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

      and

      (2.293)

      and also their lower and upper bounds, i.e. −1 and 1. It becomes apparent from inspection of Fig. 2.10b that the plot of cos x may be obtained from the plot of sin x via a horizontal translation of π/2 rad leftward; in other words,

(2.294)

      – and such a complementarity to a right angle, of amplitude π/2 rad, justifies the term cosine (with prefix ‐co standing for complementary, or adding up to a right angle). The sine is an odd function, i.e.

(2.295)

      hence, its plot is symmetrical relative to the origin of coordinates. Conversely, the cosine is an even function, i.e.

(2.296)

      – meaning that its plot is symmetrical relative to the vertical axis.

      The tangent of angle θ may be defined as the ratio of the length of the opposite leg, [AB], to the length of the adjacent leg, [OA], in triangle [OAB] – or, alternatively, as the ratio of the length of the opposite leg, [BD], to the length of the adjacent leg, [OB], in triangle [OBD], according to

(2.297)

– once more with the aid of Eq. (2.287), and as emphasized in Fig. 2.10 a; note that Eq. (2.297) may also appear as

      (2.298)

following division of both numerator and denominator by , and with the extra aid of Eqs. (2.288) and (2.290). If θ is expressed in rad, then one has

(2.299)

      in general – as plotted in Fig. 2.10c. Note that tangent is still a periodic function, but of smaller period, π rad, according to

      (2.300)

whereas combination of Eqs. (2.295), (2.296), and (2.299) implies that

(2.301)

      – so the (trigonometric) tangent is an odd function. The tangent is also a monotonically increasing function – yet it exhibits vertical asymptotes at x = kπ/2 (with relative integer k), see again Fig. 2.10c.

      The cotangent of angle θ may, in turn, be defined as the ratio of the length of the adjacent leg, [OA], to the length of the opposite leg, [AB], in triangle [OAB] – or, instead, as the tangent of the complementary of angle θ, i.e. ∠BOE, via the ratio of the length of the opposite leg, [BE], to the length of the adjacent leg, [OB], in triangle [OBE], viz.

(2.302)

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)