Article 8
Relations Among Hyperbolic Functions.

Among the six functions there are five independent relations, so that when the numerical value of one of the functions is given, the values of the other five can be found. Four of these relations consist of the four defining equations (9). The fifth is derived from the equation of the hyperbola

x12 a12 y12 b12 = 1,


cosh2u sinh2u = 1.  (10)

By a combination of some of these equations other subsidiary relations may be obtained; thus, dividing (10) successively by cosh2u,sinh2u, and applying (9), give

1 tanh2u = sech2u, coth2u 1 = csch2u. (11)

Equations (9), (10), (11) will readily serve to express the value of any function in terms of any other. For example, when tanhu is given,

cothu = 1 tanhu,sechu = 1 tanh 2 u, coshu = 1 1 tanh 2 u,sinhu = tanhu 1 tanh 2 u, cschu = 1 tanh 2 u tanhu .

The ambiguity in the sign of the square root may usually be removed by the following considerations: The functions coshu,sechu are always positive, because the primary characteristic ratio x1 a1 is positive, since the initial line O1A1 and the abscissa O1M1 are similarly directed from O1 on whichever branch of the hyperbola P1 maybe situated; but the functions sinhu,tanhu,cothu,cschu, involve the other characteristic ratio y1 b1 , which is positive or negative according as y1 and b1 have the same or opposite signs, i.e., as the measure u is positive or negative; hence these four functions are either all positive or all negative. Thus when any one of the functions sinhu,tanhu,cschu,cothu, is given in magnitude and sign, there is no ambiguity in the value of any of the six hyperbolic functions; but when either coshu or sechu is given, there is ambiguity as to whether the other four functions shall be all positive or all negative.


The hyperbolic tangent may be expressed as the ratio of two lines. For draw the tangent line AC = t; then

tanhu = y b : x a = a b y x = a b t a = t b.  (12)

The hyperbolic tangent is the measure of the triangle OAC. For

OAC OAB = at ab = t b = tanhu.  (13)

Thus the sector AOP, and the triangles AOP,POB,AOC, are proportional to u,sinhu,coshu,tanhu (eqs. 5, 13); hence

sinhu > u > tanhu. (14)
Prob. 7.
Express all the hyperbolic functions in terms of sinh u. Given cosh u = 2, find the values of the other functions.
Prob. 8.
Prove from eqs. 10, 11, that cosh u > sinh u,cosh u > 1,tanh u < 1,sech u < 1.
Prob. 9.
In the figure of Art. 1, let OA = 2,OB = 1,AOB = 60, and area of sector AOP = 3; find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyperbolic sector; and find the area of the triangle AOP. (Use tables of cos, sin, cosh, sinh.)
Prob. 10.
Show that coth u,sech u,csch u may each be expressed as the ratio of two lines, as follows: Let the tangent at P make on the conjugate axes OA,OB, intercepts OS = m,OT = n; let the tangent at B, to the conjugate hyperbola, meet OP in R, making BR = l; then
coth u = l a,sech u = m a ,csch u = n b .
Prob. 11.
The measure of segment AMP is sinh ucosh u u. Modify this for the ellipse. Modify also eqs. 10–14, and probs. 8, 10.