Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/302

 Les corrections sont expliquées en page de discussion

$\operatorname {Cot} .S=-{\frac {2p}{1+\operatorname {Cos} .a+\operatorname {Cos} .b+\operatorname {Cos} .c}},\qquad \quad$(XXXI)

$\operatorname {Tang} .(s-a)={\frac {2P}{1-\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}},\quad$(xxxii)

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

En formant les expressions analogues, relatives à $s-b,\ s-c,\ S-B,$$S-C,$ on en conclura

$\operatorname {Cot} .(s-a)+\operatorname {Cot} .(s-b)+\operatorname {Cot} .(s-c)=$

${\frac {3+\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}{2P}}={\frac {\operatorname {Cos} .^{2}{\frac {1}{2}}A+\operatorname {Cos} .^{2}{\frac {1}{2}}B+\operatorname {Cos} .^{2}{\frac {1}{2}}C}{P}},$(xxxiii)

$\operatorname {Tang} .(S-A)+\operatorname {Tang} .(S-B)+\operatorname {Tang} .(S-C)=$

${\frac {3-\operatorname {Cos} .a-\operatorname {Cos} .b-\operatorname {Cos} .c}{2p}}={\frac {\operatorname {Sin} .^{2}{\frac {1}{2}}a+\operatorname {Sin} .^{2}{\frac {1}{2}}b+\operatorname {Sin} .^{2}{\frac {1}{2}}c}{p}}.$(XXXIII)

Par les formules (5) et par la formule (xxix), on a

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

comparant ces formules aux formules (34) et (35), on pourra leur donner cette forme

$\operatorname {Sin} .{\frac {1}{2}}s=\qquad \qquad \qquad \qquad$(xxxiv)

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

$\operatorname {Cos} .{\frac {1}{2}}s=\qquad \qquad \qquad \qquad$(xxxv)

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

d’où encore