25 Graphs of Hyperbolic Functions.

Article 25
Graphs of Hyperbolic Functions.

Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, successive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corresponding values of the function to be plotted, the locus traced out by this series of points will be a graphical representation of the variation of the function as the sectorial measure varies. The equations of the curves in the ordinary cartesian notation are:

Fig.	Full Lines.	Dotted Lines.
A	$y = cosh x,$	$y = sech x;$
B	$y = sinh x,$	$y = csch x;$
C	$y = tanh x,$	$y = coth x;$
D	$y = gd x .$

Here $x$ is written for the sectorial measure $u$ , and $y$ for the numerical value of $cosh u$ , etc. It is thus to be noted that the variables $x$ , $y$ are numbers, or ratios, and that the equation $y = cosh x$ merely expresses that the relation between the numbers $x$ and $y$ is taken to be the same as the relation between a sectorial measure and its characteristic ratio. The numerical values of $cosh u, sinh u, tanh u$ are given in the tables at the end of this chapter for values of $u$ between $0$ and $4$ . For greater values they may be computed from the developments of Art. 16.

The curves exhibit graphically the relations:

\begin{matrix} sech u = \frac{1}{cosh u}, csch u = \frac{1}{sinh u}, coth u = \frac{1}{tanh u}; \\ cosh u ≮ 1, sech u ≯ 1, tanh u ≯ 1, gd u < \frac{1}{2} π, etc.; \\ sinh (- u) = - sinh u, cosh (- u) = cosh u, \\ tanh (- u) = - tanh u, gd (- u) = - gd u, etc.; \\ cosh 0 = 1, sinh 0 = 0, tanh 0 = 0, csch (0) = \infty, etc.; \\ cosh (\pm \infty) = \infty, sinh (\pm \infty) = \pm \infty, tanh (\pm \infty) = \pm 1, etc. \end{matrix}

The slope of the curve $y = sinh x$ is given by the equation $\frac{d y}{d x} = cosh x$ , showing that it is always positive, and that the curve becomes more nearly vertical as $x$ becomes inﬁnite. Its direction of curvature is obtained from $\frac{d^{2} y}{d x^{2}} = sinh x$ , proving that the curve is concave downward when $x$ is negative, and upward when $x$ is positive. The point of inﬂexion is at the origin, and the inﬂexional tangent bisects the angle between the axes.

The direction of curvature of the locus $y = sech x$ is given by $\frac{d^{2} y}{d x^{2}} =$ $sech x (2 {tanh}^{2} x - 1)$ , and thus the curve is concave downwards or upwards according as $2 {tanh}^{2} x - 1$ is negative or positive. The inﬂexions occur at the points $x = \pm {tanh}^{- 1} . 707, = \pm . 881$ , $y = . 707$ ; and the slopes of the inﬂexional tangents are $\mp \frac{1}{2}$ .

The curve $y = csch x$ is asymptotic to both axes, but approaches the axis of $x$ more rapidly than it approaches the axis of $y$ , for when $x = 3$ , $y$ is only $. 1$ , but it is not till $y = 10$ that $x$ is so small as $. 1$ . The curves $y = csch x$ , $y = sinh x$ cross at the points $x = \pm . 881$ , $y = \pm 1$ .

Prob. 66.: Find the direction of curvature, the inﬂexional tangent, and the asymptotes of the curves $y = gd x$ , $y = tanh x$ .
Prob. 67.: Show that there is no inﬂexion-point on the curves $y = cosh x$ , $y = coth x$ .
Prob. 68.: Show that any line $y = m x + n$ meets the curve $y = tanh x$ in either three real points or one. Hence prove that the equation $tanh x = m x + n$ has either three real roots or one. From the ﬁgure give an approximate solution of the equation $tanh x = x - 1$ .
Prob. 69.: Solve the equations: $cosh x = x + 2$ ; $sinh x = \frac{3}{2} x$ ; $gd x = x - \frac{1}{2} π$ .
Prob. 70.: Show which of the graphs represent even functions, and which of them represent odd ones.

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Article 25Graphs of Hyperbolic Functions.

Article 25
Graphs of Hyperbolic Functions.