warm-up: 9/14/12 find the amplitude, period, vertical asymptotes, domain, and range. sketch the...
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Warm-Up: 9/14/12Find the amplitude, period, vertical asymptotes, domain, and range. Sketch the graph.
3sec 2 24
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3sec 2 24
y x
3sec 2 24
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4.7 – Inverse Trigonometric Functions
In this section, you will learn to:
Evaluate the inverse trigonometric functions
Evaluate the composition of trigonometric functions
Functions: In order for a relation to be a function, it must
pass the vertical line test.
For a function to have an inverse, it must pass the horizontal line test.
Different values of x cannot yield the same values of y.
Function or not a function?
1)
2) 3)
1,2 1, 2 2,3 2, 5
1 2 3 4 5-1-2
1
2
3
-1
-2
-3
x
y
1 2 3-1-2-3-4-5
1
2
3
-1
-2
-3
x
y
Which function has an inverse?
1)
2) 3)
1,2 1, 2 2,3 2, 5
1 2 3 4 5-1-2
1
2
3
-1
-2
-3
x
y
1 2 3-1-2-3-4-5
1
2
3
-1
-2
-3
x
y
Inverse Sine Function: The sine function does not pass the horizontal
test, therefore it does not have an inverse.
However, if we restrict the domain, then it will pass the horizontal line test.
Inverse Sine Function:
Inverse Sine Function:
Inverse Sine Function: If we restrict the domain to the interval
, then it will pass the horizontal 2 2
line test. On this restricted interval, sin
has a unique inverse called the inverse sine
function denoted as ar
x
y x
y
1csin or sin .x y x
Definition of an Inverse Sine Function:
1
The inverse sine function is defined by
arcsin or sin if and only if
sin where 1 1 and .2 2
y x y x
y x x y
The domain of arcsin is 1,1 and the
range is , .2 2
y x
Graphing an Inverse Sine Function: To sketch the graph of an inverse sine
function, interchange the domain and the range of the original sine function.
y
sinx y 1
2
6
1
2
2
11
2
6
Graphing an Inverse Sine Function: Inverse functions are reflected about the
line y = x.
Inverse Cosine Function: The cosine function does not pass the horizontal
test, therefore it does not have an inverse.
However, if we restrict the domain, then it will pass the horizontal line test.
Inverse Cosine Function:
Inverse Cosine Function:
Inverse Cosine Function: If we restrict the domain to the interval
0 , then it will pass the horizontal
line test. On this restricted interval, cos
has a unique inverse called the inverse cosine
function denoted as arc
x
y x
y
1cos or cos . x y x
Definition of an Inverse Cosine Function:
1
The inverse cosine function is defined by
arccos or cos if and only if
cos where 1 1 and 0 .
y x y x
y x x y
The domain of arccos is 1,1 and the
range is 0, .
y x
Graphing an Inverse Cosine Function:
To sketch the graph of an inverse cosine function, interchange the domain and the range of the original cosine function.
y
cosx y 1
0 3
4
2
2
12
2
4
Graphing an Inverse Cosine Function: Inverse functions are reflected about the
line .y x
Inverse Tangent Function: The tangent function does
not pass the horizontal test, therefore it does not have an inverse.
However, if we restrict the domain, then it will pass the horizontal line test.
Inverse Tangent Function:
Inverse Tangent Function:
Inverse Tangent Function: If we restrict the domain to the interval
, then it will pass the horizontal 2 2
line test. On this restricted interval, tan
has a unique inverse called the inverse tangent
function denoted as
x
y x
y
1arctan or tan . x y x
Definition of an Inverse Tangent Function:
1
The inverse tangent function is defined by
arctan or tan if and only if
tan where and .2 2
y x y x
y x x y
The domain of arctan is , and the
range is , .2 2
y x
Graphing an Inverse Tangent Function:
To sketch the graph of an inverse tangent function, interchange the domain and the range of the original sine function.
y
tanx y .undef
2
4
1 .undef1
4
2
Graphing an Inverse Tangent Function: Inverse functions are reflected about the
line .y x
Find the exact value of the inverse functions:
21) arccos :
2
32) arctan :
3
3) arccos 1 :
Solutions:
2 31) arccos ; 0
2 4y
3) arcsin 1 ;2 2 2
y
32) arctan ;
3 6 2 2y
Composition of Functions:
1) If 1 1 and , then2 2
sin arcsin and arcsin sin
x y
x x x x
2) If 1 1 and 0 , then
cos arccos and arccos cos
x y
x x x x
3) If 1 1 and , then2 2
tan arctan and arctan tan
x y
x x x x
Composition of Function Examples:
21) sin arcsin :
2
12) cos arcsin :
2
3) tan arcsin 1 :
2
2
3cos
6 2
tan undefined2