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2.3.1 Definition and Major Features
ОглавлениеConsider a unit vector u, i.e.
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
that degenerates to
– 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
which may be rewritten as
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)
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,
– 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.
hence, its plot is symmetrical relative to the origin of coordinates. Conversely, the cosine is an even function, i.e.
– 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
– 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
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
– 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.
as outlined in Fig. 2.10a, where Eq. (2.287) was taken advantage of; Eq. (2.302) may be redone to
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
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
– as outlined in Fig. 2.10a, also at the expense of Eq. (2.287); one may rewrite Eq. (2.307) as
after taking the reciprocal of the reciprocal, in view of Eq. (2.288). Once θ is expressed in rad, Eq. (2.308) becomes
– as illustrated in Fig. 2.10c. Since this function repeats itself every 2π 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](2k − 1)π/2,(2k + 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.
that incorporates Eq. (2.287) – as depicted in Fig. 2.10 a; therefore, Eq. (2.312) may be reformulated to read
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
– as plotted in Fig. 2.10d. A period of 2π 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](2k − 1)π,2kπ[ for integer k, bounded by vertical asymptotes described by x = (2k − 1)π and x = 2kπ, respectively, and the other way round within]2kπ,(2k + 1)π[.
