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## Article 7Functional Relations for Hyperbola.

The functional relations between a sectorial measure and its characteristic ratios in the case of the hyperbola may be written in the form

 $\frac{{x}_{1}}{{a}_{1}}=cosh\frac{{S}_{1}}{{K}_{1}},\phantom{\rule{1em}{0ex}}\frac{{y}_{1}}{{b}_{1}}=sinh\frac{{S}_{1}}{{K}_{1}},$

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 $u$ for $\frac{{S}_{1}}{{K}_{1}}$ the two equations

 $\frac{{x}_{1}}{{a}_{1}}=coshu,\phantom{\rule{1em}{0ex}}\frac{{y}_{1}}{{b}_{1}}=sinhu$ (8)

serve to deﬁne the hyperbolic cosine and sine of a given sectorial measure $u$; and the hyperbolic tangent, cotangent, secant, and cosecant are then deﬁned as follows:

 $\begin{array}{ccc}\hfill tanhu=\frac{sinhu}{coshu},& \phantom{\rule{1em}{0ex}}cothu=\frac{coshu}{sinhu},\hfill & \hfill \\ \hfill sechu=\frac{1}{coshu},& \phantom{\rule{1em}{0ex}}cschu=\frac{1}{sinhu}.\hfill \end{array}}$ (9)

The names of these functions may be read “h-cosine,” ”h-sine,” “h-tangent,” etc., or “hyper-cosine,” etc.

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