The functional relations between a sectorial measure and its characteristic ratios in the case of the hyperbola may be written in the form
and these express that the ratio of the two lines on the left is a certain deﬁnite function of the ratio of the two areas on the right. These functions are called by analogy the hyperbolic cosine and the hyperbolic sine. Thus, writing for the two equations
serve to deﬁne the hyperbolic cosine and sine of a given sectorial measure ; and the hyperbolic tangent, cotangent, secant, and cosecant are then deﬁned as follows:
The names of these functions may be read “h-cosine,” ”h-sine,” “h-tangent,” etc., or “hyper-cosine,” etc.