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

+ ⋯ + x_m; the generalized formula looks indeed like

(2.282)

      where the summation in the right‐hand side is taken over all sequences of (nonnegative) integer indices k₁ through k_m, such that the sum of all k_i ’s is n. The multinomial coefficients are given by

(2.283)

      – and count the number of different ways an n‐element set can be partitioned into disjoint subsets of sizes k₁, k₂, …, k_m . For the example above, one would have been led to

(2.284)

after setting m = 3 and n = 2 in Eq. (2.282), while Eq. (2.283) would yield

(2.285)

the possibilities of integer values for (k₁,k₂,k₃) satisfying the condition placed at the bottom of the summation in Eq. (2.284) encompass (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1) – so Eq. (2.284) becomes

(2.286)

with the aid of Eq. (2.285). After realizing that 0! = 1, 2!/2! = 1, and 2!/1! = 2 – besides – one eventually retrieves Eq. (2.281), using Eq. (2.286) as departure point.

2.3 Trigonometric Functions

      Trigonometry is the branch of mathematics that studies relationships involving lengths of sides and amplitudes of angles in triangles. This field emerged in the Hellenistic world during the third century, by the hand of Euclid and Archimedes – who studied the properties of chords and inscribed angles in circles, while proving theorems equivalent to most modern trigonometric formulae; Hipparchus from Nicaea (Asia Minor) produced, however, the first tables of chords in 140 BCE – analogous to the current tables of sine values, which were completed in the second century CE by Greco‐Egyptian astronomer Ptolemy from Alexandria (Egypt). By those times, it was realized that the lengths of the sides of a right triangle and the angles between those sides satisfy fixed relationships; hence, if at least the length of one side and the amplitude of one angle is known, then all other angles and lengths can be algorithmically determined.

      2.3.1 Definition and Major Features

      Consider a unit vector u, i.e.

(2.287)

with double bars indicating length, centered at the origin of a system of coordinates, which rotates around said origin – as illustrated in Fig. 2.10a. If the angle defined by vector u (playing the role of hypotenuse in right triangle [OAB]) with the horizontal axis is denoted as θ, then cos θ equals, by definition, the ratio of the length of the adjacent leg, [OA], to the length of the hypotenuse, [OB], according to

(2.288)

      that degenerates to

(2.289)

– where Eq. (2.287) was employed to advantage; hence, cos θ is but the distance, , of the extreme point, B, of the said vector to the vertical axis. By the same token, sin θ equals, by definition, the ratio of the length of the opposite leg, [AB], to the length of the hypotenuse, according to

(2.290)

      which may be rewritten as

(2.291)

      in view again of Eq. (2.287); therefore, sin θ is given by the distance, , of the extreme point, B, of u to the horizontal axis. Sine and cosine constitute the basic trigonometric functions; they are also known as circular functions, owing to the loci of the extreme points of u describing a circle upon full rotation – as per Fig. 2.10a.

Figure 2.10 (a) Trigonometric circle, described by vector u of unit length centered at origin O, after full rotation by 2π rad around O – together with tangent to the said circle extended until crossing the axes, angle defined by u and the horizontal axis of amplitude θ, and definition of trigonometric functions as lengths of associated straight segments; and variation, with their argument x, of major trigonometric functions, viz. (b) sine (sin) and cosine (cos), (c) tangent (tan) and secant (sec), and (d) cotangent (cotan) and cosecant (cosec).

The amplitude of the aforementioned angle θ is normally reported in radian, so it will for convenience be termed x hereafter; sin x and cos x are accordingly plotted in Fig. 2.10b, as a function of x (expressed in that unit). Note their periodic nature, with period 2π rad, i.e.

      (2.292)