MATH6501 - Autumn 2016
Handout: Hyperbolic Functions
We will now introduce a new family of functions, the hyperbolic functions. They
are related to trigonometric functions, and are defined in terms of exponentials.
ex − e−x
2
x
e + e−x
cosh x =
2
ex − e−x
sinh x
tanh x = x
.
=
−x
e +e
cosh x
sinh x =
We can show from these definitions that cosh x is an even function and sinh x and
tanh x are odd functions. See Figure 1 for the graphs of these three functions.
We can also differentiate these functions by using their definitions in terms of
exponentials.
d
(sinh(x)) = cosh(x)
dx
d
(cosh(x)) = sinh(x).
dx
We can derive several identities from these functions that are analogous to trigonometric identities. The most important of these to remember is
cosh2 x − sinh2 x ≡ 1.
In the table below, we have lots of hyperbolic identities, together with their
trigonometric counterparts. Notice that they are very similar, but with some
different signs! This is called Osborne’s rule. The identities are the same, except
when we have a product of sinhs, we flip the sign. This includes cosech2 x, tanh2 x
and coth2 x as well as sinh2 x!
MATH6501 - Autumn 2016
Figure 1: The graphs of sinh, cosh and tanh.
10
sinh(θ)
cosh(θ)
tanh(θ)
5
0
-5
-10
-10
-5
0
5
10
MATH6501 - Autumn 2016
Hyperbolic
Trigonometric
coth x ≡ 1/ tanh x
cot x ≡ 1/ tan x
sech x ≡ 1/ cosh x
sec x ≡ 1/ cos x
cosech x ≡ 1/ sinh x
cosec x ≡ 1/ sin x
cosh2 x − sinh2 x ≡ 1
cos2 x + sin2 x ≡ 1
sech2 x ≡ 1 − tanh2 x
sec2 x ≡ 1 + tan2 x
cosech2 x ≡ coth2 x − 1
cosec2 x ≡ cot2 x + 1
sinh 2x ≡ 2 sinh x cosh x
sin 2x ≡ 2 sin x cos x
2
2
cosh 2x ≡ cosh x + sinh x
cos 2x ≡ cos2 x − sin2 x
cosh 2x ≡ 1 + 2 sinh2 x
cos 2x ≡ 1 − 2 sin2 x
cosh 2x ≡ 2 cosh2 x − 1
cos 2x ≡ 2 cos2 x − 1