Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/297

§. III.

Recherche des sommes ou différences d’angles ou de côtés.

La combinaison des formules (41) avec les formules (v) et (vi) donne

${\frac {\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .s}{\operatorname {Sin} .a}},\qquad {\frac {\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .(s-a)}{\operatorname {Sin} .a}}\,;$

c’est-à-dire,

$\operatorname {Sin} .s={\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}},\qquad \qquad \qquad$(xiii)

$\operatorname {Sin} .(s-a)={\frac {\operatorname {Sin} .a\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}.\qquad \qquad \qquad$(xiv)

La combinaison des formules (42) avec les formules (V) et (VI) donne ensuite

${\frac {\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}=-{\frac {\operatorname {Cos} .S}{\operatorname {Sin} .A}},\qquad {\frac {\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}={\frac {\operatorname {Cos} .(S-A)}{\operatorname {Sin} .A}},$

c’est-à-dire,

$\operatorname {Cos} .s=-{\frac {\operatorname {Sin} .A\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}},\qquad \qquad \qquad$(XIII)

$\operatorname {Cos} .(S-A)=+{\frac {\operatorname {Sin} .A\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}.\qquad \qquad \qquad$(XIV)

Mais les formules (viii, VIII) donnent (7)

$\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C={\frac {P}{2\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\qquad \qquad \qquad$(xv)