Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/301

${\begin{aligned}&\operatorname {Cos} .s={\sqrt {1-\operatorname {Sin} .^{2}s}}={\frac {\sqrt {4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Sin} .^{2}{\frac {1}{2}}B\operatorname {Sin} .^{2}{\frac {1}{2}}C-P^{2}}}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\\\\&\operatorname {Sin} .S={\sqrt {1-\operatorname {Cos} .^{2}S}}={\frac {\sqrt {4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Cos} .^{2}{\frac {1}{2}}b\operatorname {Cos} .^{2}{\frac {1}{2}}c-p^{2}}}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}\,;\end{aligned}}$

à l’aide des formules (xxvii) et (XXVII), ces dernières deviendront

$\operatorname {Cos} .s={\frac {1-\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C}{2\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}}$

$=-{\frac {1-\operatorname {Cos} .A-\operatorname {Cos} .B-\operatorname {Cos} .C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C}},\quad$(xxix)

$\operatorname {Sin} .S=-{\frac {1-\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c}{2\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}$

$={\frac {1+\operatorname {Cos} .a+\operatorname {Cos} .b+\operatorname {Cos} .c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c}}.\quad$(XXIX)

On a aussi, par les formules (xviii) et (XVIII),

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

à l’aide des formules (xxviii) et (XXVIII), ces dernières deviendront

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

$={\frac {1-\operatorname {Cos} .A+\operatorname {Cos} .B+\operatorname {Cos} .C}{4\operatorname {Sin} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C}},\qquad$(xxx)

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

$={\frac {1+\operatorname {Cos} .a-\operatorname {Cos} .b-\operatorname {Cos} .c}{4\operatorname {Cos} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c}}.\qquad$(XXX)

La comparaison des formules (xvii), (XVII), (xviii), (XXIII) aux formules (xxix), (XXIX), (xxx), (XXX), donne encore

$\operatorname {Tang} .s=-{\frac {2P}{1-\operatorname {Cos} .A-\operatorname {Cos} .B-\operatorname {Cos} .C}},\qquad$(xxxi)