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
By a combination of some of these equations other subsidiary relations may be obtained; thus, dividing (10) successively by , and applying (9), give
Equations (9), (10), (11) will readily serve to express the value of any function in terms of any other. For example, when is given,
The ambiguity in the sign of the square root may usually be removed by the following considerations: The functions are always positive, because the primary characteristic ratio is positive, since the initial line and the abscissa are similarly directed from on whichever branch of the hyperbola maybe situated; but the functions , involve the other characteristic ratio , which is positive or negative according as and have the same or opposite signs, i.e., as the measure is positive or negative; hence these four functions are either all positive or all negative. Thus when any one of the functions , is given in magnitude and sign, there is no ambiguity in the value of any of the six hyperbolic functions; but when either or 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 ; then
The hyperbolic tangent is the measure of the triangle . For
Thus the sector , and the triangles , are proportional to (eqs. 5, 13); hence