DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS

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
P ( ) ( x, y ) r
r
- r r x
- r
Definition Let P ( ) ( x, y ) r 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
r
x
cos
x
r
sec , provided that x 0
r
y
sin
y
r
csc , provided that y 0
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
x
y
tan , provided that x 0
y
x
cot , provided that y 0
where
2 2 r x y
NOTE: This is exactly the same definition that was given in Section 1 of Lesson 2 .
2 2 r x y is the distance from the point P ( ) ( x, y ) r to the origin (0, 0) ,
which is the radius of the circle whose equation is
2 2 2 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.
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.
5
4
10
8
cos
r
x
4
5
sec
5
3
10
6
sin
r
y
3
5
csc
4
3
8
6
tan
x
y
3
4
cot
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
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.
14
7
2( 7)
7
2 7
1
28
1
28
1
84
3
84
3
cos
r
x
2 7
1
cos sec 28 2 7