LESSON 3 REFERENCE ANGLES
Topics in this lesson:
1.
THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES
2.
THE REFERENCE ANGLE OF THE SPECIAL ANGLES
3.
USING A REFERENCE ANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS
1.
THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES
Definition The reference angle of the angle  , denoted by  ' , is the acute angle
determined by the terminal side of  and either the positive or negative x-axis.
Recall, an acute angle is an angle whose measurement is greater than 0  and less
than 90  .
NOTE: By definition, the reference angle is an acute angle. Thus, any angle
whose terminal side lies on either the x-axis or the y-axis does not have a reference
angle.
Examples Find the reference angle  ' for the following angles  made by rotating
counterclockwise.
1.
 is in the I quadrant.
Animation of the reference angle.
'

Relationship between  and  ' :    '
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2.
 is in the II quadrant.
Animation of the reference angle.
'

Relationship between  and  ' :
 '     OR  '  180   
OR
   '   OR    '  180 
3.
 is in the III quadrant.
Animation of the reference angle.

'
Relationship between  and  ' :
 '     OR  '    180 
OR
   '   OR 180    '  
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
4.
 is in the IV quadrant.
Animation of the reference angle.

'
Relationship between  and  ' :
 '  2    OR  '  360   
OR
   '  2  OR    '  360 
Examples Find the reference angle  ' for the following angles  made by rotating
clockwise.
1.
 is in the IV quadrant.
Animation of the reference angle.

'
Relationship between  and  ' :  '  
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2.
 is in the III quadrant.
Animation of the reference angle.
'

Relationship between  and  ' :
 '     OR  '  180   
OR
   '   OR
3.
   '  180 
 is in the II quadrant.
Animation of the reference angle.
'

Relationship between  and  ' :
 '     OR  '    180 
OR
  '  
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
OR
180    '  
4.
 is in the I quadrant.
Animation of the reference angle.
'

Relationship between  and  ' :
 '  2    OR  '  360   
OR
   '  2
OR
   '  360 
Examples Find the reference angle for the following angles.
1.
  140 
The angle  is in the II quadrant.
Animation of the reference angle.
'

 ' = 180   140   40 
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2.
   350 
The angle  is in the I quadrant.
Animation of the reference angle.
 '

 ' = 360   350   10 
3.
 
9
7
The angle  is in the III quadrant.
Animation of the reference angle.

'
' =
9
9
7
2
  =

=
7
7
7
7
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
4.

9
7
The angle  is in the II quadrant.
Animation of the reference angle.
'

' =
5.
 
9
7
9
2
  =

=
7
7
7
7
7
15
The angle  is in the IV quadrant.
Animation of the reference angle.

'
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
' 
7
15
6.
 
79 
42
The angle  is in the IV quadrant.
Animation of the reference angle.

'
 ' = 2 
7.
79 
84 
79 
5

=
=
42
42
42
42
   150 
Animation of the reference angle.
'

 ' = 180   150   30 
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
8.
  180 
The angle  is on the negative x-axis. Thus, the angle  does not have a
reference angle.
9.
 
3
2
The angle  is on the positive y-axis. Thus, the angle  does not have a
reference angle.
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2.
THE REFERENCE ANGLE OF THE SPECIAL ANGLES
The reference angle of the Special Angles of 

5
7

11
, 
, 
, and 
is
.
6
6
6
6
6
The reference angle of the Special Angles of 
3
5
7


, 
, 
, and 
is
.
4
4
4
4
4
The reference angle of the Special Angles of 
2
4
5


, 
, 
, and 
is
.
3
3
3
3
3
The reference angle of the Special Angles of
 330  is 30  .
 30  ,  150  ,  210  , and
The reference angle of the Special Angles of  45  ,  135  ,  225  , and  315 
is 45  .
The reference angle of the Special Angles of
 300  is 60  .
 60  ,  120  ,  240  , and
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Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
3.
USING A REFERENCE ANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS
To find the value of a trigonometric function of an angle in the II, III, and IV
quadrants by rotating counterclockwise, we will make use of the reference angle of
the angle. To find the value of a trigonometric function of an angle in the I, II, III,
and IV quadrants by rotating clockwise, we will make use of the reference angle of
the angle.
Theorem Let  ' be the reference angle of the angle  . Then
1.
cos    cos  '
4.
sec    sec  '
2.
sin    sin  '
5.
csc    csc  '
3.
tan    tan  '
6.
cot    cot  '
where the sign of + or  is determined by the quadrant that the angle  is in.
We will use this theorem to help us find the exact value of the trigonometric
functions of angles whose terminal side lies in the first (rotating clockwise),
second, third, and fourth quadrants. We do not need any help to find the exact
value of the trigonometric functions of angles whose terminal side lies on one of
the coordinate axes. We know the x-coordinate and the y-coordinate of the point of
intersection of these angles with the Unit Circle. So, we know the exact value of
the cosine, sine, and tangent of these angles from the x-coordinate, y-coordinate,
and the y-coordinate divided by the x-coordinate, respectively. This might be the
reason that a reference angle is not defined for an angle whose terminal side lies on
one of the coordinate axes.
The cosine, sine, and tangent of the special angles with a denominator of 6 in
radian angle measurement. The cosine, sine, and tangent of the special angles with
a denominator of 4 in radian angle measurement. The cosine, sine, and tangent of
the special angles with a denominator of 3 in radian angle measurement.
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Examples Use a reference angle to find the exact value of the six trigonometric
functions of the following angles.
1.
 
2
3
(This is the 12 0  angle in units of degrees.)
2
is in the II quadrant. The reference angle of the angle
3
2

 
is the angle  '  .
3
3
The angle  
2
is in the II quadrant, then the
3
2
 2 
 of the terminal side of the angle  
point of intersection P 
with
3
 3 
the Unit Circle is in the II quadrant. Thus, the x-coordinate of the point
 2 
P
 is negative and the y-coordinate of this point is positive. Since
 3 
2
2
 2 
cos
 , then cos
is the x-coordinate of the point P 
is negative.
3
3
 3 
2
2
 2 
 , then sin
Since sin
is the y-coordinate of the point P 
is
3
3
 3 
2
 2 
 divided by
positive. Since tan
is the y-coordinate of the point P 
3
3


2
the x-coordinate of this point, then tan
is negative since a positive
3
divided by a negative is negative.
Since the terminal side of the angle  
cos
2

1
  cos  
3
3
2
sec
2
 2
3
sin
3
2

 sin

3
3
2
csc
2
2

3
3
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
tan
2.
  210 
2

  tan
 
3
3
cot
3
(This is the
2
1
 
3
3
7
angle in units of radians.)
6
The angle   210  is in the III quadrant. The reference angle of the angle
  210  is the angle  '  30  .
Since the terminal side of the angle   210  is in the III quadrant, then the
point of intersection P ( 210  ) of the terminal side of the angle   210 
with the Unit Circle is in the III quadrant. Thus, the x-coordinate of the point
P ( 210  ) is negative and the y-coordinate of this point is negative. Since
cos 210  is the x-coordinate of the point P ( 210  ) , then cos 210  is
negative. Since sin 210  is the y-coordinate of the point P ( 210  ) , then
sin 210  is negative. Since tan 210  is the y-coordinate of the point
P ( 210  ) divided by the x-coordinate of this point, then tan 210  is positive
since a negative divided by a negative is positive.
3
cos 210    cos 30   
sin 210    sin 30   
tan 210   tan 30  
3.
 
7
4
2
1
2
1
3
3
csc 210    2
cot 210  
(This is the 315  angle in units of degrees.)
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2
sec 210   
3
7
is in the IV quadrant. The reference angle of the angle
4
7

 
is the angle  '  .
4
4
The angle  
7
is in the IV quadrant, then the
4
7
 7 
 of the terminal side of the angle  
point of intersection P 
with
4
 4 
the Unit Circle is in the IV quadrant. Thus, the x-coordinate of the point
 7 
P
 is positive and the y-coordinate of this point is negative. Since
 4 
7
7
 7 
P
cos
cos


is the x-coordinate of the point
, then
is positive.
4
4
 4 
7
7
 7 
 , then sin
Since sin
is the y-coordinate of the point P 
is
4
4
 4 
7
 7 
 divided by
negative. Since tan
is the y-coordinate of the point P 
4
4


7
the x-coordinate of this point, then tan
is negative since a negative
4
divided by a positive is negative.
Since the terminal side of the angle  
cos
2
7

 cos

4
4
2
sec
7
2


4
2
sin
2
7

  sin
 
4
4
2
csc
7
2
 
 
4
2
tan
7

  tan
 1
4
4
cot
7
 1
4
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2
2
4.
   30 
(This is the 

angle in units of radians.)
6
The angle    30  is in the IV quadrant. The reference angle of the angle
   30  is the angle  '  30  .
Since the terminal side of the angle    30  is in the IV quadrant, then the
point of intersection P (  30  ) of the terminal side of the angle    30 
with the Unit Circle is in the IV quadrant. Thus, the x-coordinate of the point
P (  30  ) is positive and the y-coordinate of this point is negative. Since
cos (  30  ) is the x-coordinate of the point P (  30  ) , then cos (  30  ) is
positive. Since sin (  30  ) is the y-coordinate of the point P (  30  ) , then
sin (  30  ) is negative. Since tan (  30  ) is the y-coordinate of the point
P (  30  ) divided by the x-coordinate of this point, then tan (  30  ) is
negative since a negative divided by a positive is negative.
cos (  30  )  cos 30  
3
2
sin (  30  )   sin 30   
tan (  30  )   tan 30   
5.

5
6
sec (  30 ) 
1
2
2
3
csc (  30  )   2
1
3
cot (  30  )  
3
(This is the  15 0  angle in units of degrees.)
5
is in the III quadrant. The reference angle of the angle
6
5



'

is the angle
.
6
6
The angle   
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
5
is in the III quadrant, then the
6
5
 5 
 of the terminal side of the angle   
point of intersection P  
6
 6 
with the Unit Circle is in the III quadrant. Thus, the x-coordinate of the point
 5 
P
 is negative and the y-coordinate of this point is negative. Since
6


 5 
 5 
 5 
cos  
 is the x-coordinate of the point P  
 is
 , then cos  
 6 
 6 
 6 
 5 
 5 
sin

P


 , then
negative. Since
is the y-coordinate of the point  
 6 
 6 
 5 
 5 
sin  
 is negative. Since tan  
 is the y-coordinate of the point
 6 
 6 
 5 
 5 
P
 is
 divided by the x-coordinate of this point, then tan  
 6 
 6 
positive since a negative divided by a negative is positive.
Since the terminal side of the angle   
6.
3

 5 
cos  
   cos  
6
2
 6 
2
 5 
sec  
  
3
 6 

1
 5 
sin  
 
   sin
6
2
 6 
 5 
csc  
  2
6



1
 5 
tan  

  tan
6
3
 6 
 5 
cot  
 
6


   225 
(This is the 
3
5
angle in units of radians.)
4
The angle    225  is in the II quadrant. The reference angle of the angle
   225  is the angle  '  45  .
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Since the terminal side of the angle    225  is in the II quadrant, then the
point of intersection P (  225  ) of the terminal side of the angle
   225  with the Unit Circle is in the II quadrant. Thus, the x-coordinate
of the point P (  225  ) is negative and the y-coordinate of this point is
positive. Since cos (  225  ) is the x-coordinate of the point P (  225  ) ,
then cos (  225  ) is negative. Since sin (  225  ) is the y-coordinate of the
point P (  225  ) , then sin (  225  ) is positive. Since tan (  225  ) is the
y-coordinate of the point P (  225  ) divided by the x-coordinate of this
point, then tan (  225  ) is negative since a positive divided by a negative is
negative.
cos (  225  )   cos 45   
sin (  225  )  sin 45  
2
2
2
2
tan (  225  )   tan 45    1
7.
 
5
3
2
sec (  225 )  
csc (  225 ) 
2
2
2
 

2
2
cot (  225 )   1
(This is the  30 0  angle in units of degrees.)
5
is in the I quadrant. The reference angle of the angle
3
5

 

'

is the angle
.
3
3
The angle   
5
is in the I quadrant, then the
3
5
 5 
 of the terminal side of the angle   
point of intersection P  
3
 3 
with the Unit Circle is in the I quadrant. Thus, the x-coordinate of the point
Since the terminal side of the angle   
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
 5 
P
 is positive and the y-coordinate of this point is positive. Since
 3 
 5 
 5 
 5 
cos  
 is the x-coordinate of the point P  
 is
 , then cos  
3
3
3






 5 
 5 
 is the y-coordinate of the point P  
 , then
positive. Since sin  
 3 
 3 
 5 
 5 
sin  
 is positive. Since tan  
 is the y-coordinate of the point
 3 
 3 
 5 
 5 
P
 is
 divided by the x-coordinate of this point, then tan  
 3 
 3 
positive since a positive divided by a positive is positive.
8.

1
 5 
cos  

  cos
3
2
 3 
 5 
sec  
  2
3


3

 5 
sin  

  sin
3
2
 3 
2
 5 
csc  
 
3
 3 

 5 
tan  

  tan
3
 3 
1
 5 
cot  
 
3
 3 
  240 
3
(This is the
4
angle in units of radians.)
3
The angle   240  is in the III quadrant. The reference angle of the angle
  240  is the angle  '  60  .
cos 240    cos 60   
1
2
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
sec 240    2
sin 240    sin 60   
tan 240   tan 60  
9.
 
5
6
3
2
csc 240   
2
cot 240  
3
3
1
3
(This is the 15 0  angle in units of degrees.)
5
is in the II quadrant. The reference angle of the angle
6
5

 

'

is the angle
.
6
6
The angle  
10.
cos
3
5

  cos
 
6
6
2
sec
5
2
 
6
3
sin
5

1
 sin

6
6
2
csc
5
 2
6
tan
5

1
  tan
 
6
6
3
cot
5
 
6
   45 
(This is the 
3

angle in units of radians.)
4
The angle    45  is in the IV quadrant. The reference angle of the angle
   45  is the angle  '  45  .
cos (  45  )  cos 45  
2
2
sec (  45 ) 
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2
2

2
sin (  45  )   sin 45   
2
csc (  45 )  
2
tan (  45  )   tan 45    1
11.
   330 
2
2
 
2
cot (  45 )   1
(This is the 
11
angle in units of radians.)
6
The angle    330  is in the I quadrant. The reference angle of the angle
   330  is the angle  '  30  .
3
cos (  330  )  cos 30  
sin (  330  )  sin 30  
2
1
2
tan (  330  )  tan 30  
12.

5
4
sec (  330 ) 
2
3
csc (  330  )  2
1
cot (  330  ) 
3
3
(This is the 225  angle in units of degrees.)
5
is in the III quadrant. The reference angle of the angle
4
5


is the angle  '  .
4
4
The angle  
cos
2
5

  cos
 
4
4
2
sec
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
5
2
 
 
4
2
2
13.
 
sin
2
5

  sin
 
4
4
2
csc
5
2
 
 
4
2
tan
5

 tan
1
4
4
cot
5
1
4
2
3
2
(This is the  12 0  angle in units of degrees.)
2
is in the III quadrant. The reference angle of the angle
3
2

 
is the angle  '  .
3
3
The angle   
14.


1
 2 
cos  
 
   cos
3
2
 3 
 2 
sec  
  2
 3 
3

 2 
sin  
 
   sin
3
2
 3 
2
 2 
csc  
  
3
 3 

 2 
tan  

  tan
3
3


1
 2 
cot  
 
3
 3 
11
6
3
(This is the 33 0  angle in units of degrees.)
11
is in the IV quadrant. The reference angle of the angle
6
11


is the angle  '  .
6
6
The angle  
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
15.
 
cos
3
11

 cos

6
6
2
sec
11
2

6
3
sin
11

1
  sin
 
6
6
2
csc
11
 2
6
tan
11

1
  tan
 
6
6
3
cot
11
 
6
7
4
3
(This is the  315  angle in units of degrees.)
7
is in the I quadrant. The reference angle of the angle
4
7

 
is the angle  '  .
4
4
The angle   
2

 7 
cos  
  cos 
4
2
 4 
 7 
sec  
 
4



2

 7 
sin  

  sin
4
 4 
2
 7 
csc  

 
4
2


2
2
2

 7 
tan  
1
  tan
4
4


16.
 
4
3
2
2
 7 
cot  
 1
4


(This is the  240  angle in units of degrees.)
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
4
is in the II quadrant. The reference angle of the angle
3
4

 
is the angle  '  .
3
3
The angle   

1
 4 
cos  
   cos  
3
2
 3 
 4 
sec  
  2
 3 
3

 4 
sin  

  sin
3
2
 3 
2
 4 
csc  
 
3
 3 

 4 
tan  
 
   tan
3
 3 
17.
 
3
4
1
 4 
cot  
  
3
 3 
3
(This is the 135  angle in units of degrees.)
3
is in the II quadrant. The reference angle of the angle
4
3

 

'

is the angle
.
4
4
The angle  
cos
2
3

  cos
 
4
4
2
sec
3
2
 
 
4
2
sin
2
3

 sin

4
4
2
csc
3
2


4
2
tan
3

  tan
 1
4
4
cot
3
 1
4
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2
2
18.
  300 
5
angle in units of radians.)
3
(This is the
The angle   300  is in the IV quadrant. The reference angle of the angle
  300  is the angle  '  60  .
cos 300   cos 60  
1
2
sin 300    sin 60   
tan 300    tan 60   
19.
 
7
6
sec 300   2
3
2
3
csc 300   
2
cot 300   
1
3
3
(This is the  21 0  angle in units of degrees.)
7
is in the II quadrant. The reference angle of the angle
6
7

 

'

is the angle
.
6
6
The angle   
3

 7 
cos  
 
   cos
6
2
 6 
2
 7 
sec  
  
3
 6 

1
 7 
sin  

  sin
6
2
 6 
 7 
csc  
  2
6



1
 7 
tan  
 
   tan
6
3
 6 
 7 
cot  
  
 6 
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
3
20.
   135 
(This is the 
3
angle in units of radians.)
4
The angle    135  is in the III quadrant. The reference angle of the angle
   135  is the angle  '  45  .
cos (  135  )   cos 45   
sin (  135  )   sin 45   
2
2
2
2
tan (  135  )  tan 45   1
21.
 

3
sec (  135 )  
2
csc (  135 )  
2
2
2
 
2
 
2
cot (  135 )  1
(This is the  60  angle in units of degrees.)

is in the IV quadrant. The reference angle of the angle
3


   is the angle  '  .
3
3
The angle   

1
 
cos     cos 
3
2
 3
 
sec     2
 3
3

 
sin      sin
 
3
2
 3
2
 
csc     
3
 3

 
tan      tan
 
3
 3
1
 
cot     
3
 3
3
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Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330