DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS

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 (30 )
6
, (45 )
4
, AND (60 )
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
P ( ) ( x, y ) r
- r r x
r
- r
2 2 2 x y r
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
Definition Let P ( ) ( x, y ) r be the point of intersection of the terminal side of
the angle with the circle whose equation is
2 2 2 x y r . 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
x
y
tan , provided that x 0
y
x
cot , provided that y 0
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
x
1
sec , provided that x 0
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
sin y
y
1
csc , provided that y 0
x
y
tan , provided that x 0
y
x
cot , provided that y 0
y
1
s
P( ) ( x, y)
- 1 1 x
1
- 1 1 2 2 x y (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