Download Math 36 Fall T08 6.1 Inverse Trigonometric Functions

Math 36 "Fall ’08"
6.1 "Inverse Trigonometric Functions"
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Skills Objectives:
*
Develop inverse trigonometric functions
*
Find values of inverse trigonometric functions
*
Graph inverse trigonometric functions
Conceptual Objectives:
*
Understand the di¤erent notations for inverse trigonometric functions
*
Understand why domain restrictions on trigonometric functions are needed for inverse trigonometric functions
*
Extend properties of inverse functions to develop inverse trigonometric identities
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Preliminaries:
A function is one-to-one if it passes the horizontal line test.
Properties of inverse functions:
1: If f is a one to one function, then the inverse function, f
1
2: The domain of f
The range of f
3:
f
1
1
1
; exists.
= the range of f:
= the domain of f .
(f (x)) = x for all x in the domain of f:
f f
1
(x) = x for all x in the domain of f
4: The graphs of f and f
1
1
:
are symmetric about the line y = x
Page: 1
Bibiana Lopez
Math35 (Fall 2008)
6.1 Inverse Trigonometric Functions
Inverse Sine Function
De…nition:
Inverse Sine Function:
The function y = sin
1
(x) or y = arcsin (x)
Important Facts:
(in terms of angles)
is the inverse of the function y = sin (x)
1: Domain: [ 1; 1]
2: Range:
2; 2
WARNING!!!
The inverse sine function should not be interpreted as a reciprocal
Example 1: (Finding exact values of an inverse sine function)
Find the exact value of the following expressions:
1
a)
sin
b)
arcsin
1
2
p
3
2
Page: 2
Bibiana Lopez
Math35 (Fall 2008)
c)
sin
1
6.1 Inverse Trigonometric Functions
(0)
Sine-Inverse Sine Identities:
Example 2: (Using inverse identities to evaluate expressions involving inverse sine functions)
Find exact values of the trigonometric expressions:
a)
sin sin
b)
sin
1
c)
sin
1
p
1
sin
sin
2
2
4
5
6
Page: 3
Bibiana Lopez
Math35 (Fall 2008)
6.1 Inverse Trigonometric Functions
Inverse Cosine Function
De…nition:
Inverse Cosine Function:
The function y = cos
1
(x) or y = arccos (x)
Important Facts:
(in terms of angles)
is the inverse of the function y = cos (x)
1: Domain: [ 1; 1]
2: Range: [0; ]
Example 3: (Finding exact values of an inverse cosine function)
Find the exact value of the following expressions:
p
2
2
1
a)
cos
b)
arccos (0)
c)
cos
1
1
2
Page: 4
Bibiana Lopez
Math35 (Fall 2008)
6.1 Inverse Trigonometric Functions
Cosine-Inverse Cosine Identities:
Example 4: (Using inverse identities to evaluate expressions involving inverse cosine functions)
Find exact values of the trigonometric expressions:
a)
cos cos
b)
cos
1
c)
cos
1
1
2
1
cos
4
cos 74
Inverse Tangent Function
De…nition:
Inverse Tangent Function:
The function y = tan
1
(x) or y = arctan (x)
Important Facts:
(in terms of angles)
is the inverse of the function y = tan (x)
1: Domain: ( 1; 1)
2: Range:
2; 2
Example 5: (Finding exact values of an inverse tangent function)
Find the exact value of the following expressions:
p
a)
tan 1
3
b)
arctan ( 1)
Page: 5
Bibiana Lopez
Math35 (Fall 2008)
c)
tan
1
6.1 Inverse Trigonometric Functions
(0)
Tangent-Inverse Tangent Identities:
Example 6: (Using inverse identities to evaluate expressions involving inverse tangent functions)
Find exact values of the trigonometric expressions:
a)
tan tan
b)
tan
1
1
17
tan 23
Remaining Inverse Trigonometric Functions
Page: 6
Bibiana Lopez
Math35 (Fall 2008)
Inverse Function
6.1 Inverse Trigonometric Functions
Domain
Range
Page: 7
Graph
Bibiana Lopez
Math35 (Fall 2008)
6.1 Inverse Trigonometric Functions
Example 7: (Finding the exact value of inverse trigonometric functions)
Find the exact value of the following expressions:
p
a)
cot 1 3
b)
csc
1
p
c)
sec
1
( 2)
2
Inverse Secant, Cosecant and Cotangent Identities:
Example 8: (Using inverse identities)
Find the exact value of
sec
1
2
Page: 8
Bibiana Lopez
Math35 (Fall 2008)
6.1 Inverse Trigonometric Functions
Finding Exact Values for Expressions Involving Inverse Trigonometric functions
Now we will …nd exact values of trigonometric expressions that involve inverse trigonometric functions.
Example 9: (Finding exact values of trigonometric expressions involving inverse trigonometric functions)
Find the exact value of:
1
a)
tan
b)
cos sin
1
2
3
c)
sin cos
1
1
3
sin
2
Page: 9
Bibiana Lopez