(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[.
2.3.2 Angle Transformation Formulae
Referring again to Fig. 2.10a, one may label as u1 the unit vector centered at the origin, defining an angle θ1 with the horizontal axis – with coordinates (cos θ1, sin θ1) as per Eqs. (2.288) and (2.290); and likewise as u2 the unit vector centered at O but defining an angle θ2 – with coordinates (cos θ2, sin θ2), with θ2 > θ1 for simplicity. Under these circumstances, the scalar product of u1 and u2 (to be discussed later) reads
as per its defining algorithm, and because ‖ u1‖ = ‖ u2‖ = 1 by hypothesis; Eq. (2.317) readily simplifies to
where θ2 − θ1 > 0 represents the amplitude of the angle defined by vectors u1 and u2, i.e. ∠ u1, u2. As will be duly proven below, u1 · u2 may instead be calculated via
so elimination of u1· u2 between Eqs. (2.318) and (2.319) unfolds
this is equivalent to
after having relabeled θ1 and θ2 to y and x, respectively. Equation (2.321), known as the basic angle transformation formula, permits calculation of the cosine of a difference of angles based on knowledge of sine and cosine of the individual angles – and is actually valid, irrespective of the relative amplitude of angles θ1 and θ2. If θ2 is set equal to π/2 in particular, then Eq. (2.320) becomes
(2.322)
which reduces to
because cos π/2 is nil and sin π/2 is unity; Eq. (2.323) confirms that sine and cosine are complementary functions – so a change of variable to x ≡ π/2 − θ1 (implying θ1 = π/2 − x) allows retrieval of Eq. (2.294), as expected.
After rewriting Eq. (2.321) as
(2.324)
at the expense of Eqs. (2.295) and (2.296), and changing notation of −y to y (in view of its being a dummy variable), one obtains
Eq. (2.325) permits rapid calculation of the cosine of a sum of two arguments, again based on the sine and cosine of the individual arguments. On the other hand, Eqs. (2.294) and
