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A2HCh0606 Inverse Trigonometric Functions
Homework and Reading
Read p4505–510
(1) HW p511 #1– 47 odd, skip 17
(2) HW p512 #49–73 odd. 91, 97,
skip 63
Goal
Inverse Sine Function
p1
Students evaluate and graph the inverse sine and other trigonometric functions, and
evaluate and graph the compositions of trigonometric functions.
Inverse Sine Function
Def. of Inverse Sine Function
The inverse sine function is defined by
y = sin x
Horizontal
Line Test
1
! !
! !
-!
!
"
2
!
2
0
(
!
!
2!
where ! 1 " x " 1 and !# 2 " y " # 2 . The domain of
y = arcsin x is [ !1, 1] , and the range is $% ! # 2 , # 2 &' .
!
#
"
#
$
-1
If we restrict the domain to this
interval, we have an inverse to sin x
Example
A
The following holds true for the interval #$ !" 2 , " 2 %&
sin !1 x = y iff sin y = x
II y = sin x takes on its full range of values. ! 1 ' sin x ' 1
III y = sin x is one – to – one.
Example
B
Find sin !1 (1) ( Find the arcsine of 1)
sin !1 (1) = y " sin y = 1
sin # 2 = 1
y = sin x is increasing.
I
)
y = arcsin x y = sin !1 x if and only if sin y = x
( )
Find arcsin 1 2
arcsin x = y iff sin y = x
( )
arcsin 1 2 = y ! sin y = 1 2
sin " 6 = 1 2 see p474 common angles
Try : Find the following
B
4 or 45º
# " 4 or # 45º
" or 180º
#1
C
Other Inverse Trigonometric
Functions
"
sin #1 2 2
arcsin $ # 2 2 &
%
'
A
sin 0
Other Inverse Trigonometric Functions
Example
A
Def. of the Inverse Trigonometric Functions
Domain
Range
y = arcsin x iff sin y = x
Function
!1" x "1
y = arccos x iff cos y = x
!1" x "1
!# 2 " y "# 2
y = arctan x iff tan y = x
!$"x"$
π
cos !1 3 2 = y " cos y = 3 2
cos # 6 = 3 2 see p474 common angles
0" y"#
! # 2 < y2π< # 2
Try : Find the following
!
2
π
π
A
!
2
-1 0
!
1
-4
"
2
-3
-2
-1 0
!
-1 0
y = arccos x
y = arcsin x
Using a Calculator
1
"
2
Use a calculator to approximate the values
A sin !1 0.5524 (use radians)
2nd sin !1 .5524 enter
sin !1 0.5524 " 0.5852
arccos 0.5 (use degrees)
2nd cos !1 .5 enter
cos !1 0.5 = 60º
Try : Find the following
A
B
C
tan !1 3 3
(deg )
#
%
3
arccos !
2 & (rad)
$
arctan 0
(deg )
30º
about 2.617
180º
1
-2π
2
#
tan !1 1
(
arccos ! 1 2
3
4
C arctan 0
B
-π
y = arctan
x
Example
B
Find cos !1 3 2
cos !1 x = y iff cos y = x
)
4 or 45º
5# or 120º
6
# or 0 or 180º
A2HCh0606 Inverse Trigonometric Functions
Composition of Functions
Composition of Functions
Evaluate each
A sin ( arcsin 0.12 )
QR : §2.7
For all x in the domain of f and f !1 , inverse function
have the properties : f f !1 ( x ) = x and f !1 ( f ( x )) = x
(
if ! 1 " x " 1, sin ( arcsin x ) = x
)
if ! 1 " 0.12 " 1, sin ( arcsin 0.12 ) = 0.12
Inverse Properties of Trigonometric Functions
If ! 1 " x " 1 and ! # " y " # , then
2
2
sin ( arcsin x ) = x and arcsin ( sin y ) = y
(
arctan tan 5# 6
B
( ( ))
If x is a real number and ! # 2 " y " # 2 , then
tan ( arctan x ) = x and arctan ( tan y ) = y
(
)
if ! # 2 " y " # 2 , arctan ( tan y ) = y
# < 5# , Find the coterminal, 5# ! # = ! #
2
6
6
6
If ! 1 " x " 1 and 0 " y " # , then
cos ( arccos x ) = x and arccos ( cos y ) = y
Try : Evaluate each
A arccos cos 7! 2
p2
Example
arctan tan ! # 6 = ! # 6
)
if 0 " y " ! , arccos ( cos y ) = y
! < 7! 2 , Find the coterminal, 7! 2 # 3! = ! 2
(
)
arccos cos ! 2 = ! 2
B
sin ( arcsin ( #0.2 ))
if # 1 " x " 1, sin ( arcsin x ) = x
if # 1 " #0.2 " 1, sin ( arcsin ( #0.2 )) = #.2
Evaluating Composition of
Functions
Example
Example
Evaluating Composition of Functions
A Find the exact value of cos sin !1 3 5
sin !1 x = y iff sin y = x
opp
sin !1 3 5 = y " sin y = 3 5 =
hyp
let u = sin !1 3 5 , find cos ( u )
adj
Fact cos =
, use a triangle.
hyp
B
(
)
(
y
5
a2 + b2 = c2
a = c !b
2
2
( ( ))
Find the exact value of sin tan !1 ! 1 2
tan !1 x = y iff tan y = x
opp
tan !1 ! 1 2 = y " tan y = ! 1 2 =
adj
let u = tan !1 ! 1 2 , find sin ( u )
opp
Fact sin =
, use a triangle.
hyp
!
a
3
Example
sin ( u ) = !
y
c
3x
x
1
c
c = 9x + 1
4$
!
Find the exact value of sec # arcsin &
5%
"
arcsin x = y iff sin y = x
opp
arcsin 4 5 = y ' sin y = 4 5 =
hyp
let u = arcsin 4 5 , find sec ( u )
hyp
Fact. sec ( =
, use a triangle
adj
a2 + b2 = c2
y
5
!
a
4 x
a 2 = 25 ) 16 = 9
a=3
5
sec ( u ) =
3
Try :
Find sin cos !1 x
Try :
5%
"
Find the exact value of csc $ arctan! '
12 &
#
arctan x = y iff tan y = x
(
(
)
cos !1 x = y iff cos y = x
x adj
cos !1 x = y " cos y = =
1 hyp
)
opp
arctan ! 5 12 = y ( tan y = ! 5 12 =
adj
let u = arctan ! 5 12 , find csc ( u )
hyp
Fact. csc ) =
, use a triangle
opp
)
1
5
a2 = c2 ) b2
c = 9x 2 + 1
3x
sin ( u ) =
9x 2 + 1
c 2 = 144 + 25 = 169
c = 13
13
csc ( u ) = !
5
!1
Try :
!
2
c2 = a2 + b2
x
2
!
c2 = 5
a2 + b2 = c2
)
(
y
c= 5
( 3x )2 + 12 = c 2
(
)
)
12 + 2 2 = c 2
2
Find the exact value of sin ( arctan 3x )
arctan x = y iff tan y = x
3x opp
arctan 3x = y ! tan y =
=
1
adj
let u = arctan 3x, find sin ( u )
opp
Fact sin =
, use a triangle.
hyp
2
(
(
a2 + b2 = c2
x
a 2 = 25 ! 9 = 16
a=4
4
cos ( u ) =
5
C
)
let u = cos !1 x, find sin ( u )
opp
Fact. sin # =
, use a triangle
hyp
y
x
12
!
c
!5
y
1
c2 = a2 + b2
12 = a 2 + x 2
a2 = 1 ! x 2
a = 1 ! x2
sin ( u ) =
1 ! x2
= 1 ! x2
1
!
a
x
x