have that
11.
cos 9840
cos 120
sin 9840
sin 120
tan 9840
tan 120
1
2
cos 60
sin 60
tan 60
sec 9840
3
csc 9840
2
cot 9840
3
2
2
3
1
3
900
Subtracting two times 360 , we obtain that 900
720
= 180 .
900 and 180 are coterminal. The angle 180
Thus, the two angles of
is on the negative x-axis. Thus, using Unit Circle Trigonometry, we have that
cos 900
cos 180
1
sec 900
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1
12.
sin 900
sin 180
tan 900
tan 180
0
0
csc 900
undefined
cot 900
undefined
1290
Adding three times 360 , we obtain that
1290
1080 =
210 .
1290 and 210 are coterminal. The angle
Thus, the two angles of
210 is in the II quadrant and has a reference angle of 30 . Thus, by the
theorem above, we have that
cos ( 1290 )
cos ( 210 )
3
2
cos 30
2
3
sec ( 1290 )
sin ( 1290 )
sin ( 210 )
tan ( 1290 )
tan ( 210 )
sin 30
tan 30
1
2
csc ( 1290 )
2
1
3
cot ( 1290 )
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3