hyperbolic functions. the hyperbolic sine, hyperbolic cosine & hyperbolic tangent
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Hyperbolic Functions
The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent
xx
xx
xx
xx
ee
ee
x
xx
eex
eex
cosh
sinhtanh
2sinh
2cosh
The Inverse Hyperbolic Cotangent, Hyperbolic Secant &
Hyperbolic Cosecant
xx
xx
xx
xx
eexhx
eexhx
ee
ee
x
x
xx
2
sinh
1csc
2
cosh
1sec
sinh
cosh
tanh
1coth
Values
).2csc(.4
)3coth(ln.3
)0tanh(.2
)2cosh(ln.1
:followingtheofeachexactlyFind
1
2
1
21
22)2csc(3
4
5
8
10
31
3
31
3)3coth(ln3
011
110tanh2
4
5
221
2
2)2cosh(ln.1
4
2
2
4
22
22
3ln3ln
3ln3ln
00
00
2ln2ln
e
e
ee
eeee
ee
ee
ee
ee
ee
Solution
Remember Logarithmic Identities ( Calculus I )
xaI xa log.
7.3
.2
75.1
:
7ln
ln
7log5
e
xe
Examples
x
Logarithmic Identities II
naa xxnII loglog.
xxx
Examples
11
2512
2
32
33
lnlnln.4
ln5ln5ln2.3
25ln5ln5ln2.2
25log5log5log2.1
:
Identities
xhxh
xhx
xx
xx
xxx
xxx
xx
22
22
2
2
22
22
csc1coth)7(
1sectanh)6(
2
12coshsinh)5(
2
12coshcosh)4(
sinhcosh2cosh)3(
coshsinh22sinh)2(
1sinhcosh)1(
ProofsIdentity (1)
1]4[4
1
)]2()2[(
)2
()2
(
sinhcosh
222241
22
22
xxxx
xxxx
eeee
eeee
xx
ProofsIdentity (2)
x
ee
ee
eeee
xx
xx
xx
xxxx
2sinh
)(2
1
)]([2
)2
()2
(2
coshsinh2
22
2241
ProofsIdentity (3)
x
ee
ee
eeee
eeee
xx
xx
xx
xxxx
xxxx
2cosh
)(2
1
)22[(
)]2()2[(
)2
()2
(
sinhcosh
22
2241
222241
22
22
ProofsIdentity (4)
2
12coshcosh
1cosh2
)1(coshcosh
sinhcosh
2cosh
2
2
22
22
xx
x
xx
xx
x
ProofsIdentity (5)
2
12coshsinh
1sinh2
sinh)1(sinh
sinhcosh
2cosh
2
2
22
22
xx
x
xx
xx
x
ProofsIdentity (6)
xhx
xx
x
x
x
xx
22
22
2
2
2
22
sectanh1
cosh
1
cosh
sinh
cosh
cosh
1sinhcosh
ProofsIdentity (7)
xhx
xx
x
x
x
xx
22
22
2
2
2
22
csc1coth
sinh
1
sinh
sinh
sinh
cosh
1sinhcosh
Derivatives of Hyperbolic Functions
xhxhx
xhxhx
xhx
xhx
xx
xx
cothcsc)(csc)6(
tanhsec)(sec)5(
csc)(coth)4(
sec)(tanh)3(
sinh)(cosh)2(
cosh)(sinh)1(
2
2
Proofs(1) (sinhx)’ = coshx (2) (coshx)’ = sinhx
x
ee
ee
x
xx
xx
cosh
)(
)2
(
)(sinh
21
x
ee
ee
x
xx
xx
sinh
)[(
)2
(
)(cosh
21
Proofs(3) (tanhx)’ = sech2x
xh
x
x
xx
x
xxxxx
x
x
2
2
2
22
2
sec
cosh
1cosh
sinhcosh
cosh
sinhsinhcoshcosh
)cosh
sinh(
)(tanh
Proofs(4) (cothx)’ = -csch2x
xh
xx
xx
x
xx
x
xxxxx
x
x
2
22
22
2
22
2
csc
cosh
1
cosh
)sinh(cosh
cosh
coshsinh
sinh
coshcoshsinhsinh
)sinh
cosh(
)(coth
Proofs(5) (sechx)’ = - sechx tanhx
xhx
x
x
xx
x
x
hx
tanhseccosh
sinh
cosh
1cosh
sinh
sinhcosh
)(cosh
)cosh
1(
)(sec
2
2
1
Proofs(5) (cschx)’ = - cschx cothx
xhx
x
x
xx
x
x
hx
cothcscsinh
cosh
sinh
1sinh
cosh
coshsinh
)(sinh
)sinh
1(
)(csc
2
2
1
Integrals Involving hyperbolic Functions
chhxdxxhx
chhxdxxhx
cxdxxh
cxdxxh
cxdxx
cxdxx
csccothcsc.6
sectanhsec.5
cothcsc.4
tanhsec.3
coshsinh.2
sinhcosh.1
2
2
Examples I
x
x
x
hx
xy
hxy
xx
xxy
hxxy
y
xy
exxy
yFind
4tanh
1002coth
22
93
94
sec68
)3(cosh.7
)]7(cosharcsin5)in(sinh[sinh(arcs.6
coscosh
sinsinh.5
csccoth.4
)]2[cosh(sinarctan.3
)][ln(coshtanh.2
coshsinh.1
Examples II
dxxxhx
dxxx
dxxxhx
dxx
hx
dxxx
egralsfollowingtheofeachfollowingtheEvaluatet
)3cot(arcsin)3(arcsincsc91
1.5
)2nsinh(arcta41
1.4
)tanh()(sec1
.3
)1(sec
1.2
)cosh(ln1
.1
:int
2
2
42
5
Examples III
dxxx
dxhx
dxx
dxx
dxxh
xxxh
dxx
x
dxx
xh
egralsfollowingtheofeachEvaluatet
x7cosh76
2
2
2
5sinh7
csc.6
coth.5
tanh.4
5csc49
5coth5csc3
5sinh49
5cosh.2
5tanh49
5sec.1
:int
Examples IV
dxx
dxxh
dxxhx
dxxxh
dxx
dxxx
dxxx
dxx
egralsfollowingtheofeachfollowingtheEvaluatet
4
6
4100
101
4
22
5100
3
tanh.8
sec.7
csccoth.6
tanhsec.5
cosh.4
sinhcosh.3
sinhcosh.2
cosh.1
:int
Examples V
dxxe
dxxx
dxxx
dxxx
egralsfollowingtheofeachfollowingtheEvaluatet
x cosh.4
cosh.3
sinh.2
cosh.1
:int
2
3
2
Graphs of Hyperbolic Functions
Graphs of Exponential Functions Functions
Graphs of Exponential Functions
f(x) = Cosh x
f(x) = cosh x = (½)ex + (½) e-x
as x → ∞ the values f(x) → ∞ following (½)ex as x → - ∞ the values f(x) → ∞ following (½) e-x
f(x) = sinh x
f(x) = tanh x
f(x) = secx