Calculus: Analytic Geometry and Calculus, with Vectors

9.3 Hyperbolic functions 507

nometric functions, the first and last are reciprocals, the next to the first
and the next to the last are reciprocals, and the middle two are reciprocals.
With the aid of the fact that exe-x = 1, we can square cosh x and sinh x
and obtain the first of the formulas

(9.33) cosh2 x - sinh2 x = 1
(9.331) tanh2 x + sech2 x = 1
(9.332) coth2 x - csch2 x = 1.

To obtain the second (or third) formula, we can divide by cosh2 x (or
by sinh2 x) and transpose some terms. Graphs of the first three hyper-

bolic functions are shown in Figures 9.34 and 9.341, and the others are
easily drawn.

-2 -

-1

1'

y=tank x

X

Figure 9.34 Figure 9.341

To work out formulas for the inverses of the first hyperbolic functions,
we let
e : e
(9.35) x=cosh t=e

2e-,

y=sinht=e

2e

and observe that cosh t is increasing over the interval t > 0 and sinh t
is increasing over the whole infinite interval. The equations (9.35) can
be put in the forms

(9.351) ell - 2xe= + 1 = 0, ell - 2ye° - 1 = 0.

These equations are quadratic in a°, and solving for e° gives

(9.352) ee = x + x2 - 1, et = y + \/-y2+ 1,

it being necessary to choose the positive sign in each case because et

is positive and increasing as x and y increase. Taking logarithms and
changing y to x gives the first items in the first two of the following for-
mulas. Similar methods and differentiation give the remaining items.