8 Relations Among Hyperbolic Functions.

Article 8
Relations Among Hyperbolic Functions.

Among the six functions there are ﬁve independent relations, so that when the numerical value of one of the functions is given, the values of the other ﬁve can be found. Four of these relations consist of the four deﬁning equations (9). The ﬁfth is derived from the equation of the hyperbola

\begin{matrix} \frac{x_{1}^{2}}{a_{1}^{2}} - \frac{y_{1}^{2}}{b_{1}^{2}} = 1, \end{matrix}

giving

\begin{matrix} {cosh}^{2} u - {sinh}^{2} u = 1 . & (10) \end{matrix}

By a combination of some of these equations other subsidiary relations may be obtained; thus, dividing (10) successively by ${cosh}^{2} u, {sinh}^{2} u$ , and applying (9), give

\begin{aligned} 1 - {tanh}^{2} u & = {sech}^{2} u, \\ {coth}^{2} u - 1 & = {csch}^{2} u . \end{aligned}\}

(11)

Equations (9), (10), (11) will readily serve to express the value of any function in terms of any other. For example, when $tanh u$ is given,

\begin{matrix} coth u = \frac{1}{tanh u}, sech u = \sqrt{1 - {tanh}^{2} u}, \\ cosh u = \frac{1}{\sqrt{1 - {tanh}^{2} u}}, sinh u = \frac{tanh u}{\sqrt{1 - {tanh}^{2} u}}, \\ csch u = \frac{\sqrt{1 - {tanh}^{2} u}}{tanh u} . \end{matrix}

The ambiguity in the sign of the square root may usually be removed by the following considerations: The functions $cosh u, sech u$ are always positive, because the primary characteristic ratio $\frac{x_{1}}{a_{1}}$ is positive, since the initial line $O_{1} A_{1}$ and the abscissa $O_{1} M_{1}$ are similarly directed from $O_{1}$ on whichever branch of the hyperbola $P_{1}$ maybe situated; but the functions $sinh u, tanh u, coth u, csch u$ , involve the other characteristic ratio $\frac{y_{1}}{b_{1}}$ , which is positive or negative according as $y_{1}$ and $b_{1}$ have the same or opposite signs, i.e., as the measure $u$ is positive or negative; hence these four functions are either all positive or all negative. Thus when any one of the functions $sinh u, tanh u, csch u, coth u$ , is given in magnitude and sign, there is no ambiguity in the value of any of the six hyperbolic functions; but when either $cosh u$ or $sech u$ is given, there is ambiguity as to whether the other four functions shall be all positive or all negative.

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The hyperbolic tangent may be expressed as the ratio of two lines. For draw the tangent line $A C = t$ ; then

\begin{array}{l} tanh u & = \frac{y}{b} : \frac{x}{a} = \frac{a}{b} \cdot \frac{y}{x} \\ = \frac{a}{b} \cdot \frac{t}{a} = \frac{t}{b} . & (12) \end{array}

The hyperbolic tangent is the measure of the triangle $O A C$ . For

\begin{matrix} \frac{O A C}{O A B} = \frac{a t}{a b} = \frac{t}{b} = tanh u . & (13) \end{matrix}

Thus the sector $A O P$ , and the triangles $A O P, P O B, A O C$ , are proportional to $u, sinh u, cosh u, tanh u$ (eqs. 5, 13); hence

sinh u > u > tanh u .

(14)

Prob. 7.

Express all the hyperbolic functions in terms of

sinh u

. Given

cosh u = 2

, ﬁnd the values of the other functions.

Prob. 8.

Prove from eqs. 10, 11, that

cosh u > sinh u, cosh u > 1, tanh u < 1, sech u < 1

Prob. 9.

In the ﬁgure of Art. 1, let

O A = 2, O B = 1, A O B = 6 0^{\circ}

, and area of sector

A O P = 3

; ﬁnd the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyperbolic sector; and ﬁnd the area of the triangle

A O P

. (Use tables of cos, sin, cosh, sinh.)

Prob. 10.

Show that

coth u, sech u, csch u

may each be expressed as the ratio of two lines, as follows: Let the tangent at

P

make on the conjugate axes

O A, O B

, intercepts

O S = m, O T = n

; let the tangent at

B

, to the conjugate hyperbola, meet

O P

R

, making

B R = l

; then

coth u = \frac{l}{a}, sech u = \frac{m}{a}, csch u = \frac{n}{b} .

Prob. 11.

The measure of segment

A M P

sinh u cosh u - u

. Modify this for the ellipse. Modify also eqs. 10–14, and probs. 8, 10.

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Article 8Relations Among Hyperbolic Functions.

Article 8
Relations Among Hyperbolic Functions.