Inverse Trigonometric Functions: Differentiation
Section 5.8
Section 5.8
259
Inverse Trigonometric Functions: Differentiation
1. y = arcsin x
(a)
X
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y
-1.571
-0.927
-0.644
-0.412
-0.201
0
0.201
0.412
0.644
0.927
1.571
(o)-•EB•
(b)
(d) Symmetric about origin:
arcsin(- x) = -arcsin x
Intercept: (0, 0)
-2
1
5. arcsin 2
3. False.
1
arccos2
7T
=
6
7T
=
3
since the range is [0, 7T].
1
7. arccos 2
=
7T
9. arctan ..)3 _ 7T
3 -6
3
11. arccsc(- ../2)
= -
~
13. arccos(- 0.8)
= 2.50
15. arcsec(l.269) =
arccosC.~69)
= 0.66
17. (a) sin( arctan~)
=
~
19. (a) co{ arcsin(
-k)J
=
cot(-~) = - ..J3
v'3
. 54)
(b) sec(arcsm
~-·
=
35
LJ·
(b) esc[ arctan(-
8 = arcsin 2x
y = cos 8 = .J1 - 4x2
~"
= -
1
:
12
~->
3
21. y = cos(arcsin 2x)
5
)]
12
23. y
=
sin(arcsec x)
7T
(J =I=8 = arcsec x, 0<8~7T,
2
y1-4x2
. y= smfJ-
R-=1
~v?=l
Jxl .L__j
I
The absolute value bars on x are necessary because of
the restriction 0 ~ 8 ~ 7T, 8 =I= 7T/2, and sin 8 for this
domain must always be nonnegative.
Logarithmic, Exponential, and Other Transcendental Functions
Chapter 5
260
25. y =tan(
arcsec~)
(} = arcsec ~
3
y =
_fi)
27. y = esc( arctan
(}=arctan~
.j2
.Jx2 - 9
tan (} = ....:....:.'--::-----'--
y = esc (} =
3
.Jr + 2
....:....:..:...._--=.
X
.h4r
2
-2EI;d2
29. sin( arctan 2x) =
31. arcsin(3x - 7T) = !
+
1
3x-
7T
=sin{!}
x = Hsin(!} + 7T]
= 1.207
E1=J
-2
Asymptotes: y = ± 1
arctan 2x = (}
tan(}= 2x
ZJ
2x
8
•
ll
Slnu=
2x
~
v 1 + 4r
33. arcsin~= arccos
I
Jx
~ = sin( arccos
Jx)
~=~.O~x~1
2x=1-x
3x
1
=
1
x=3
35. (a) arccsc x = arcsin.!, lxl :2: 1
(b) arctan x
X
Let y = arccsc x. Then for
7T
-2
~ y
1
Lety = arctanx
< 0 and 0 < y ~
7T
2•
7T
+ arctan:; = 2· x >
+ arctan(l/x). Then,
tan(arctan x)
tan y -
0
+ tan[arctan(1/x)]
-~--;---'---,----":---.o...,.:......,=.,
- 1 - tan(arctan x) tan[arctan(1/x)]
x + (1/x)
1 - x(1/x)
esc y = x => sin y = 1/x. Thus, y = arcsin(1/x).
Therefore, arccsc x = arcsin(1/x).
=
x
+ (1/x)
0
. .
(wh1ch IS undefined).
Thus, y = 7T/2. Therefore, arctan x
37.
f(x) = arcsin(x - 1)
39.
x- 1 = siny
x = 1
+ siny
Domain: [0, 2]
Range:
[-f. f]
f(x) is the graph of arcsin x
shifted 1 unit to the right.
f(x) = arcsec 2x
2x = secy
(q)
2"
1
x=2secy
-I
-2
"
_,
+ arctan(l/x) = 7T/2.
(o. -~)
Domain: (
-=. -~]. [~. =)
H·"l
J.
------2
-2
-1
Section 5.8
110. Given family (exponential functions): y
Ce"
=
535
4
·EE·
y' = Ce" = y
1
y'= - -
Orthogonal trajectory (parabolas):
Inverse Trigonometric Functions: Differentiation
y
-4
fydy = - fdx
y2
2
=
-x
+ KI
yz = -2x
+K
112. The number of initial conditions matches the number of
constants in the general solution.
114. Two families of curves are mutually orthogonal if each
curve in the first family intersects each curve in the
second family at right angles.
116. True
dy = (x - 2)(y
dx
+
1)
118. True
xz
+ yz
2Cy
=
xz
+ yz
dy=~
dx
c- y
Kx-x2
2Kx- 2x2
2y 2
c- y
K-x
y
Section 5.8
=
2Kx
dy=K-x
dx
y
- - • - - = Cy - yz
x
2. y
=
2Cy-
x2
x2
+ y2 _
+ y2 _
2x2
y2 _ x2
= --- = - 1
2y2
x2 _ y2
-::-~'--::----=
Inverse Trigonometric Functions: Differentiation
arccos x
(a)
-1
X
-0.8
3.142
y
2.499
(b)
-0.6
2.214
-0.4
1.982
-0.2
1.772
0
1.571
(<).rn.
0.2
0.4
0.6
0.8
1
1.369
1.159
0.927
0.634
0
(d) Intercepts: ( 0,
No symmetry
-I
4.
(_,~) = (1,~)
(_,
-~) =
(-
~, -~)
(-A,_)= (-A,-~)
6. arcsin 0
=
0
f)
and (1, 0)
536
Chapter 5
8. arccos 0 =
12.
Logarithmic, Exponential, and Other Transcendental Functions
5
27T
10. arc cot(- .J3} = ;
arccos(-~) =
18. (a) tan(
5
7T
6
14. arcsin(- 0.39)
arccos~) =tan(~) =
= -0.40
16. arctan(- 3)
(b) cos( arcsin
1
5
) =
13
~~
= - 1.25
d::15
12
20. (a) sec[
arctan(-~)]= ~
22. y = sec( arc tan 4x)
8 =arctan 4x
+ 16x2
y = sec 8 = .J1
;::vl·
24. y = cos(arccot x)
8 = arccotx
I
y = cos 8 =
X
---=====
R+1
h)
_(
. X28• y = cos
arcsm - r -
26. y = sec[arcsin(x - 1)]
8 = arcsin(x - 1)
.
X-
h
8 = arcsm--
1
r
y = sec 8 = ---:===
.J2x- xz
30.
~-'
-- [ . (-65)] = -5ffi
11
(b) tan arcsm
.Jr2 - (x- h)2
y = cos 8 = ....:...._----"-'------"'r
32. arctan(2x - 5) = - 1
5
'~'
2x- 5 = tan(-1)
1
x = 2(tan( -1)
-5
Asymptote: x = 0
X
arccos2 = 8
cosO=~
2
.J4- x
tan 8 = ....:...._.:..._....:..:....
2
X
+ 5) = 1.721