## Article 7

Functional
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:

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

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