Finding Values of the Trigonometric Functions

Download Report

Transcript Finding Values of the Trigonometric Functions

Chapter 4
Trigonometric
Functions
4.2 Trigonometric
Functions: The Unit Circle
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
1
Objectives:
• Use a unit circle to define trigonometric functions of real
numbers.
• Recognize the domain and range of sine and cosine
functions.

• Find exact values of the trigonometric functions at .
4
• Use even and odd trigonometric functions.
• Recognize and use fundamental identities.
• Use periodic properties.
• Evaluate trigonometric functions with a calculator.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
2
The Unit Circle
A unit circle is a circle of radius 1, with its center at the
origin of a rectangular coordinate system. The equation
x2  y 2  r 2.
of this circle is
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
3
The Unit Circle (continued)
In a unit circle, the radian measure of the central
angle is equal to the length of the intercepted arc.
Both are given by the same real number t.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
4
The Six Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
5
Definitions of the Trigonometric Functions in Terms of a
Unit Circle
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
6
Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric
functions at t.
1
sin t  y 
2
3
cost  x 
2
1
1
1 3
3
y
2



tan t  
x
3
3
3 3 3
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
7
Example: Finding Values of the Trigonometric Functions
Use the figure to find the values of the trigonometric
functions at t.
1 1
1
1
2
2 3 2 3
csct    2
sect  



y 1
x
3
3
3
3 3
2
2
3
x
cot t   2  3
1
y
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
8
The Domain and Range of the Sine and Cosine Functions
y = cos x
y = sin x
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
9
Exact Values of the Trigonometric Functions at t 

4

Trigonometric functions at t 
occur frequently. We use
4
the unit circle to find values of the trigonometric functions at
 We see that point P = (a, b)
t .
4
lies on the line y = x. Thus, point P
has equal x- and y-coordinates: a = b.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
10

Exact Values of the Trigonometric Functions at t 
4
(continued)
We find these coordinates as follows:
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
11

Exact Values of the Trigonometric Functions at t 
4
(continued)
We have used the unit circle to find the coordinates of point
P = (a, b) that correspond to t   .
4
 2 2
P  
,

2
2 

Copyright © 2014, 2010, 2007 Pearson Education, Inc.
12
Example: Finding Values of the Trigonometric Functions

at t 
4
Find csc

4

1
1
2 2
2
csc  

 2

4 y
2
2
2
2
Find sec
 2 2
P  
,

 2 2 

4

1
2 2
2
1

 2

sec  
2
2
4 x
2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
13
Example: Finding Values of the Trigonometric Functions

at t 
4
Find cot

4
2
 x
cot   2  1
4 y
2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
 2 2
P  
,

 2 2 
14

Trigonometric Functions at
4
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
15
Even and Odd Trigonometric Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
16
Example: Using Even and Odd Functions to Find Values
of Trigonometric Functions
Find the value of each trigonometric function:




sec     sec    2
 4
4


2


sin      sin    
2
 4
4
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
17
Fundamental Identities
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
18
Example: Using Quotient and Reciprocal Identities
5
2
Given sin t  and cos t 
find the value of each of
3
3
the four remaining trigonometric functions.
2
2 5 2 5
2 3
2
sin t  3
= i =
= i =
tan t 
5
5
3 5
cos t
5 5
5
3
1 3
1
 
csc t 
sin t 2 2
3
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
19
Example: Using Quotient and Reciprocal Identities
(continued)
5
2
Given sin t  and cos t 
find the value of each of
3
3
the four remaining trigonometric functions.
3 5 3 5
1
3
1
= i =


sec t 
5
cos t
5
5
5 5
3
1
5
1
5
5 5 5
5


cot t 
=
i =
=
tan t
2 5 2 5 2 5 5 2i5 2
5
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
20
The Pythagorean Identities
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
21
Example: Using a Pythagorean Identity
1

Given that sin t  and 0  t  , find the value of
2
2
cost using a trigonometric identity.
sin t  cos t  1
2
2
2
1
cos t  1 
4
2
 1   cos 2 t  1
 
2
3
cos t 
4
1
 cos 2 t  1
4
3
3
cos t 

4 2
2
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
22
Definition of a Periodic Function
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
23
Periodic Properties of the Sine and Cosine Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
24
Periodic Properties of the Tangent and Cotangent
Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
25
Example: Using Periodic Properties
Find the value of each trigonometric function:


5


 cot      cot  1
cot
4
4
4


9 
9 

2




cos     cos    cos   2   cos 
4
2
 4 
 4 
4

Copyright © 2014, 2010, 2007 Pearson Education, Inc.
26
Repetitive Behavior of the Sine, Cosine, and Tangent
Functions
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
27
Using a Calculator to Evaluate Trigonometric Functions
To evaluate trigonometric functions, we will use the
keys on a calculator that are marked SIN, COS, and
TAN. Be sure to set the mode to degrees or radians,
depending on the function that you are evaluating. You
may consult the manual for your calculator for specific
directions for evaluating trigonometric functions.
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
28
Example: Evaluating Trigonometric Functions with a
Calculator
Use a calculator to find the value to four decimal places:
sin

4
 0.7071
csc1.5  1.0025
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
29