Since [BD] and [AD] are consecutive straight segments, Eq. (2.439) is equivalent to
(2.440)
this relationship coincides with Eq. (2.431), because
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
(2.441)
or else
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.
– 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
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
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
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
By the same token, a perpendicular can be dropped from corner C to side [AB] in
