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In this section, we will introduce the
inverse trigonometric functions and
construct their derivative formulas.
A function is called one-to-one if whenever f ( x1
) =
f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
A function is called one-to-one if whenever f ( x1
) =
f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
5
2.5
-2.5
This function is not one-to-one. 0
2.5
A function is called one-to-one if whenever f ( x1
) =
f ( x2 )
it must be true that x1 = x2.
That is, an output value cannot come from two
different input values.
Significance:
Only one-to-one functions have inverse functions.
Below is shown the graph of
y = f ( x ) = sin ( x )
-5
-2.5
0
2.5
5
This function is not one-to-one and so has no inverse function.
Below is shown the graph of
y = f ( x ) = sin ( x )
-5
-2.5
0
2.5
This function has an inverse.
5
Consider restricting the domain
of the sine function to:
[! !2 , !2 ]
This is the function in blue
shown to the left. The function f (
x ) = arcsin
( x ) is the inverse of the sine
function with restricted domain
[! !2 , !2 ].
That is, the arcsin(x) is the angle θ in the interval [! !2 , !2 ]
with sin
(! ) = x .
Below is shown the graph of
y = f ( x ) = cos ( x )
-5
0
5
This function is not one-to-one and so has no inverse function.
Below is shown the graph of
y = f ( x ) = cos ( x )
-5
0
This function has an inverse.
5
Consider restricting the domain of
the cosine function to:
[ 0, ! ! ]
This is the function in blue shown
to the left. The function f (
x ) = arccos
( x ) is the inverse of the cosine
function with restricted domain
[ 0, ! ! ] .
That is, the arccos(x) is the angle θ in the interval [ 0, ! ! ]
with cos
(! ) = x .
Below is shown the graph of
y = f ( x ) = tan ( x )
-5
0
5
This function is not one-to-one and so has no inverse function.
Below is shown the graph of
y = f ( x ) = tan ( x )
-5
0
5
This function has an inverse function.
Consider restricting the domain of
the tangent function to:
(! !2 , !2 )
This is the function in blue shown
to the left. The function f (
x ) = arctan
( x ) is the inverse of the tangent
function with restricted domain
(! !2 , !2 ).
That is, the arctan(x) is the angle θ in the interval (! !2 ,
with tan
(! ) = x .
!
2
)
Use the definitions of the section to find the exact value of tan ( arcsin
( 97 )) . Use the definitions of the section to find the exact value of csc ( arccos
(! 114 )) . Use the definitions of the section to find the exact
(
)
value of sec arctan
( x 2 ) . Use the definitions of the section to find the exact
(
value of cos arcsin
( x )). The following are true:
f ( x ) = arcsin x
! f "( x) =
f ( x ) = arccos x
! f "( x) =
f ( x ) = arctan x
! f "( x) =
1
1# x 2
#1
1# x 2
1
1+ x 2
The following are true:
f ( x ) = arcsec x
! f "( x) =
f ( x ) = arccsc x
! f "( x) =
f ( x ) = arccot x
! f "( x) =
1
x
x 2 #1
#1
x
x 2 #1
#1
1+ x 2
Find the derivative of the function f ( x ) = arctan ( x 2 ) . Find the derivative of the function f ( x ) = arcsin ( e3x ) . Find the derivative of the function f ( x ) = arccos ( ln x ).