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

as outlined in Fig. 2.10a, where Eq. (2.287)was taken advantage of; Eq. (2.302)may be redone to

(2.303)

again after dividing numerator and denominator by , and recalling Eqs. (2.288)and (2.290). For a general argument x (in rad ), one may accordingly state

(2.304)

following comparative inspection of Eqs. (2.298)and (2.303)– which varies with argument x as depicted in Fig. 2.10d. Once again, a period of π rad is apparent, i.e.

(2.305)

while Eqs. (2.301)and (2.304)imply

(2.306)

– meaning that cotangent is also an odd function. The cotangent always decreases when x increases, and is driven by vertical asymptotes described by x = kπ (with relative integer k ) as can be perceived in Fig. 2.10d.

With regard to secant of angle θ , it follows from the ratio of the length of the hypotenuse, [ OB ], to the length of the adjacent leg, [ OA ], in triangle [ OAB ] – or, alternatively, as the ratio of the length of the hypotenuse, [ OD ], to the length of the adjacent leg, [ OB ], in triangle [ OBD ], according to

(2.307)

– as outlined in Fig. 2.10a, also at the expense of Eq. (2.287); one may rewrite Eq. (2.307)as

(2.308)

after taking the reciprocal of the reciprocal, in view of Eq. (2.288). Once θ is expressed in rad , Eq. (2.308)becomes

(2.309)

– as illustrated in Fig. 2.10c. Since this function repeats itself every 2 π rad , i.e.

(2.310)

it can be claimed as periodic; furthermore, its definition as per Eq. (2.309)entails

(2.311)

with the aid of Eq. (2.296), so the secant is an even function and thus symmetrical with regard to the vertical axis. The secant is not a monotonic function; it decreases and then increases within](2 k − 1) π /2,(2 k + 1) π /2[ for even integer k , with vertical asymptotes at the extremes, or vice versa with odd integer k .

Finally, the cosecant of angle θ is given by the ratio of the length of the hypotenuse, [ OB ], to the length of the opposite leg, [ AB ], in triangle [ OAB ] – or, equivalently, as the secant of the complementary angle of θ , i.e. the ratio of the length of the hypotenuse, [ OE ], to the length of the adjacent leg, [ OB ], in triangle [ OBE ], i.e.

(2.312)

that incorporates Eq. (2.287)– as depicted in Fig. 2.10a; therefore, Eq. (2.312)may be reformulated to read

(2.313)

upon taking the reciprocal of the reciprocal, owing to Eq. (2.290). For θ expressed in rad , Eq. (2.313)will in general look like

(2.314)

– as plotted in Fig. 2.10d. A period of 2 π rad is again found, viz.

(2.315)

in addition, Eq. (2.314)has it that

(2.316)

with the aid of Eq. (2.295), should the argument be replaced by its negative – so the cosecant is symmetrical with regard to the origin of the axes, as per its odd behavior. Note that the cosecant increases and then decreases within](2 k − 1) π, 2 kπ [ for integer k , bounded by vertical asymptotes described by x = (2 k − 1) π and x = 2 kπ , respectively, and the other way round within]2 kπ, (2 k + 1) π [.

Referring again to Fig. 2.10a, one may label as u 1the unit vector centered at the origin, defining an angle θ 1with the horizontal axis – with coordinates (cos θ 1, sin θ 1) as per Eqs. (2.288)and (2.290); and likewise as u 2the unit vector centered at O but defining an angle θ 2– with coordinates (cos θ 2, sin θ 2), with θ 2 > θ 1for simplicity. Under these circumstances, the scalar product of u 1and u 2(to be discussed later) reads

(2.317)

as per its defining algorithm, and because ‖ u 1‖ = ‖ u 2‖ = 1 by hypothesis; Eq. (2.317)readily simplifies to

(2.318)

where θ 2− θ 1 > 0 represents the amplitude of the angle defined by vectors u 1and u 2, i.e. ∠ u 1, u 2. As will be duly proven below, u 1 · u 2may instead be calculated via