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
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
