Chapter 1 | Functions and Graphs
109
2 t +2+ e −2 t 4
e 2 t −2+ e −2 t 4
cosh 2 t −sinh 2 t = e
− =1. This identity is the analog of the trigonometric identity cos 2 t +sin 2 t =1. Here, given a value t , the point ( x , y ) = (cosh t , sinh t ) lies on the unit hyperbola x 2 − y 2 =1 ( Figure 1.50 ).
Figure 1.50 The unit hyperbola cosh 2 t −sinh 2 t =1.
Graphs of Hyperbolic Functions To graph cosh x and sinh x , we make use of the fact that both functions approach (1/2) e x as x →∞, since e − x →0 as x →∞. As x →−∞, cosh x approaches 1/2 e − x , whereas sinh x approaches −1/2 e − x . Therefore, using the graphs of 1/2 e x , 1/2 e − x , and −1/2 e − x as guides, we graph cosh x and sinh x . To graph tanh x , we use the fact that tanh(0) = 0, −1 < tanh( x ) <1 for all x , tanh x →1 as x →∞, and tanh x →−1 as x →−∞. The graphs of the other three hyperbolic functions can be sketched using the graphs of cosh x , sinh x , and tanh x ( Figure 1.51 ).
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