Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/295

 Les corrections sont expliquées en page de discussion

$\operatorname {Sin} .A={\frac {2p}{\operatorname {Sin} .b\operatorname {Sin} .c}},\quad (viii)\qquad \operatorname {Sin} .a={\frac {2P}{\operatorname {Sin} .B\operatorname {Sin} .C}}.\quad (VIII)$

Par les formules (v, V), (vi, VI), on a

$\left.{\begin{array}{ll}\operatorname {Sin} .{\frac {1}{2}}B={\sqrt {\frac {\operatorname {Sin} .(s-c)\operatorname {Sin} .(s-a)}{\operatorname {Sin} .c\operatorname {Sin} .a}}},&\operatorname {Cos} .{\frac {1}{2}}B={\sqrt {\frac {\operatorname {Sin} .s\operatorname {Sin} .(s-b)}{\operatorname {Sin} .c\operatorname {Sin} .a}}},\\\\\operatorname {Sin} .{\frac {1}{2}}C={\sqrt {\frac {\operatorname {Sin} .(s-a)\operatorname {Sin} .(s-b)}{\operatorname {Sin} .a\operatorname {Sin} .b}}},&\operatorname {Cos} .{\frac {1}{2}}C={\sqrt {\frac {\operatorname {Sin} .s\operatorname {Sin} .(s-c)}{\operatorname {Sin} .a\operatorname {Sin} .b}}}\,;\end{array}}\right\}(41)$

$\left.{\begin{array}{ll}\operatorname {Sin} .{\frac {1}{2}}b={\sqrt {-{\frac {\operatorname {Cos} .S\operatorname {Cos} .(S-B)}{\operatorname {Sin} .C\operatorname {Sin} .A}}}},&\operatorname {Cos} .{\frac {1}{2}}b={\sqrt {\frac {\operatorname {Cos} .(S-C)\operatorname {Cos} .(S-A)}{\operatorname {Sin} .C\operatorname {Sin} .A}}},\\\\\operatorname {Sin} .{\frac {1}{2}}c={\sqrt {-{\frac {\operatorname {Cos} .S\operatorname {Cos} .(S-C)}{\operatorname {Sin} .A\operatorname {Sin} .B}}}}\,&\operatorname {Cos} .{\frac {1}{2}}c={\sqrt {\frac {\operatorname {Cos} .(S-A)\operatorname {Cos} .(S-B)}{\operatorname {Sin} .A\operatorname {Sin} .B}}}\,;\end{array}}\right\}(42)$

De ces dernières on conclut, à l’aide des formules (1, 2, 3, 4, 7)

${\begin{aligned}&\operatorname {Sin} .{\frac {1}{2}}(B\pm C)={\frac {\operatorname {Sin} .(s-c)\pm \operatorname {Sin} .(s-b)}{\operatorname {Sin} .a}}.{\sqrt {\frac {\operatorname {Sin} .s\operatorname {Sin} .(s-a)}{\operatorname {Sin} .b\operatorname {Sin} .c}}}\\&\qquad \qquad \qquad ={\frac {\operatorname {Sin} .(s-c)\pm \operatorname {Sin} .(s-b)}{2\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}a}}\operatorname {Cos} .{\frac {1}{2}}A,\\\\&\operatorname {Cos} .{\frac {1}{2}}(B\pm C)={\frac {\operatorname {Sin} .s\mp \operatorname {Sin} .(s-a)}{\operatorname {Sin} .a}}.{\sqrt {\frac {\operatorname {Sin} .(s-b)\operatorname {Sin} .(s-c)}{\operatorname {Sin} .b\operatorname {Sin} .c}}}\\&\qquad \qquad \qquad ={\frac {\operatorname {Sin} .s\mp \operatorname {Sin} .(s-a)}{2\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}a}}\operatorname {Sin} .{\frac {1}{2}}A,\\\\&\operatorname {Sin} .{\frac {1}{2}}(b\pm c)={\frac {\operatorname {Cos} .(S-B)\pm \operatorname {Cos} .(S-C)}{\operatorname {Sin} .A}}.{\sqrt {-{\frac {\operatorname {Cos} .S\operatorname {Cos} .(S-A)}{\operatorname {Sin} .B\operatorname {Sin} .C}}}}\\&\qquad \qquad \qquad ={\frac {\operatorname {Cos} .(S-B)\pm \operatorname {Cos} .(S-C)}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}A}}\operatorname {Sin} .{\frac {1}{2}}a,\\\\&\operatorname {Cos} .{\frac {1}{2}}(b\pm c)={\frac {\operatorname {Cos} .(S-A)\mp \operatorname {Cos} .S}{\operatorname {Sin} .A}}.{\sqrt {\frac {\operatorname {Cos} .(S-B)\operatorname {Cos} .(S-C)}{\operatorname {Sin} .B\operatorname {Sin} .C}}}\\&\qquad \qquad \qquad ={\frac {\operatorname {Cos} .(S-A)\mp \operatorname {Sin} .S}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}A}}\operatorname {Cos} .{\frac {1}{2}}a.\\\\\end{aligned}}$

dédoublant et décomposant les seconds membres, à l’aide des formules (8), (9), (10), (11), on trouvera également, soit par les deux premières, soit par les deux dernières équations,