Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/294

Le texte de cette page a été corrigé et est conforme au fac-similé.

{\begin{array}{rll}\operatorname {Sin} .{\frac {1}{2}}A&={\sqrt {\frac {1-\operatorname {Cos} .A}{2}}}&={\sqrt {-{\frac {\operatorname {Cos} .a-\operatorname {Sin} .b\operatorname {Sin} .c-\operatorname {Cos} .b\operatorname {Cos} .c}{2\operatorname {Sin} .b\operatorname {Sin} .c}}}},\\\\\operatorname {Sin} .{\frac {1}{2}}a&={\sqrt {\frac {1-\operatorname {Cos} .a}{2}}}&={\sqrt {-{\frac {\operatorname {Cos} .A-\operatorname {Sin} .B\operatorname {Sin} .C+\operatorname {Cos} .B\operatorname {Cos} .C}{2\operatorname {Sin} .B\operatorname {Sin} .C}}}},\\\\\operatorname {Cos} .{\frac {1}{2}}A&={\sqrt {\frac {1+\operatorname {Cos} .A}{2}}}&={\sqrt {+{\frac {\operatorname {Cos} .a+\operatorname {Sin} .b\operatorname {Sin} .c-\operatorname {Cos} .b\operatorname {Cos} .c}{2\operatorname {Sin} .b\operatorname {Sin} .c}}}},\\\\\operatorname {Cos} .{\frac {1}{2}}a&={\sqrt {\frac {1+\operatorname {Cos} .a}{2}}}&={\sqrt {+{\frac {\operatorname {Cos} .A+\operatorname {Sin} .B\operatorname {Sin} .C+\operatorname {Cos} .B\operatorname {Cos} .C}{2\operatorname {Sin} .B\operatorname {Sin} .C}}}},\end{array}}

d’où, en vertu des formules (18), (20), (21), (22),

{\begin{array}{lr}\operatorname {Sin} .{\frac {1}{2}}A={\sqrt {\frac {\operatorname {Sin} .(s-b)\operatorname {Sin} .(s-c)}{\operatorname {Sin} .b\operatorname {Sin} .c}}},&(v)\\\\\operatorname {Sin} .{\frac {1}{2}}a={\sqrt {\frac {-\operatorname {Cos} .S\operatorname {Cos} .(S-A)}{\operatorname {Sin} .B\operatorname {Sin} .C}}},&(V)\\\\\operatorname {Cos} .{\frac {1}{2}}A={\sqrt {\frac {\operatorname {Sin} .s\operatorname {Sin} .(s-a)}{\operatorname {Sin} .b\operatorname {Sin} .c}}},&(vi)\\\\\operatorname {Cos} .{\frac {1}{2}}a={\sqrt {\frac {\operatorname {Cos} .(S-B)\operatorname {Cos} .(S-C)}{\operatorname {Sin} .B\operatorname {Sin} .C}}}.&(VI)\end{array}}

En posant, pour abréger,

{\begin{array}{lr}p^{2}\ =+\operatorname {Sin} .s\operatorname {Sin} .(s-a)\operatorname {Sin} .(s-b)\operatorname {Sin} .(s-c),&(vii)\\P^{2}=-\operatorname {Cos} .S\operatorname {Cos} .(S-A)\operatorname {Cos} .(S-B)\operatorname {Cos} .(S-C),&(VII)\end{array}}

on tire encore de là, au moyen de la formulé (7),