4.4 inverse circular functions
35
Chapter 4.4 Inverse Circular Functions 1
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Transcript of 4.4 inverse circular functions
Chapter 4.4 Inverse Circular
Functions
1
Recall
For an inverse of a function to be in itself a
function, the function must be 1-1.
Are circular functions 1-1?
2
x
y
Restricting the Domain
Given sin we consider , as2 2
the restricted domain.
f x x
2
2
3
Inverse Sine Function
2 2
2 2
inverse sine functio
Let be the sine function with domain , .
Then the is defined as
Arcsin if and only i
n
f sin
where 1,1 and , .
f
y x x y
x y
4
Inverse Sine Function
2
2
1
1
11
2
2
5
Inverse Sine Function
2
2
1
1
6
Example 4.4.1
Evaluate the following.
1. Arcsin 0 0
2. Arcsin 12
13. Arcsin
2 6
34. Arcsin
2 3
25. Arcsin
2 4
7
Restricting the Domain
Given cos we consider 0, as
the restricted domain.
f x x
8
Inverse Cosine Function
inverse cosine funct
Let be the cosine function with domain 0, .
Then the is defined as
Arccos if and only if cos
where 1,1 an
ion
d 0, .
f
y x x y
x y
9
Inverse Cosine Function
0
1
111 0
10
Inverse Cosine Function
0
1
1
11
Example 4.4.2
Evaluate the following.
1. Arccos 02
2. Arccos 1 0
1 23. Arccos
2 3
34. Arccos
2 6
25. Arccos
2 4
12
Restricting the Domain
2 2Given tan we consider , as
the restricted domain.
f x x
2
2
13
Inverse Tangent Function
2 2
2 2
inverse tangent fu
Let be the tangent function with domain , .
Then the is defined as
Arctan if and only if tan
where and , .
nction
f
y x x y
x y
14
Inverse Tangent Function
2
2
2
2
15
Inverse Tangent Function
16
Example 4.4.3
Evaluate the following.
1. Arctan 0 0
2. Arctan 14
33. Arctan
3 6
4. Arctan 33
17
Restricting the Domain
Given cot we consider 0, as
the restricted domain.
f x x
0
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Inverse Cotangent Function
inverse cotangent
Let be the cotangent function with domain 0, .
Then the is defined as
Arccot if and only if cot
where a
functio
n 0, .
n
d
f
y x x y
x y
19
Inverse Cotangent Function
0
0
20
Inverse Cotangent Function
21
Example 4.4.4
Evaluate the following.
31. Arccot 1
4
3 22. Arccot
3 3
3. Arccot 36
4. Arccot 0 =2
22
Restricting the Domain
2 2Given sec we consider 0, , as
the restricted domain.
f x x
0
2
1
1
23
Inverse Secant Function
2 2
2 2
inverse secant function
Let be the secant function with domain 0, , .
Then the is defined as
Arcsec if and only if sec
where , 1 1, and 0, , .
f
y x x y
x y
24
Inverse Secant Function
0
2
1
1 101
2
25
Inverse Secant Function
26
Example 4.4.5
Evaluate the following.
1. Arcsec 0 is undefined.
2. Arcsec 1 0
33. Arcsec 2
4
4. Arcsec 23
27
Restricting the Domain
2 2Given csc we consider ,0 0, as
the restricted domain.
f x x
0
2
2
1
1
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Inverse Cosecant Function
2 2
2 2
inverse cosecant functi
Let be the cosecant function with domain ,0 0, .
Then the is defined as
Arccsc if and only if csc
where , 1 1, and
o
,0
n
0, .
f
y x x y
x y
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Inverse Cosecant Function
2
2
1
1
10
01
2
2
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Inverse Cosecant Function
31
Example 4.4.5
Evaluate the following.
2 31. Arccsc
3 3
2. Arccsc 12
3. Arccsc 24
4. Arccsc 26
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Challenge!
Evaluate the following.
1 31. Arccos Arcsin
2 2
2. Arccos cos3
33. Arcsec sin
2
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SUMMARY Function Domain Range
Arcsinx
Arccos x
Arctan x
Arccot x
Arccsc x
Arcsec x
,2 2
,2 2
0,
0,
, 02 2
0,2
1,1
1,1
, 1 1,
, 1 1,
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End of Chapter 4.4
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