however, the first tables of chords in 140 BCE – analogous to the current tables of sine values, which were completed in the second century CE by Greco‐Egyptian astronomer Ptolemy from Alexandria (Egypt). By those times, it was realized that the lengths of the sides of a right triangle and the angles between those sides satisfy fixed relationships; hence, if at least the length of one side and the amplitude of one angle is known, then all other angles and lengths can be algorithmically determined.
2.3.1 Definition and Major Features
Consider 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,
which may be rewritten as
in view again of Eq. (2.287); therefore, sin θ is given by the distance,
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
