,
, FUNDAMENTALS OF
ENGINEERING MATHEMATICS
DIYA SATISH KUMAR
,
,
,1.DIFFERENTIAL
CALCULUS
, HYPERBOLIC FUNCTIONS :
1) Sin ho
exe
=
2) coshx =
exe
= xumdb
3) tank = sinh
cos ha
4)
cosecho=
inter
5)
seche
6) cothx =
T
Note :
cos hoc + sinho e
coshx-sin ha = e-
* Parametric form of hyperbolic equation
x
y = 1 [x = coshe , y
=
sinhot]
coshe-sinh 1 -
hyperbolic function
=
·
-y =
cosh-sinhx
=
(ey_(
=+ex exa ex 1
=
+ 2 -
+ 2 -
= =
4
* Formulas
cosho-sinh
↓
2
1) = / + cost
I-tanhn sehn + sinhon
2) =
3) cothoc-1 = cosechs
,*
Graphs
1) sinh(x) 2 =
cosh(x)3)y =
tanh(x)
y
=
y
~ -
-
[0 8)
D( 0
,
0) D( 0
,
8) + ,
R( -
0
,
8) R (1 ,
8) RC -
11)
4)
y
= cosech(x) Sly sechtoo)
= a)
y
= coth(x)
-
- -
-
&
D(-, 0u(0 , 0) D( - 0
,
0) D( 0 0) , v(o ,
x)
REV(0) R(0 ,
1] R( -
0 -
,
1) u(i , 8)
,* Properties
coshl-x) =
cosh(x) even func symm abt y axis
sinh(- >) = -
sinh(o) odd func symmabt origin
tanh(-x) =
-tanh(x)
cosech(-o =
-cosech(x)
sech(x) = sech(o)
coth( xc) -
=
-
coth(x)
cosh(2x) =
1 + 2 sinh-(o)
= 2cosh(x) -
1
cosh(2x) = cosh" ( + sinh (c)
sinh (2x) = 2 Sinhloccosh(x)
sin ho = -isinis
cosha = cos ix
tan hx =
-
itan is
cot hx = i cot ix
sechx =
sec ic
sin ix = is in ha
Hyperbolic Functions not
are
periodic
, 1) cosho-sinh = /
2) I-tanhn = sehn
3) cothoc-1 = cosechs
Prove the above formulas
al cos h2x - sin 42x = /
MY cosh-sinhx
=
(ey_(
=+ex exa ex 1
=
+ 2 -
+ 2 -
= =
4
MI cosO + sinE = 1
(Trigonometric Identity)
Let o =
ix
sin = : cosix= coshe
(shoc + (isinho = 1 sinixisinha
cos hi-simh=
6) I-tanh =
sech
cosh-sinhx =
- coshx
,
1 - tan hex = sech
=> sech" + tankx
=
c) coth x-cosech x =
/
cosho -
sin hx = /
= sin x ,
coth cosech
=
=
x
-
1
=> cot hx-cosechc =
1
, FUNDAMENTALS OF
ENGINEERING MATHEMATICS
DIYA SATISH KUMAR
,
,
,1.DIFFERENTIAL
CALCULUS
, HYPERBOLIC FUNCTIONS :
1) Sin ho
exe
=
2) coshx =
exe
= xumdb
3) tank = sinh
cos ha
4)
cosecho=
inter
5)
seche
6) cothx =
T
Note :
cos hoc + sinho e
coshx-sin ha = e-
* Parametric form of hyperbolic equation
x
y = 1 [x = coshe , y
=
sinhot]
coshe-sinh 1 -
hyperbolic function
=
·
-y =
cosh-sinhx
=
(ey_(
=+ex exa ex 1
=
+ 2 -
+ 2 -
= =
4
* Formulas
cosho-sinh
↓
2
1) = / + cost
I-tanhn sehn + sinhon
2) =
3) cothoc-1 = cosechs
,*
Graphs
1) sinh(x) 2 =
cosh(x)3)y =
tanh(x)
y
=
y
~ -
-
[0 8)
D( 0
,
0) D( 0
,
8) + ,
R( -
0
,
8) R (1 ,
8) RC -
11)
4)
y
= cosech(x) Sly sechtoo)
= a)
y
= coth(x)
-
- -
-
&
D(-, 0u(0 , 0) D( - 0
,
0) D( 0 0) , v(o ,
x)
REV(0) R(0 ,
1] R( -
0 -
,
1) u(i , 8)
,* Properties
coshl-x) =
cosh(x) even func symm abt y axis
sinh(- >) = -
sinh(o) odd func symmabt origin
tanh(-x) =
-tanh(x)
cosech(-o =
-cosech(x)
sech(x) = sech(o)
coth( xc) -
=
-
coth(x)
cosh(2x) =
1 + 2 sinh-(o)
= 2cosh(x) -
1
cosh(2x) = cosh" ( + sinh (c)
sinh (2x) = 2 Sinhloccosh(x)
sin ho = -isinis
cosha = cos ix
tan hx =
-
itan is
cot hx = i cot ix
sechx =
sec ic
sin ix = is in ha
Hyperbolic Functions not
are
periodic
, 1) cosho-sinh = /
2) I-tanhn = sehn
3) cothoc-1 = cosechs
Prove the above formulas
al cos h2x - sin 42x = /
MY cosh-sinhx
=
(ey_(
=+ex exa ex 1
=
+ 2 -
+ 2 -
= =
4
MI cosO + sinE = 1
(Trigonometric Identity)
Let o =
ix
sin = : cosix= coshe
(shoc + (isinho = 1 sinixisinha
cos hi-simh=
6) I-tanh =
sech
cosh-sinhx =
- coshx
,
1 - tan hex = sech
=> sech" + tankx
=
c) coth x-cosech x =
/
cosho -
sin hx = /
= sin x ,
coth cosech
=
=
x
-
1
=> cot hx-cosechc =
1
