Download Finding Values of the Trigonometric Functions

Chapter 4
Trigonometric
Functions
4.2 Trigonometric
Functions: The Unit Circle
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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.
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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
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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.
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The Six Trigonometric Functions
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Definitions of the Trigonometric Functions in Terms of a
Unit Circle
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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
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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
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The Domain and Range of the Sine and Cosine Functions
y = cos x
y = sin x
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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.
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
Exact Values of the Trigonometric Functions at t 
4
(continued)
We find these coordinates as follows:
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
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 

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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
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Example: Finding Values of the Trigonometric Functions

at t 
4
Find cot

4
2
 x
cot   2  1
4 y
2
2
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 2 2
P  
,

 2 2 
14

Trigonometric Functions at
4
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Even and Odd Trigonometric Functions
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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
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Fundamental Identities
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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
sin t  3 = 2 i 3 = 2 = 2 i 5 = 2 5
tan t 
5
5
3 5
cos t
5 5
5
3
1 3
1
 
csc t 
sin t 2 2
3
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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


=
i =
=
cot t 
tan t
2 5 2 5 2 5 5 2i5 2
5
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The Pythagorean Identities
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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
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Definition of a Periodic Function
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Periodic Properties of the Sine and Cosine Functions
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Periodic Properties of the Tangent and Cotangent
Functions
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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

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Repetitive Behavior of the Sine, Cosine, and Tangent
Functions
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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.
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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
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