Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/304

${\begin{aligned}&\operatorname {Sin} .{\frac {1}{2}}(s-a)={\sqrt {\frac {1-\operatorname {Cos} .(s-a)}{2}}}=\\&{\sqrt {-{\frac {1+\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C-2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}}}},\\\\&\operatorname {Cos} .{\frac {1}{2}}(s-a)={\sqrt {\frac {1+\operatorname {Cos} .(s-a)}{2}}}=\\&{\sqrt {\frac {1+\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C+2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}}}\,;\end{aligned}}$

comparant ces formules aux formules (35) et (36), on pourra leur donner cette forme

$\operatorname {Sin} .{\frac {1}{2}}(s-a)=\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$(xxxvii)

${\sqrt {-{\frac {\operatorname {Sin} .{\frac {1}{2}}\left({\frac {1}{2}}\varpi -S\right)\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-B)\right\}\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-C)\right\}}{\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}}}},$

$\operatorname {Cos} .{\frac {1}{2}}(s-a)=\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$(xxxviii)

${\sqrt {\frac {\operatorname {Cos} .{\frac {1}{2}}\left({\frac {1}{2}}\varpi -S\right)\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-B)\right\}\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-C)\right\}}{\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}}}\,;$

d’où encore

$\operatorname {Tang} .{\frac {1}{2}}(s-a)=\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$(xxxix)

${\sqrt {-\operatorname {Tang} .{\frac {1}{2}}\left({\frac {1}{2}}\varpi -S\right)\operatorname {Cot} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}\operatorname {Tang} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-B)\right\}\operatorname {Tang} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-C)\right\}}}.$

Par les mêmes formules (5) et par la formule (XXX), on trouve

${\begin{aligned}&\operatorname {Sin} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}={\sqrt {\frac {1-\operatorname {Sin} .(S-A)}{2}}}=\\&{\sqrt {\frac {-1+\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c-2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}},\\\\&\operatorname {Cos} .{\frac {1}{2}}\left\{{\frac {1}{2}}\varpi -(S-A)\right\}={\sqrt {\frac {1+\operatorname {Sin} .(S-A)}{2}}}=\\&{\sqrt {\frac {1+\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c+2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}}\,;\end{aligned}}$

comparant ces formules aux formules (31) et (32), on pourra leur donner cette forme

de sorte que les formules (XIII), (XVII), (XX), (XXIV), (XXIX), (XXXI), (XXXIV), (XXXV), (XXXVI), offrent autant de moyens d’obtenir l’aire d’un triangle sphérique en fonction de trois de ses parties. On reconnaîtra en particulier la dernière pour celle de M. Lhuilier, donnés, par M. Legendre, dans ses Élémens de Géométrie.