TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
04
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
1
( )
,
,
2
2
( )
,
,
[1, )
2
( )
3 tan ( )
,
,
( 1,1)
( )
( )
4 cot ( )
,
{0} ,
{0}
( )
1
2
5
( )
,
,
(0,1]
( )
x
x
f
f
x
x
f
f
x
x
f
f
x
x
x
x
f
f
x
x
f
f
x
x
e
e
Sinh x
D
R
e
e
Cosh x
D
R
Sinh x
e
e
h x
D
R
Cosh x
e
e
Cosh x
e
e
h x
D
R
Sinh x
e
e
Sech x
D
R
Cosh x
e
e
1
2
6 csc ( )
,
{0} ,
[ 1,1]
( )
f
f
x
x
h x
D
R
Sinh x
e
e
ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
ـــــــــــــــــ
Some rules :
2
2
2
2
2
2
2
2
1 cosh( )
sinh( )
2 cosh( ) sinh( )
3 cosh ( ) sinh ( )
1
(
)
4 1 tanh ( )
sec
( )
5 coth ( ) 1
csc
( )
1
1
6 cosh ( )
(cosh 2
1)
7 sinh ( )
(cosh 2
1)
2
2
8 sinh(
)
sinh
cosh
sinh
cosh
sinh(2 )
2sinh
x
x
x
x
e
x
x
e
x
x
main rule
x
h
x
x
h
x
x
x
x
x
x
y
x
y
y
x
x
2
2
2
cosh
9 cosh(
)
cosh
cosh
sinh sinh
cosh(2 )
cosh
sinh
tanh
tanh
2 tanh
10 tanh(
)
tanh(2 )
1 tanh
tanh
1 tanh
x
y
x
y
x
y
x
y
x
x
x
x
y
x
x
y
x
x
y
x
The proof:
1) cosh( )
sinh( )
2
2
2
2
2) cosh( ) sinh( )
2
2
2
2
x
x
x
x
x
x
x
x
x
x
x
x
e
e
e
e
x
x
e
e
e
e
e
e
x
x
e
e
TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
04
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
2
2
0
3) cosh ( ) sinh ( )
[cosh( ) sinh( )][cosh( ) sinh( )]
1
x
x
x
x
x
x
x
x
e
e
e
Find the value of cosh(0) , sinh(0)
:
x
E
0
0
0
0
cosh(0)
1
2
sinh(0)
0
2
e
e
e
e
HYPERBOLIC TRIG. FUN. DERIVATIVE :
2
2
2
2
1
sinh
cosh
.
sinh
cosh
2
cosh
sinh
.
cosh
sinh
3
tanh
sech
.
tanh
sech
4
coth
csch
.
coth
csch
5
sech
sec
tanh
.
sech
sec
d
d
x
x
Gen
u
u du
dx
dx
d
d
x
x
Gen
u
u du
dx
dx
d
d
x
x
Gen
u
u du
dx
dx
d
d
x
x
Gen
u
u du
dx
dx
d
d
x
hx
x
Gen
u
hu
dx
dx
tanh
6
csch
csc
coth
.
csch
csc
coth
u du
d
d
x
hx
x
Gen
u
hu
u du
dx
dx
PROOF:
2
2
2
2
2
2
2
2
2
1
sinh
cosh
2
2
(
)(
)
(
)(
)
3
tanh
[
]
(
)
(
2
)
(
2
)
(
)
4
2
[
]
(
)
sec
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
d
d e
e
e
e
x
x
dx
dx
d
d e
e
e
e
e
e
e
e
e
e
x
dx
dx e
e
e
e
e
e
e
e
e
e
e
e
e
e
h x
TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
04
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
to the functions :
dy
dx
Find
:
Exam
2
2
2
sinh 3
sinh 3
1
tanh(
)
2 sec
(
)
2
cosh 3
3
3
(1 cosh 4 )
4sinh 4
1 cosh 4
4
(csc
)
(csc
)
(csc
)
1
csc
coth
(csc
)
csc
(csc
)
[
coth
]
(csc
x
x
Lnx
Lnx
y
x
y
x
h
x
y
e
y
e
x
y
Ln
x
x
y
x
y
hx
Lny
Ln
hx
Lnx Ln
hx
hx
x
Ln
hx
y
Lnx
y
hx
x
Ln
hx
y
y Lnx
x
x
y
(csc
)
)
[ coth
]
Lnx
Ln
hx
hx
x Lnx
x
:
GRATION
NTE
I
PERBOLIC TRIG. FUN.
Y
H
2
2
1
sinh
cosh
2
cosh
sinh
3
sech
tanh
4
csch
coth
5
sec
tanh
sech
6
csc
coth
csch
u du
u
C
u du
u
C
u du
u
C
u du
u
C
hu
u du
u
C
hu
u du
u
C
2
2
1
1
cosh 7
sinh 7
7
1
2
tanh sec
tanh
2
sinh(
)
3
cosh(
)
xdx
x
c
x
h xdx
x
c
Lnx
dx
Lnx
c
x
TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
04
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
INVERSE HYPERBOLIC TRIG. FUN.
1
2
1
2
1
1
2
1
2
1
1 sinh ( )
(
1)
2 cosh ( )
(
1)
1
1
3 tanh ( )
(
)
2
1
1
1
4 coth ( )
(
)
2
1
1
1
5 sech ( )
(
)
1
1
6
ch ( )
(
)
x
Ln x
x
x
Ln x
x
x
x
Ln
x
x
x
Ln
x
x
x
Ln
x
x
cs
x
Ln
x
Some rules :
1
1
1
1
1
1
1
csch ( )
sinh ( )
1
s
h ( )
cosh ( )
1
coth ( )
tanh ( )
x
x
ec
x
x
x
x
DERIVATIVE :
PERBOLIC TRIG. FUN.
Y
H
INVERSE
1
2
1
2
1
2
1
2
1
2
1
2
1
1
sinh ( )
1
1
2
cosh ( )
1
1
3
tanh ( )
1
1
4
coth ( )
1
1
5
sech ( )
1
1
6
ch ( )
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
cs
u
dx
dx
u
u
TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
00
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
1
2
1
2
1
2
1
5
1
4
2
:
1)
sinh (3 )
3
1 9
2)
cosh (
)
1
3)
sech (ln )
1
ln
1 ln
4)
(coth (
))
5(coth (
)) [
]
1
x
x
x
x
x
x
x
exam
y
x
y
x
y
e
e
y
e
y
x
y
x
x
x
y
e
e
y
e
e
:
NTEGRATION
I
PERBOLIC TRIG. FUN.
Y
H
INVERSE
1
2
2
1
2
2
1
2
2
1
1
2
2
1
2
2
1
sinh ( )
2
cosh ( )
1
tanh ( )
3, 4
1
coth ( )
1
5
sech
1
6
ch
du
u
C
a
a
u
du
u
C
a
u
a
u
du
a
a
C
u
a
u
a
a
du
u
C
a
a
u a
u
du
u
cs
C
a
a
u
a
u
1
2
1
2
:
1
1)
sinh (2 )
2
1 4
1
4
2)
sinh (
)
4
3
9 16
exam
dx
x
C
x
dx
x
C
x
TRIGONOMETRY HYPERBOLIC FUNCTIONS
LEC : 8
04
MOHAMED SABAH AL TAEE \ MOSUL UNIVERSITY \ MATHEMATICS SCIENCES
1
2
1
2
3)
tanh (
)
1
1
4)
tanh
3
3
9
x
x
x
e
dx
e
C
e
dx
x
C
x
1
2
2
1
3
2
2
3
1
sinh( )
,
tanh(4 )
4
4
1
2
tanh( )
,
: sinh( )
cosh( )
5
3
3
cosh(ln 4) ?
4
:
cosh 2
cosh(tan
)
sec
(
)
csc
(sin
)
,
,
,
sinh 2
1
1
1
sec (
csc
coth
,
x
x
if
x
Find
x
if
x
show that
x
x
Find the value of
Evaluate
x
x
h e
h
x
dx
dx
dx
dx
x
x
e
x
x
h
h x
x
x
dx
x
2
2
tanh(ln )
3
tan(sec
)
2
4
1
) tanh(
)
1
1
(
1)
5
:
(10)
,
sec
(tan )
,
6
:
,
4
4
x
hx
x
x
dx
x
Find y to
y
y
h
x
y
e
Evaluate
dx
dx
x
x
x
MOHAMED SABAH AL TAEE
M.SC / MATHEMATICS
E-MAIL : msmt_80@yahoo.com
2013 -2014