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

(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 uto 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 udescribing 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 2 π rad around O – together with tangent to the said circle extended until crossing the axes, angle defined by uand 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 2 π rad , i.e.

(2.292)

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. 2.10a; 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)