Calculus: Analytic Geometry and Calculus, with Vectors

9.3 Hyperbolic functions 509

to discover therole of the parameter t is to let r be the vector running from
the origin to a particle which occupies the position (x(t),y(r)) at time t.
Then

(9.381) r = cosh ti + sinh tj

and differentiation gives the velocity and acceleration

(9.382) V = sinh tj + cosh tj
(9.383) a = cosh ti + sinh tj.

Thus a = r, so the particle is accelerated directly away from the origin,
and the magnitude of the acceleration is proportional to (actually equal
to) the first power of the distance. It is then known from classical
physics that the particle must (as it does) move on a conic. Moreover,
because the force is a central force which is always directed away from
or toward the origin, the angular momentum of the particle must be
constant and the vector from 0 to P must sweep over regions of equal
area in equal time intervals. With (or perhaps even without) the aid

of principles of physics that tell us to examine areas, we can calculate
the area of the shaded region of Figure 9.371 and show that the area is
cosh-1 x and hence is t. The problems show how the details can be
handled. Thus the geometrical similarity between trigonometric func-
tions (which used to be called circular functions) and hyperbolic functions
(which still are called hyperbolic functions) is exposed. Trigonometric
functions of t are "functions of the sector of area t of the unit circle shown
in Figure 9.372." Hyperbolic functions of t are "functions of the sector
of area t of the unit hyperbola shown in Figure 9.371." Those who have
not peered into ancient mathematical tomes and are more accustomed
to "sines of angles" than to "sines of arcs" and "sines of sectors" should
quietly observe that the sector of the unit circle of Figure 9.372 has
area t when the arc from 14 to P has length t and hence when the angle
110P "contains" t radians. Among other things, this little excursion
into history explains the antics of those who write arcsin x in place of
sin-' x and talk about "the arc whose sine is x."

Problems 9.39

1 Verify the six formulas for derivatives of hyperbolic functions.

2 Verify the six formulas for derivatives of inverse hyperbolic functions.

3 Textbooks on differential equations show that if a flexible homogeneous

cable or chain is suspended from its two ends and sags under the influence of a

parallel force field (an idealized gravitational field) then the cable or chain must

occupy a part of the graph of the equation

(1) Y = Zk (e' + e-') =

k

cosh kx,