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X; HYPERBOLIC RATIOS.
46. Definì́tions of the Hyperbolic Ratì́os. An important class of exponentials, Whic-h because of their relation to the equilateral hyperbola are called the hyperbolic s-i-ne,-cosine, tangent, Cotangent, secant and cosecant, and are Symbolized by the abbreviations sinh, cosh, tanh; Coth, Sech, csch, are defined by the following identities: ”
48. Geometrical Construction for Hyperbolìc Rat́ì́os. For the r(-:presentation of the hyperbolic ratios the equilateral hyperbola is employed. Its equation ín_ Cartesian co-ordinates is
Let OX, O Y be its axes, 0J an asymptote, P any point on the curve, x and y its co-ordinates, AQB the quadrant of
a circle with centre at the origin and radius a, JVQ a tangent to the circle from the foot of the ordinate y, PS a tangent to the hyperbola parallel to the chord PA, a, /? the co-ordinates of Q, 6 the arc-ratio of the angle AOQ.
It is obvious from its definition that cosh u has 1 for its smallest and + oo for its largest value corresponding to u = o and oo respectively, and if the variations of x / a be confined to the right hand branch of the hyperbola its range of values is likewise between 1 and -f- oo; hence we may assume
—. cosh u,
and by virtue of the relations cosh* u — sinh2 u = 1, and x* / a' —y I a' = 1,
- = s1nh u.
Also, since x* — y* = a* and x* — NQ* — a2, therefore
and we have
- = sec 6 = cosh u, - — tan 6 = sinh u,
and the co-ordinates of Q being a, /?, also
0 y . , , a a „ . := - = - = sin V — tanh u, - — - = cos 0 = sech «,
and finally, since OB K is similar to ONE and a2 =j> . LO*
BK x , LO a .
= — = coth u, — = = csch u.
Hence if a be made the denominator in each of the hyperbolic ratios, their numerators will be six straight lines, drawn from 0, A, B, or P, which may be indicated thus:
a sinh u = NP, perpendicular distance of P from OX, a cosh u = ON, distance from centre to foot of NP, a tanh u = AH, distance along a tangent from A to OP, a coth u = BK, distance along a tangent to the conjugate
hyperbola from B to OP, a sech u ~ OM, intercept of tangent at P upon OX, a csch u = LO, — (intercept of tangent at P upon OY).
This construction gives pertinence to the name ratio as
applied to the six analogues of the goniometric ratios. Compare these with the constructions of Art. 42.
49. Agenda. Properties of the Equilateral Hyperbola. Prove the following propositions concerning the equilateral hyperbola. (Fig. 25 of Art. 48.)
(1) . The tangent to the hyperbola at P passes through M, the foot of the ordinate to Q.
(2) . The locus of /, the intersection of the tangents NQ and MP, is A J the common tangent to the hyperbola and circle.
(3) . The line OIV bisects the angle 6 and the area OA VP and intersects the hyperbola at its point of tangency with PS.
(4) . A straight line through P and Q passes through the left vertex of the hyperbola and is parallel to OV.
(5) . The angle APN= one-half the angle QOIST.
50. The Gudermannian. When 6 is defined as a function of u by the relation tan 6 = sinh u (Art. 48) it is called the Gudermannian of »* and is written gd u. Sin 6, cos 6 and tan 6 are then regarded as functions of u and are written sg u, eg u and tg u.'
51. Agenda. From the definitions of the Gudermannian functions prove the formulae: »
* By Cayley, Elliptic Functions, p. 56, where the equation of definition is u — In tan ( J + J
Since tan (i V + J $) -= 1 + si" i and sinh-' tan 6 - In 1 + si" g (Art. 56), the cos Q cos Q
equivalence of the two definitions is obvious. The name is given in honor of
Gudermann, who first studied these functions.