4.4.5 general inverse circular functions (3)
TRANSCRIPT
Chapter 4.4.5 Inverse Circular
Functions
1
SUMMARY Function Domain Range
Arcsinx
Arccos x
Arctan x
Arccot x
Arccsc x
Arcsec x
2
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,
3
Recall
For an inverse of a function to be in itself a
function, the function must be 1-1.
Are circular functions 1-1?
4
x
y
Restricting the Domain
Given sin we consider , as2 2
the restricted domain.
f x x
2
2
5
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
6
Inverse Sine Function
2
2
1
1
11
2
2
7
Inverse Sine Function
2
2
1
1
8
Inverse Sine Function
2 2
2 2
Recall: 1,1 and , for sin .
Form: sin
1,1 and sin ,
Hence,
1 1Dom , and ,
2 2
x y y Arc x
y f x aArc bx
bx aArc bx
a af Rng f
b b
9
Illustration
Consider 3 sin2
2,2
3 3,
2 2
xf x Arc
Dom f
Rng f
10
x -2 0 2
y 0 3
2
3
2
22
3
2
3
2
Restricting the Domain
Given cos we consider 0, as
the restricted domain.
f x x
11
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
12
Inverse Cosine Function
0
1
111 0
13
Inverse Cosine Function
0
1
1
14
Inverse Cosine Function
Recall: 1,1 and 0, for cos .
Form: cos
1,1 and cos 0,
1 1Dom ,
0, , 0
,0 , 0
x y y Arc x
y f x aArc bx
bx aArc bx
fb b
a aRng f
a a15
Illustration
Consider 2 cos 2
1 1,
2 2
2 ,0
f x Arc x
Dom f
Rng f
16
x -1/2 0 ½
y 0 2
1
21
2
2
0
Restricting the Domain
2 2Given tan we consider , as
the restricted domain.
f x x
2
2
17
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 an
nct
d
io
.
n
,
f
y x x y
x R y
18
Inverse Tangent Function
2
2
2
2
19
Inverse Tangent Function
20
Inverse Tangent Function
2 2Recall: and , for tan .
Form: f tan
Dom R and ,2 2
x R y y Arc x
x y aArc bx
a af Rng f
21
Illustration
1Consider tan 3
1 1,
2 2
f x Arc x
Dom f R
Rng f
22
x -1/3 0 1/3
y 1/4 0 -1/4
1
2
1
2
End of Chapter 4.4.5
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