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

(2.341)

recalling again the definition of cotangent as reciprocal of tangent, i.e. Eq. (2.304), it is possible to transform Eq. (2.341)to

(2.342)

Equations (2.339)and (2.342)accordingly convey a tool for the calculation of cotan{ x ± y } knowing solely cotan x and cotan y .

Upon ordered addition of Eqs. (2.321)and (2.325), one gets

(2.343)

whereas ordered subtraction thereof gives rise to

(2.344)

by the same token, ordered addition of Eqs. (2.328)and (2.330)generates

(2.345)

and ordered subtraction unfolds

(2.346)

One may now define two auxiliary variables, X and Y , according to

(2.347)

and

(2.348)

respectively; ordered addition of Eqs. (2.347)and (2.348)yields

(2.349)

or else

(2.350)

upon isolation of x – whereas ordered subtraction of Eq. (2.348)from Eq. (2.347)leads to

(2.351)

which in turn conveys

(2.352)

when solved for y . Insertion of Eqs. (2.347), (2.348), (2.350), and (2.352)transforms Eq. (2.343)to

(2.353)

and Eq. (2.344)likewise supports

(2.354)

one may similarly transform Eq. (2.345)to

(2.355)

while Eq. (2.346)yields

(2.356)

Equations (2.353)– (2.356)may be useful whenever logarithms of their left‐hand sides are to be handled, because there is no formula to calculate the logarithm of an algebraic sum – unlike happens with the logarithm of a product, see Eq. (2.20).

In attempts to generate further useful relationships involving transformation of arguments of trigonometric functions, it is convenient to resort to the complex domain – and recall that complex numbers, say z 1and z 2, may be defined by Cartesian (or rectangular) coordinates as

(2.357)

and

(2.358)

with denoting imaginary unit, besides a i’s and b i’s denoting real numbers. Since z iis represented by two coordinates in a plane – a ias abscissa (on the real axis) and b ias ordinate (on the imaginary axis), one may denote as θ ithe angle of vector of coordinates ( a i ,b i) with the real axis, and by ρ iits length; hence, Eq. (2.357)may appear alternatively as

(2.359)

and

(2.360)

respectively. Note that

(2.361)

and

(2.362)

were used as defining relationships, in agreement with Eqs. (2.288)and (2.290), complemented by Fig. 2.10a; and after replacing ‖ u‖ by ρ i, by a i, and by b i. The product of two complex numbers reads

(2.363)

in view of Eqs. (2.359)and (2.360), or else

(2.364)