Download Lesson 5

LESSON 5 DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS
DETERMINED BY A POINT AND A LINE IN THE xy-PLANE
Topics in this lesson:
1.
DEFINITION AND EXAMPLES OF THE SIX TRIGONOMETRIC
FUNCTIONS DETERMINED BY A POINT IN THE xy-PLANE
2.
EXAMPLES OF THE SIX TRIGONOMETRIC FUNCTIONS
DETERMINED BY A LINE IN THE xy-PLANE
1.
DEFINITION AND EXAMPLES OF THE SIX TRIGONOMETRIC
FUNCTIONS DETERMINED BY A POINT IN THE xy-PLANE
y
r
Pr ( )  ( x , y )

r
-r
r
x
-r
Definition Let Pr ( )  ( x , y ) be a point on the terminal side of the angle 
which is not the origin ( 0 , 0 ) . 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
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
tan  
y
, provided that x  0
x
where r 
cot  
x
, provided that y  0
y
x2  y2
NOTE: This is exactly the same definition that was given in Section 1 of Lesson 2 .
r
x 2  y 2 is the distance from the point Pr ( )  ( x , y ) to the origin ( 0 , 0 ) ,
which is the radius of the circle whose equation is x  y  r . The definition
above is saying that in order to define any one of the six trigonometric functions,
you only need to know the coordinates of any point on the terminal side of the
angle  , which is not the origin ( 0 , 0 ) , and you do not need the circle. However,
hopefully, you would agree that the Unit Circle has been helpful for us and that you
would continue to make use of it. I know that I use it all the time to help me.
2
2
2
Notice that for the examples below, we will not have to make use of a reference
angle for the problem. The sign of the answer will follow from the definition of the
trigonometric function.
Examples Find the exact value of the six trigonometric functions of the angle  if
the given point is on the terminal side of  .
1.
(  8, 6 )
This point is in the second quadrant and lies on a circle of radius
r  64  36  100  10 . Animation of the point and the angle.
cos  
8
x
4

 
r
10
5
sec   
sin  
y
6
3


r
10
5
csc  
tan  
y
6
3

 
x
8
4
cot   
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
5
4
5
3
4
3
NOTE: In Lesson 9, we will find that the angle  is approximately 143.1 
or any angle coterminal to this angle.
2.
( 3 ,  9)
This point is in the fourth quadrant and lies on a circle of radius
r  3  81  84  2 21 . Animation of the point and the angle.
cos  
x

r
cos  
1
sin  
2
y
 
r
sin   
tan  
7
3
84

3

84
 sec  
1

28
7
7
1
1



2(7)
14
28
2 7
28  2 7
9 21
3 21
3 21
9
9
 
 
 
 
2 ( 21 )
2(7)
14
84
2 21
2 21
9
 csc   
2 21
9
9 3
 9
y

 
 3 3
x
3
3
cot   
1
3 3
 
3
9
NOTE: In Lesson 9, we will find that the angle  is approximately 280 .9 
or any angle coterminal to this angle.
3.
(  24 , 0 )
This point is on the negative x-axis and lies on a circle of radius
r 
(  24 ) 2  0 
24 2  24 . Animation of the point and the angle.
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
cos  
 24
x

 1
r
24
sec    1
sin  
y
0

 0
r
24
csc  = undefined
tan  
y
0

 0
x
 24
cot  = undefined
NOTE: We already know that the angle  is 180  or any angle coterminal
to this angle.
4.
(  12 ,  18 )
This point is in the third quadrant and lies on a circle of radius
r 
(  12 ) 2  (  18 ) 2 
6 2 4  6 29 
angle.
12 2  18 2 
6 2 ( 4  9 )  6 13 .
(6  2) 2  ( 6  3 ) 2 
Animation of the point and the
cos  
 12
x
2

 
r
6 13
13
sec   
sin  
 18
y
3

 
r
6 13
13
csc   
tan  
 18
y
3


x
 12
2
cot  
13
2
13
3
2
3
NOTE: In Lesson 9, we will find that the angle  is approximately 236 .3 
or any angle coterminal to this angle.
Back to Topics List
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2.
EXAMPLES OF THE SIX TRIGONOMETRIC
DETERMINED BY A LINE IN THE xy-PLANE
FUNCTIONS
Example The terminal side of the angle  is in the fourth quadrant and lies on the
3
y


x . Find the exact value of the six trigonometric functions of the angle
line
7
.
Pick a point on the given line that lies in the fourth quadrant. Since x-coordinates
are positive in the fourth quadrant, you will need to choose a positive number for
3
y


( 7 )   3 . Thus, the point
the x-coordinate of the point. If x = 7, then
7
3
( 7 ,  3 ) lies on the line y   x in the fourth quadrant. Since the terminal side
7
3
of  lies on the line y   x , then the point ( 7 ,  3 ) lies on the terminal side of
7
 . This point lies on a circle of radius r  49  9  58 . Animation of the
line, point, and the angle.
cos  
7
58
3
sin   
tan   
58
3
7
sec  
58
7
58
csc   
cot   
3
7
3
NOTE: In Lesson 9, we will find that the angle  is approximately 336 .8  or any
angle coterminal to this angle.
Question: What was the slope of the given line y  
3
3
x ? Answer: 
7
7
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
What was the tangent of the angle  ? Answer: 
3
7
Example The terminal side of the angle  is in the third quadrant and lies on the
line 10 x  8 y  0 . Find the exact value of the six trigonometric functions of the
angle  .
First, take the equation for the line and solve for y.
10 x  8 y  0 
10 x  8 y 
y
10
x 
8
y
5
x
4
Pick a point on this line that lies in the third quadrant. Since x-coordinates are
negative in the third quadrant, you will need to choose a negative number for the x5
y

(  4 )   5 . Thus, the point
x


4
coordinate of the point. If
, then
4
5
(  4 ,  5 ) lies on the line y  x in the third quadrant. Since the terminal side of
4
5
 lies on the line y  x , then the point (  4 ,  5 ) lies on the terminal side of
4
 . This point lies on a circle of radius r  16  25  41 . Animation of the
line, point, and the angle.
cos   
4
sin   
5
tan  
5
4
41
41
sec   
csc   
cot  
41
4
41
5
4
5
NOTE: In Lesson 9, we will find that the angle  is approximately 231.3  or any
angle coterminal to this angle.
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
5
5
x ? Answer:
4
4
5
What was the tangent of the angle  ? Answer:
4
Question: What was the slope of the line y 
Example The terminal side of the angle  is in the second quadrant and lies on the
line y   4 x . Find the exact value of the six trigonometric functions of the angle
.
Pick a point on the given line that lies in the second quadrant. Since x-coordinates
are negative in the second quadrant, you will need to choose a negative number for
the x-coordinate of the point. If x   1 , then y   4 (  1 )  4 . Thus, the point
(  1, 4 ) lies on the line y   4 x in the second quadrant. Since the terminal side
of  lies on the line y   4 x , then the point (  1, 4 ) lies on the terminal side of
 . This point lies on a circle of radius r 
line, point, and the angle.
1
cos   
sin  
17
4
17
tan    4
1  16 
17 . Animation of the
sec    17
csc  
cot   
17
4
1
4
NOTE: In Lesson 9, we will find that the angle  is approximately 104 .0  or any
angle coterminal to this angle.
Question: What was the slope of the given line y   4 x ? Answer:  4
What was the tangent of the angle  ? Answer:  4
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Given the graph of the line y  m x and the angle  below, then tan   m . This
will be proved in the next lesson.
y
y  mx 

x
Back to Topics List
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330