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162 Chapter Two

x
x
sinà x (e e )/2
x
x
cosh x (e e )/2
x
x
x
x
tanà x sinà x/cosh x (e e )/(e e )
x
x
csch x 1/sinà x 2/(e e )
x
x
sech x 1/cosh x 2/(e e )
x
x
x
x
cotà x cosh x/sinà x (e e )/(e e )
Hyperbolic functions as series
Let x be a real number. The exclamation symbol (!) denoteð fac-
torial. The following equationð hold:

7
5
3
sinà x x x /3! x /5! x /7! ...
2
6
4
cosh x 1 x /2! x /4! x /6! ...
Graph of hyperbolic sinł function
Figure 2.69 is an approximate grapà of the function y sinà x;
botà itð domain and range extend over the entire set of real
numbers.

Figure 2.69 Approximate
grapà of the hyperbolic sine
function.