Math 36 "Fall ’08"
3.4 "De…nition 3 of Trigonometric Functions: Unit Circle Approach"
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Skills Objectives:
*
Draw a unit circle with special angles and label cosine and sine values.
*
Determine the domain and range of circular functions.
*
Classify circular functions as even or odd.
Conceptual Objectives:
*
De…ne trigonometric functions using the unit circle approach.
*
Relate x-coordinate and y-coordinate of points on a unit circle to the values of cosine and sine functions.
*
Visualize periodic properties of circular functions.
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Preliminaries:
Recall that the …rst de…nition of trigonometric functions we developed was in terms of ratios of sides of right triangles
(Section 1.3).
Then, in Section 2.2 we superimposed right triangles in the Cartesian plane, which led to a second de…nition
of trigonometric functions (for any angle) in terms of ratios of x and y coordinates of a point and the distance from the
origin to that point.
In this section we inscribe the right triangles into the unit circle in the Cartesian plane, which will yield a third de…nition
of trigonometric functions.
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Trigonometric Functions and the Unit Circle
Recall that the equation for a unit circle centered at the origin is given by:
1
This is how we obtained the following de…nition:
De…nition:
If
"Sine and Cosine as Coordinates in the Unit Circle"
is an angle in standard position and (x; y) is the point of intersection of
the terminal side and the unit circle then:
Example 1: ( Evaluating sine and cosine at multiples of
2)
Example 2: ( Evaluating sine and cosine at multiples of
4
2
)
Example 3: ( Evaluating sine and cosine at multiples of
6)
Circular Functions
De…nition (3) :
"Trigonometric Functions"
Unit Circle Approach
Let (x; y) be any point on the unit circle. If
is a real number that represents the distance from
the point (1; 0) along the circumference to the point (x; y) ; then
The coordinates of the points along the unit circle can be written as _________________
and since
is a real number, the trigonometric functions are often called circular functions.
3
Example 4:
(Finding exact circular function values)
Find the exact values for:
a)
tan 23
b)
cot 32
c)
sec 116
d)
csc 34
Example 5:
(Solving equations involving circular functions)
Use the unit circle to …nd all values of
a)
sin =
b)
cos =
c)
tan = 1
, 0
2 , for which:
1
2
p
2
2
4
Properties of Circular Functions
Words
Math
Domains and Ranges of the Circular Functions
For any real numbers, ; and integer, n,
Function
Domain
Range
sin
cos
tan
csc
sec
cot
Recall from algebra that even and odd functions have both an algebraic and graphical interpretation
Algebraically:
-
Even Function:
-
Odd Function:
Graphically:
5
(Using properties of circular function)
Example 6:
Evaluate the following:
a)
cos
5
6
b)
sin
4
3
c)
sin
3
2
d)
cos
Example 7:
7
4
(Even and odd circular functions)
Show that secant is an even function
6