with Eqs. (2.288) and (2.290), complemented by Fig. 2.10 a; and after replacing ‖ u ‖ by ρi, in view of Eqs. (2.359) and (2.360), or else
after lumping ρ1 and ρ2, and applying the distributive property to the product of sums of trigonometric functions; since ι2 = −1 by definition, Eq. (2.364) becomes
along with convenient factoring out of ι. Insertion of Eqs. (2.325) and (2.328) supports transformation of Eq. (2.365) to
in the case of n complex numbers, Eq. (2.366) readily generalizes to
via consecutive application of the transformation of Eq. (2.363) to Eq. (2.366). Should, in addition, ρ1 = ρ2 = ⋯ = ρn = ρ and θ1 = θ2 = ⋯ = θn = θ, then Eq. (2.367) degenerates to
– in view of the functional form conveyed by either Eq. (2.359) or Eq. (2.360), and the definition of power; if ρ is further set equal to unity, then Eq. (2.368) simplifies to
usually known as Moivre's formula – and valid for any positive or negative integer n (as well as for rational numbers). For instance, Eq. (2.369) yields
in the case of n = −1, which breaks down to merely
in view of Eqs. (2.295) and (2.296) – and with z defined
combination of Eqs. (2.370) and (2.371) obviously looks like
Ordered addition of Eqs. (2.371) and (2.372) produces
(2.374)
that may be solved for cos θ as
if Eq. (2.371) is subtracted from Eq. (2.372), then one gets
(2.376)
which gives rise to
after isolation of sin θ. By the same token, one gets
after raising both sides of Eq. (2.372) to the nth power, or else
