INVERSE TRIGOMETRIC FUNCTIONS
LEC : 7
63
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \MATHEMATICS SCIENCES
DEF.: The inverse sine fun. Denoted by
1
sin x
is defined to be the inverse
sin
2
2
x
x
of restricted sine fun.
DEF.: The inverse cosine fun. Denoted by
1
cos x
is defined to be the inverse
cos
0
x
x
of restricted cosine fun.
DEF.: The inverse tangent fun. Denoted by
1
tan
x
is defined to be the inverse
tan
2
2
x
x
of restricted tangent fun.
DEF.: The inverse secant fun. Denoted by
1
sec x
is defined to be the inverse
sec
0
,
2
x
x
x
of restricted secant fun.
1
1
sin
sin
x
x
:
NOTE
:
EXAM
1
1
1
(1)
sin
,
2
1
sin
sin
sin
2
if x
find value of
x
x
x
1
sin
1
1
1
1
2
1
2
1
2
2
1
1
sin
2
2
6
1
(2)
sin(sin
)
2
1
1
1
sin
sin(sin
)
sin( )
2
6
2
6
2
1
3
(3)
cos(sin
)
2
2
cos
1 sin
1
1
3
cos(sin
)
1 sin sin
1 sin
1 ( )
2
6
2
2
x
x
simplify
show that
x
x
x
Inverse Trigometric Function Properties
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1) sin (
)
sin ( )
5) sec ( )
cos ( )
2) tan (
)
tan ( )
6) sec (
)
sec ( )
1
3) cos ( )
sin ( )
7) cot ( )
tan ( )
2
1
4) cot ( )
tan ( )
8) csc ( )
sin ( )
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
INVERSE TRIGOMETRIC FUNCTIONS
LEC : 7
63
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \MATHEMATICS SCIENCES
PROOF :
1
1)
sin (
)
sin
sin
let y
x
y
1
sin
1
1
1
1
1
1
1
1
1
1
1
1
1
(
)
sin
sin(
)
sin
sin
sin (
)
sin
3)
sin ( )
2
sin ( )
sin(
)
cos
cos
2
2
cos
sin ( )
2
1
8)
sin ( )
1
1
sin
csc
csc
sin
1
csc
sin ( )
x
x
y
x
y
x
y
x
y
x
x
x
let y
x
x
y
x
y
y
y
x
x
x
let y
x
y
x
y
y
x
x
y
x
x
: مشتقات الدوال المثلثية العكسية
1
2
1
2
1
2
1
2
1
2
1
2
1
1
sin
1
1
2
cos
1
1
3
tan
1
1
4
cot
1
1
5
sec
1
1
6
csc
1
d
du
u
dx
dx
u
d
du
u
dx
dx
u
d
du
u
dx
dx
u
d
du
u
dx
dx
u
d
du
u
dx
dx
u
u
d
du
u
dx
dx
u
u
INVERSE TRIGOMETRIC FUNCTIONS
LEC : 7
63
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \MATHEMATICS SCIENCES
to :
dy
dx
Find
:
EXAM
1
1
1
2
1
tan
tan
2
2
1
2
1
2
sin 2
1 4
3
1
tan 3
1 9
1
( sin )
sin
cos cos
1
sin
1 cos
cos
x
x
dy
y
x
dx
x
dy
y
x
e
e
dx
x
x
dy
x
x
y
x
dx
x
x
OR
cos
1
1
2
1
1
sec
sec
1
x
x
x
dy
x
x
dx
dy
y
e
x
e
x e
dx
x
x
: تكامالت الدوال المثلثية العكسية
1
2
2
1
1
2
2
1
1
2
2
1
1
sin
1
cos
1
tan
1
cot
1
sec
1
csc
u
a
a
du
C
a
u
u
a
a
u
a
a
du
C
a
u
u
a
a
u
a
a
du
C
u u
a
u
a
a
EXAM
1
2
2
1
2
1
4
1
tan (4 )
4
4
1 16
1 (4 )
1 1
2
tan
2 3
3
9
4
dx
dx
x
C
x
x
dx
x
C
x
INVERSE TRIGOMETRIC FUNCTIONS
LEC : 7
63
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \MATHEMATICS SCIENCES
1
2
2
1
2
2
1
1
2
2
1
1
4
2 2
tan (
)
1
1 (
)
sec
1
1
1
2
2
sin
sin
2
2
2
4
2
4
1
2
1 1
1
tan
tan
2
2
3
3 (
)
3
3
2 3
3
x
x
x
x
x
x
x
x
x
x
e dx
e dx
e
C
e
e
dx
e dx
e
C
e
e
e
dx
dx
x
x
C
x
x
x
x
x
x
x
x
dx
dx
C
x
x
1) Find
dy
dx
to
1
1
1
1
(cos
) ,
cot
,
(tan )
,
cot
y
Ln
x
y
x
y
x
y
x
2) Find
dy
dx
to
3
1
1
1
tan
,
sin (
)
(
)
y
x
x
y
e
xy
cos
x
y
3) Find
2
2
2
2
2
2
2
2
sec
,
,
1 tan
1
1
sin
1 tan
sin
cos
,
,
cos
1
1 cos (2 )
1 tan
x
x
x
e
dx
dx
dx
x
e
x
Ln x
x
x
x
x
dx
dx
dx
x
x
x
4) Find the area of the reigon bounded by the curve
2
4
12
0
x y
y
and x-axis .
5) Find the volume generated by rotation about the x-axis , the reigon bounded by the
1
2
2
5(
1)
,
0
,
0 ,
4
y
x
y
from
x
x
curve
6) Find the length of the curve :
2
1
,
3
3
1
y
x
x
MOHAMED SABAH AL TAEE
M.SC / MATHEMATICS
E-MAIL : msmt_80@yahoo.com
2013 -2014