Article 9
Variations of the Hyperbolic Functions.


Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their variations, to consider only sectors of one half of a rectangular hyperbola, whose conjugate radii are equal, and to take the principal axis OA as the common initial line of all the sectors. The sectorial measure u assumes every value from , through 0, to + , as the terminal point P comes in from infinity on the lower branch, and passes to infinity on the upper branch; that is, as the terminal line OP swings from the lower asymptotic position y = x, to the upper one, y = x. It is here assumed, but is proved in Art. 17, that the sector AOP becomes infinite as P passes to infinity.

Since the functions coshu,sinhu,tanhu, for any position of OP, are equal to the ratios of x,y,t, to the principal radius a, it is evident from the figure that

cosh0 = 1,sinh0 = 0,tanh0 = 0, (15)

and that as u increases towards positive infinity, coshu,sinhu are positive and become infinite, but tanhu approaches unity as a limit; thus

cosh = ,sinh = ,tanh = 1. (16)

Again, as u changes from zero towards the negative side, coshu is positive and increases from unity to infinity, but sinhu is negative and increases numerically from zero to a negative infinite, and tanhu is also negative and increases numerically from zero to negative unity; hence

cosh() = ,sinh() = ,tanh() = 1. (17)

For intermediate values of u the numerical values of these functions can be found from the formulas of Arts. 16, 17, and are tabulated at the end of this chapter. A general idea of their manner of variation can be obtained from the curves in Art. 25, in which the sectorial measure u is represented by the abscissa, and the values of the functions coshu, sinhu, etc., are represented by the ordinate.

The relations between the functions of u and of u are evident from the definitions, as indicated above, and in Art. 8. Thus

cosh(u) = +coshu,sinh(u) = sinhu, sech(u) = +sechu,csch(u) = cschu, tanh(u) = tanhu,coth(u) = cothu. (18)
Prob. 12.
Trace the changes in sech u,coth u,csch u, as u passes from to + . Show that sinh u,cosh u are infinites of the same order when u is infinite. (It will appear in Art. 17 that sinh u,cosh u are infinites of an order infinitely higher than the order of u.)
Prob. 13.
Applying eq. (12) to figure, page §, prove tanh u1 = tan AOP.