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Chapter 1 | Functions and Graphs
Figure 1.49 The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: “Mtpaley”, Wikimedia Commons)
Definition Hyperbolic cosine
x + e − x 2
cosh x = e
Hyperbolic sine
x − e − x 2
sinh x = e
Hyperbolic tangent
x − e − x e x + e − x
tanh x = sinh x cosh x
= e
Hyperbolic cosecant
csch x = 1
= 2
e x − e − x
sinh x
Hyperbolic secant
sech x = 1
= 2
e x + e − x
cosh x
Hyperbolic cotangent
x + e − x e x − e − x
= e
coth x = cosh x sinh x
The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh , sech , csch , and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively. Using the definition of cosh( x ) and principles of physics, it can be shown that the height of a hanging chain, such as the one in Figure 1.49 , can be described by the function h ( x ) = a cosh( x / a )+ c for certain constants a and c . But why are these functions called hyperbolic functions ? To answer this question, consider the quantity cosh 2 t −sinh 2 t . Using the definition of cosh and sinh, we see that
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