Download Example: Finding Values of the Trigonometric Functions

Chapter 5
Trigonometric
Functions
5.4 Trigonometric
Functions of Real
Numbers; Periodic
Functions
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1
Objectives:
Use a unit circle to define trigonometric functions of
real numbers.
Recognize the domain and range of sine and cosine
functions.
Use even and odd trigonometric functions.
Use periodic properties.
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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 of this circle is x 2  y 2  r 2 .
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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.
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4
Definitions of the Trigonometric Functions in Terms of a
Unit Circle
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5
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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6
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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7
The Domain and Range of the Sine and Cosine Functions
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8
Even and Odd Trigonometric Functions
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9
Example: Using Even and Odd functions to Find Exact
Values
Find the exact value of each trigonometric function:
1
a. cos(60)  cos60 
2


3


b. tan      tan    
3
 6
6
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10
Definition of a Periodic Function
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11
Periodic Properties of the Sine and Cosine Functions
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12
Example: Using Periodic Properties to Find Exact Values
Find the exact value of each trigonometric function:
2
a. cos 405  cos(45  360)  cos 45 
2
3

7



 sin   2   sin 
b. sin
2
3
3
3

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13
Periodic Properties of the Tangent and Cotangent
Functions
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14
Repetitive Behavior of the Sine, Cosine, and Tangent
Functions
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15