F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

after lumping ρ 1and ρ 2, and applying the distributive property to the product of sums of trigonometric functions; since ι 2 = − 1 by definition, Eq. (2.364)becomes

(2.365)

along with convenient factoring out of ι . Insertion of Eqs. (2.325)and (2.328)supports transformation of Eq. (2.365)to

(2.366)

in the case of n complex numbers, Eq. (2.366)readily generalizes to

(2.367)

via consecutive application of the transformation of Eq. (2.363)to Eq. (2.366). Should, in addition, ρ 1 = ρ 2 = ⋯ = ρ n = ρ and θ 1 = θ 2 = ⋯ = θ n = θ , then Eq. (2.367)degenerates to

(2.368)

– 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

(2.369)

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

(2.370)

in the case of n = − 1, which breaks down to merely

(2.371)

in view of Eqs. (2.295)and (2.296)– and with z defined

(2.372)

combination of Eqs. (2.370)and (2.371)obviously looks like

(2.373)

Ordered addition of Eqs. (2.371)and (2.372)produces

(2.374)

that may be solved for cos θ as

(2.375)

if Eq. (2.371)is subtracted from Eq. (2.372), then one gets

(2.376)

which gives rise to

(2.377)

after isolation of sin θ . By the same token, one gets

(2.378)

after raising both sides of Eq. (2.372)to the n th power, or else

(2.379)

once reciprocals are taken of both sides; Eq. (2.379)may then be rewritten as

(2.380)

given the rule of composition of powers – where combination with Eq. (2.369)yields

(2.381)

together with Eq. (2.373)upon replacement of θ by nθ . Since Eq. (2.378)may be rewritten as

(2.382)

as per Eqs. (2.369)and (2.372), one concludes that

(2.383)

following ordered addition of Eqs. (2.381)and (2.382), together with cancelation of symmetrical terms afterward; by the same token, ordered subtraction of Eq. (2.381)from Eq. (2.382)generates

(2.384)

In view of Eq. (2.375), one may calculate the power of a cosine via

(2.385)

and Eq. (2.377)similarly supports

(2.386)

after retrieving Newton’s binomial as per Eq. (2.236), it is possible to reformulate Eq. (2.385)to

(2.387)

where the powers of z and of its reciprocal may be lumped to yield

(2.388)

If the exponent of the cosine function is an even integer, say, 2 n , then Eq. (2.388)can be redone to

(2.389)