F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

In attempts to solve equations explicitly involving trigonometric functions, one may to advantage resort to their inverse functions; this includes sin −1 x , which develops graphically as outlined in Fig. 2.13a. Only the portion of sin −1 x between −π /2 and π /2 – corresponding to x within interval [ − 1,1], is normally considered, as indicated in bold in Fig. 2.13a; otherwise a function would not result, since more than one value would be taken by the dependent variable for any given value of the independent variable. By the same token, one may define the inverse of cosine, cos −1 x – as also plotted in Fig. 2.13a; in this case, one usually restricts to cos −1 x between 0 and π , which corresponds to x spanning also interval [ − 1,1]. Note that the curves representing inverse functions may be obtained through rotation of the original curves by π rad around the bisectrix of the odd quadrants – see Fig. 2.13a vis‐à‐vis with Fig. 2.10b.

A similar rationale may be pursued to obtain the inverse tangent and cotangent, i.e. tan −1 x and cotan −1 x ; these are plotted in Fig. 2.13b. In both cases, argument x spans the whole real axis – but one usually restricts attention to the portion comprised between −π /2 and π /2 in the case of tan −1 x , and to the portion comprised between 0 and π in the case of cotan −1 x , so as to back up true (i.e. single‐valued) functions.

Exponential functions are quite useful in process engineering problems; solutions to differential equations involving an exponential of a given argument, and simultaneously of its negative are indeed frequently found. Therefore, a set of functions termed hyperbolic functions has been designed to assist in the associated modeling; coincidentally, they satisfy most operational relationships of trigonometric functions, and have accordingly also been termed hyperbolic trigonometric functions.

The two basic hyperbolic functions are the hyperbolic sine, sinh x , defined as

(2.472)

and the hyperbolic cosine, cosh x , abiding to

(2.473)

the plots of Eqs. (2.472)and (2.473)are provided in Fig. 2.14a. Note the even nature of cosh x , i.e.

(2.474)

with the aid of Eq. (2.473); in contrast to the odd nature of sinh x , according to

(2.475)

as per Eq. (2.472). The curves representing these two functions overlap at large x , i.e.

(2.476)

stemming from Eqs. (2.472)and (2.473), as emphasized in Fig. 2.14a (with the exact concept of limit coming soon); while there is a unit minimum value of cosh x at x = 0, viz.

(2.477)

based on Eq. (2.473)– with derivation rules to be introduced in due course.

Figure 2.14 Variation, with their argument x , of major hyperbolic functions, (a) hyperbolic sine (sinh) and cosine (cosh), (b) hyperbolic tangent (tanh) and cotangent (cotanh), and (c) hyperbolic secant (sech) and cosecant (cosech).

On the other hand, ordered addition of Eqs. (2.472)and (2.473), viz.

(2.478)

along with cancelation of symmetrical terms lead to

(2.479)

while ordered subtraction of Eq. (2.472)from Eq. (2.473)yields

(2.480)

since e x/2 cancels out with its negative – or simply

(2.481)

Equations (2.479)and (2.481)are expected, in view of cosh x being an even function and sinh x being an odd function – as per Eqs. (2.474)and (2.475), respectively, coupled with Eq. (2.1).

The remaining functions of practical interest include the hyperbolic tangent, tanh x , defined as

(2.482)

– at the expense of Eqs. (2.472)and (2.473), after dropping 2 from both numerator and denominator; as well as its reciprocal, the hyperbolic cotangent, cotanh x , according to

(2.483)

with the aid of Eq. (2.482)– both of which are depicted in Fig. 2.14b. Note the monotonically increasing pattern of tanh x , with

(2.484)

serving as leftward horizontal asymptote based on Eq. (2.482), complemented by

(2.485)

serving as rightward horizontal asymptote; along with the decreasing behavior of cotanh x , despite the discontinuity at x = 0, i.e.

(2.486)

stemming from Eq. (2.483)– which accordingly justifies why the vertical axis plays the role of vertical asymptote.