appendix: hyperbolic trigonometric functions

Definitions (a) cosh(x) = (e^x+e^-x)/2, (b) sinh(x) = (e^x-e^-x)/2, (c) tanh(x) = sinh(x)/cosh(x) (= (e^x-e^-x)/(e^x+e^-x)). From the definitions, we can easily deduce
(1) cosh²(x) - sinh²(x) = 1, (2) sinh(2x) = 2cosh(x)sinh(x), (3) cosh(2x) = cosh²(x) + sinh²(x). From (1) and (3), (3') cosh(2x) = 2cosh²(x) - 1. Using (1) and (2),(3), we get (4) sinh(2x) = 2tanh(x)/(1-tanh²(x)), (5) cosh(2x) = (1+tanh²(x))/(1-tanh²(x)), (6) tanh(2x) = 2tanh(x)/(1+tanh²(x)). Again from the definition, after some calculation, (7) tanh(x+y) = (tanh(x)+tanh(y))/(1+tanh(x)tanh(y)).
Note that, by (c), tanh(x) = (1-e^-2x)/(1+e^-2x) = 1 - 2/(e^2x+1), so that tanh(0) = 0. Also, for x > 0, e^2x > 1, and increases with x, so that tanh(x) increases on [0,∞), and tanh(x) tends to 1 as x tends to ∞. Thus tanh(x) is increasing, and maps [0,∞) to [0,1). It follows that the inverse function arctanh(x) is an increasing function, mapping [0,1) to [0,∞).
return to main hypeerbolic menu