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

x1 a1 = cosh S1 K1,y1 b1 = sinh S1 K1,

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 S1 K1 the two equations

x1 a1 = coshu,y1 b1 = 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:

tanhu = sinhu coshu,cothu = coshu sinhu, sechu = 1 coshu,cschu = 1 sinhu. (9)

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