4.2 TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE

advertisement
4.2
TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE
Copyright © Cengage Learning. All rights reserved.
What You Should Learn
• Identify a unit circle and describe its relationship
to real numbers.
• Evaluate trigonometric functions using the unit
circle.
• Use the domain and period to evaluate sine and
cosine functions.
• Use a calculator to evaluate trigonometric
functions.
2
The Unit Circle
3
The Unit Circle
The unit circle
x2 + y2 = 1
4
The Unit Circle
Each real number t also corresponds to a central angle 
(in standard position) whose radian measure is t.
The real number t is the (directional) length of the arc
intercepted by the angle  , given in radians.
5
The Trigonometric Functions
6
The Trigonometric Functions
7
The Trigonometric Functions
The unit circle has been divided into eight equal arcs,
corresponding to t-values of
8
The Trigonometric Functions
The unit circle has been divided into 12 equal arcs,
corresponding to t-values of
9
Example 1 – Evaluating Trigonometric Functions
Evaluate the six trigonometric functions at each real
number.
a.
b.
c. t = 0 d. t = 
Solution:
a.
corresponds to the point (x, y) =
.
10
Example 1 – Solution
cont’d
11
Example 1 – Solution
b.
corresponds to the point (x, y) =
cont’d
.
12
Example 1 – Solution
cont’d
13
Example 1 – Solution
cont’d
c. t = 0 corresponds to the point (x, y) = (1, 0).
sin 0 = y = 0
cos 0 = x = 1
tan 0 =
=
=0
14
Example 1 – Solution
cont’d
is undefined.
is undefined.
15
Example 1 – Solution
cont’d
d. t =  corresponds to the point (x, y) = (–1, 0).
sin  = y = 0
cos  = x = –1
tan  =
=
=0
16
Example 1 – Solution
cont’d
is undefined.
is undefined.
17
Domain and Period of Sine and Cosine
18
Domain and Period of Sine and Cosine
The domain of the sine and cosine functions is the set of all
real numbers.
–1 
y 1
–1  Sin t  1
–1 
x 1
–1  cos t  1
19
Domain and Period of Sine and Cosine
sin (t + 2 n) = sin t
cos (t + 2 n) = cos t
20
Domain and Period of Sine and Cosine
21
Example 3 – Using the Period to Evaluate the Sine and Cosine
a. Because
b. Because
, you have
, you have
22
Example 3 – Using the Period to Evaluate the Sine and Cosine
c. For
odd.
,
cont’d
because the sine function is
23
Download