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

Since [ BD ] and [ AD ] are consecutive straight segments, Eq. (2.439)is equivalent to

(2.440)

this relationship coincides with Eq. (2.431), because , , and , i.e. the sides of a right triangle, as per comparative inspection of Fig. 2.11a and c.

When a , b , and c in Eq. (2.431)represent sides [ OA ], [ AB ], and [ OB ] (or u, for that matter), respectively, in Fig. 2.10a, with lengths , , and , one may resort to Eqs. (2.289)and (2.291)to reformulate Eq. (2.431)as

(2.441)

or else

(2.442)

after taking Eq. (2.287)into account and replacing θ by x (as usual); Eq. (2.442)is known as fundamental theorem of trigonometry.

Figure 2.12 Graphical representation of generic triangle [ ABC ] – with indication of corners A , B , and C , lengths of opposite sides a (corresponding to [ BC ]), b (corresponding to [ AC ]), and c (corresponding to [ AB ]), and angles of adjacent sides (a, b, c) α (formed by [ AB ] and [ AC ]), (a,b) β (formed by [ AB ] and [ BC ]), and (a,b) γ (formed by [ AC ] and [ BC ]) – after drawing an altitude (b) from A to [ BC ], B to [ AC ], or C to [ AB ], or (c) from the center, O , of circumcircle ( ABC ) to point D on [ BC ].

The Pythagorean theorem is a special case of a more general theorem relating the lengths of the sides of any triangle (not necessarily containing a right angle), viz.

(2.443)

– which degenerates to Eq. (2.431)when γ (i.e. the angle formed by sides of length a and b ) equals π /2, since the corresponding cosine is nil); this is usually known as cosine formula (or cosine rule), and abides to the nomenclature in Fig. 2.12a. Equation (2.443)is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known; it was explicitly stated only in the fifteenth century, by Arab mathematician Jamshid al‐Kashi. By changing which sides of the triangle are denoted as a , b , and c , Eq. (2.443)may appear as

(2.444)

or else

(2.445)

– encompassing angles α and β , respectively. To prove the validity of Eq. (2.443), one may to advantage drop the perpendicular from corner A onto side [ BC ], as illustrated in Fig. 2.12b – so the definition of cosine as per Eq. (2.288)allows one to write

(2.446)

where the first term in the right‐hand side represents the length of the portion of [ BC ] closer to B , and the second term represents the remainder of [ BC ] closer to C , upon multiplication of both sides by a , Eq. (2.446)becomes

(2.447)

A similar rationale may be followed with regard to the perpendicular from corner B to side [ AC ] – see Fig. 2.12b, where again the definition of cosine supports

(2.448)

here the first term in the right‐hand side represents the length of the portion of [ AC ] closer to A , and the second term represents the length of the remainder of [ AC ] closer to C . Multiplication of both sides by b then converts Eq. (2.448)to

(2.449)

By the same token, a perpendicular can be dropped from corner C to side [ AB ] in Fig. 2.12b to yield

(2.450)

where the first term in the right‐hand side represents the length of the portion of [ AB ] closer to A , while the second term represents the length of the remainder of [ AB ] closer to B ; c may then multiply both sides to produce

(2.451)

Ordered addition of Eqs. (2.447)and (2.449)gives rise to

(2.452)