LESSON 2 DEFINITION OF THE SIX TRIGONOMETRIC
FUNCTIONS USING THE UNIT CIRCLE
Topics in this lesson:
1.
DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A
CIRCLE OF RADIUS r
2.
DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE
UNIT CIRCLE
3.
THE SPECIAL ANGLES IN TRIGONOMETRY
4.
TEN THINGS EASILY OBTAINED FROM UNIT CIRCLE
TRIGONOMETRY
5.
THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL
ANGLES IN THE FIRST QUADRANT BY ROTATING
COUNTERCLOCKWISE
6.
ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL

ANGLES OF
6
( 30  ) ,


( 45  ) , AND
( 60  )
4
3
7.
THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE
SPECIAL ANGLES
1.
DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING
A CIRCLE OF RADIUS r
y
r
s
Pr ( )  ( x , y ) 

-r
r
x
r
-r
x2  y2  r 2
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Definition Let Pr ( )  ( x , y ) be the point of intersection of the terminal side of
2
2
2
the angle  with the circle whose equation is x  y  r . Then we define the
following six trigonometric functions of the angle 
cos  
x
r
sec  
r
, provided that x  0
x
sin  
y
r
csc  
r
, provided that y  0
y
tan  
y
, provided that x  0
x
cot  
x
, provided that y  0
y
NOTE: By definition, the secant function is the reciprocal of the cosine function.
The cosecant function is the reciprocal of the sine function. The cotangent function
is the reciprocal of the tangent function.
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2.
DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING
THE UNIT CIRCLE
Since you can use any size circle to define the six trigonometric functions, the best
circle to use would be the Unit Circle, whose radius r is 1. Using the Unit Circle,
we get the following special definition.
Definition Let P ( )  ( x , y ) be the point of intersection of the terminal side of
the angle  with the Unit Circle. Since r  1 for the Unit Circle, then by the
definition above, we get the following definition for the six trigonometric functions
of the angle  using the Unit Circle
cos   x
sec  
1
, provided that x  0
x
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sin   y
tan  
y
, provided that x  0
x
csc  
1
, provided that y  0
y
cot  
x
, provided that y  0
y
y
1
s
P ( )  ( x , y ) 

-1
1
x
1
-1
x 2  y 2  1 (The Unit Circle)
NOTE: The definition of the six trigonometric functions of the angle  in terms of
the Unit Circle says that the cosine of the angle  is the x-coordinate of the point of
intersection of the terminal side of the angle  with the Unit Circle. This definition
also says that the sine of the angle  is the y-coordinate of the point of intersection
of the terminal side of the angle  with the Unit Circle. The tangent of the angle 
is the y-coordinate of the point of intersection of the terminal side of the angle 
with the Unit Circle divided by the x-coordinate of the point of intersection. The
secant function is still the reciprocal of the cosine function, the cosecant function is
still the reciprocal of the sine function, and the cotangent function is still the
reciprocal of the tangent function.
Examples Find the exact value of the six trigonometric functions for the following
angles.
1.
  0
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P ( 0  )  (1, 0 )
cos 0   x  1
sec 0  
1
1
 1
cos 0 
1
sin 0   y  0
csc 0  
1
1

= undefined
sin 0 
0
cot 0  
1
1

= undefined
tan 0 
0
tan 0  
2.
 
y
0
 0
x
1

2
NOTE: This is the 9 0  angle in units of degrees.
 
P    ( 0 , 1)
2
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1
1
 
sec   


undefined
0
 
2
cos  
2
 
cos    x  0
2
3.
 
sin    y  1
2
1
1
 
csc   
 1
1
 
2
sin  
2
y
1
 
tan   

= undefined
x
0
2
1
x
0
 
cot   


 0
y
1
 
2
tan  
2
 
NOTE: This is the 18 0  angle in units of degrees.
P (  )  (  1, 0 )
cos   x   1
sec  
1
1

 1
cos 
1
sin   y  0
csc  
1
1

= undefined
sin 
0
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tan  
4.
y
0

0
x
1
cot  
1
1

= undefined
tan 
0
  270 
NOTE: This is the
3
angle in units of radians.
2
P ( 270  )  ( 0 ,  1)
sec 270  
sin 270    1
csc 270    1
tan 270  
5.
1
= undefined
0
cos 270   0
1
= undefined
0
cot 270  
  2
NOTE: This is the 36 0  angle in units of degrees.
Copyrighted by James D. Anderson, The University of Toledo
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0
 0
1
P ( 2 )  (1, 0 )
cos 2   1
sec 2   1
sin 2   0
csc 2  
1
= undefined
0
cot 2  
1
= undefined
0
tan 2  
6.
0
0
1
   90 
NOTE: This is the 

angle in units of radians.
2
P (  90  )  ( 0 ,  1)
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sec (  90  ) 
sin (  90  )   1
csc (  90  )   1
tan (  90  ) 
7.
1
= undefined
0
cos (  90  )  0
1
= undefined
0
cot (  90  ) 
0
 0
1
   180 
NOTE: This is the   angle in units of radians.
P (  180  )  (  1, 0 )
cos (  180  )   1
sec (  180  )   1
sin (  180  )  0
csc (  180  ) 
1
= undefined
0
cot (  180  ) 
1
= undefined
0
tan (  180  ) 
0
0
1
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8.
  
3
2
NOTE: This is the  27 0  angle in units of degrees.
 3 
P
  ( 0 , 1)
2


9.
 3 
cos  
  0
 2 
1
 3 
sec  
 
= undefined
0
 2 
 3 
sin  
 1
 2 
 3 
csc  
 1
 2 
1
 3 
tan  
 
= undefined
0
 2 
0
 3 
cot  
 0
 
1
 2 
   360 
NOTE: This is the  2  angle in units of radians.
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P (  360  )  (1, 0 )
cos (  360  )  1
sec (  360  )  1
sin (  360  )  0
csc (  360  ) 
1
= undefined
0
cot (  360  ) 
1
= undefined
0
tan (  360  ) 
0
0
1
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3.
THE SPECIAL ANGLES IN TRIGONOMETRY
The Special Angles (in radians) in trigonometry are
0, 
2
3
5




,  ,  ,  , 
, 
, 
, ,
6
4
3
2
3
4
6

7
5
4
3
5
7
11
, 
, 
,
, 
, 
, 
,  2
6
4
3
2
3
4
6
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The Special Angles (in degrees) in trigonometry are
0  ,  30  ,  45  ,  60  ,  90  ,  120  ,  135  ,  150  ,  180  ,
 210  ,  225  ,  240  ,  270  ,  300  ,  315  ,  330  ,  360 
Here are the coordinates of the point of intersection of the terminal side of the
positive Special Angles with the Unit Circle. These coordinates could be used to
find the exact value of the six trigonometric functions of these Special Angles.
Here are the coordinates of the point of intersection of the terminal side of the
negative Special Angles with the Unit Circle. These coordinates could be used to
find the exact value of the six trigonometric functions of these Special Angles.
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4.
TEN THINGS EASILY
TRIGONOMETRY
1.
The x-coordinate of any point on the Unit Circle satisfies the condition
 1  x  1 . Since we get the cosine of any angle  from the x-coordinate
of the point of intersection of the terminal side of the angle  with the Unit
Circle, then  1  cos   1 for all  . Thus, the range of the cosine
function is the closed interval [  1, 1 ] .
2.
The y-coordinate of any point on the Unit Circle satisfies the condition
 1  y  1 . Since we get the sine of any angle  from the y-coordinate of
the point of intersection of the terminal side of the angle  with the Unit
Circle, then  1  sin   1 for all  . Thus, the range of the sine function
is also the closed interval [  1, 1 ] .
3.
OBTAINED
FROM
UNIT
CIRCLE
y
, where x is the x-coordinate and y is the yx
coordinate of the point of intersection of the terminal side of the angle  with
By definition, tan  
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the Unit Circle. Since cos   x and sin   y by definition, then
sin 
tan  
for all  .
cos 
4.
Similarly, since sec  
1
1
by definition, then sec  
for all  .
cos 
x
5.
Similarly, since csc  
1
1
by definition, then csc  
for all  .
sin 
y
6.
Similarly, since cot  
cos 
x
cot


by definition, then
for all  .
sin 
y
7.
The equation of the Unit Circle is x  y  1 . This equation says that if
you take the x-coordinate and the y-coordinate of a point on the Unit Circle,
square them and then add, you will get the value of 1. Since we get the
cosine of any angle  from the x-coordinate and the sine of the angle  from
the y-coordinate of the point of intersection of the terminal side of the angle
 with the Unit Circle, then we obtain the equation
2
2
( cos  ) 2  ( sin  ) 2  1 ,
which we write as
cos 2   sin 2   1 for all  .
This equation is the first of the three Pythagorean Identities.
8.
2
2
Taking the equation cos   sin   1 and dividing both sides of the
2
equation by cos  we obtain the second Pythagorean Identity:
cos 2   sin 2 
1

cos 2 
cos 2 

cos 2 
sin 2 
1


cos 2 
cos 2 
cos 2 
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
2
 sin  
 1 
  

1  
cos

cos





2

1  tan 2   sec 2 

sec 2   tan 2   1 for all  .
9.
2
2
Similarly, taking the equation cos   sin   1 and dividing both sides
2
of the equation by sin  you will obtain the third Pythagorean Identity:
csc 2   cot 2   1 for all  .
10.
The sign of the cosine, sine, and tangent functions in the Four Quadrants:
(  ,  )
( cos  , sin  )
tan  
()
y

 ()
x
()
(  ,  )
( cos  , sin  )
tan  
()
y

 ()
x
()
y
(  ,  )
( cos  , sin  )
tan  
()
y

 ()
x
()
(  ,  )
( cos  , sin  )
tan  
x
()
y

 ()
x
()
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5.
THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE
SPECIAL ANGLES IN THE FIRST QUADRANT BY ROTATING
COUNTERCLOCKWISE
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
THE POINT OF INTERSECTION OF THE ANGLE
6
( 30  ) WITH
THE UNIT CIRCLE
y
1
    3 1 
P   
,

6
2
2 
  
1
1
x
1
This picture shows us that the x-coordinate is greater than the y-coordinate of the

 
( 30  ) with the
P


point
, which is the point of intersection of the angle
6
6
Unit Circle. Thus,
cos


6

1
sin

6
2
3
2
sec
csc

6

6

2
 2
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3
tan

6

1
2 
3
1
cot
3

6

3
2

THE POINT OF INTERSECTION OF THE ANGLE
4
( 45  ) WITH THE
UNIT CIRCLE
y
1
2
   2

P   
,

2 
4  2
1
1
x
1
This picture shows us that the x-coordinate is same as the y-coordinate of the

 
( 45  ) with the
point P   , which is the point of intersection of the angle
4
4
 
Unit Circle. Thus,
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
cos
sin
4

tan
4

4

2
2

2

sec
csc
2
2
2 1
2
2
cot

4

4

4

2

2
2
2

2

2
1

THE POINT OF INTERSECTION OF THE ANGLE
3
( 60  ) WITH THE
UNIT CIRCLE
y
1
3
   1

P    ,


3
2
2
  

1
1
1
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x
This picture shows us that the x-coordinate is less than the y-coordinate of the

 
( 60  ) with the
P


point
, which is the point of intersection of the angle
3
3
Unit Circle. Thus,
cos
sin
tan


3

3

3
1
2
sec

3
2

3
2 
1
2
csc
3
cot

 2
3


2

1
3

3
3
3
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6.
ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL

ANGLES OF
6
( 30  ) ,


( 45  ) , AND
( 60  )
4
3
You will need to remember the three numbers 1 ,

angles
6
( 30  ) ,

1
3 , and
3 . Arrange the

( 45  ) , and
( 60  ) from smallest to largest. Then
4
3
arrange the three numbers from smallest to largest:
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
6
( 30  )

( 45  )
4

( 60  )
3
|
|
|

1
3


3
1

In other words, the tangent of the smallest angle of
number of
number of
Tangent
6
( 30  ) is the smallest
1

( 60  ) is the largest
3 , the tangent of the largest angle of 3
3 , and the tangent of the middle angle of

4
( 45  ) is the middle
number of 1 .
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7.
THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE
SPECIAL ANGLES
To find the trigonometric functions for the special angles in the II, III, and IV
quadrants by rotating counterclockwise and for all the special angles in the IV, III,
II, and I quadrants by rotating clockwise, we make use of the reference angle of
these angles. Reference angles will be discussed in Lesson 3.
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