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
y x
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3sec 2 24
y x
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3sec 2 24
y x
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4.7 – Inverse Trigonometric Functions
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In this section, you will learn to:
Evaluate the inverse trigonometric functions
Evaluate the composition of trigonometric functions
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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.
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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
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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
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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.
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Inverse Sine Function:
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Inverse Sine Function:
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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
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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
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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
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Graphing an Inverse Sine Function: Inverse functions are reflected about the
line y = x.
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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.
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Inverse Cosine Function:
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Inverse Cosine Function:
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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
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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
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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
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Graphing an Inverse Cosine Function: Inverse functions are reflected about the
line .y x
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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.
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Inverse Tangent Function:
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Inverse Tangent Function:
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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
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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
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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
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Graphing an Inverse Tangent Function: Inverse functions are reflected about the
line .y x
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Find the exact value of the inverse functions:
21) arccos :
2
32) arctan :
3
3) arccos 1 :
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Solutions:
2 31) arccos ; 0
2 4y
3) arcsin 1 ;2 2 2
y
32) arctan ;
3 6 2 2y
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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
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Composition of Function Examples:
21) sin arcsin :
2
12) cos arcsin :
2
3) tan arcsin 1 :
2
2
3cos
6 2
tan undefined2