Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/299

$\operatorname {Sin} .s={\frac {2p}{P}}\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B\operatorname {Cos} .{\frac {1}{2}}C,\qquad \qquad$(xx)

$\operatorname {Cos} .S=-{\frac {2P}{p}}\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b\operatorname {Sin} .{\frac {1}{2}}c\,;\qquad \qquad$(XX)

$\operatorname {Sin} .(s-a)={\frac {2p}{P}}\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B\operatorname {Sin} .{\frac {1}{2}}C,\qquad \qquad$(xxi)

$\operatorname {Cos} .(S-A)={\frac {2P}{p}}\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b\operatorname {Cos} .{\frac {1}{2}}c.\qquad \qquad$(XXI)

Si l’on multiplie respectivement ces quatre valeurs par les quatre qui les précèdent immédiatement, on aura encore

$\operatorname {Sin} .^{2}s=p\operatorname {Cot} .{\frac {1}{2}}A\operatorname {Cot} .{\frac {1}{2}}B\operatorname {Cot} .{\frac {1}{2}}C,\qquad \qquad$(xxii)

$\operatorname {Cos} .^{2}S=P\operatorname {Tang} .{\frac {1}{2}}a\operatorname {Tang} .{\frac {1}{2}}b\operatorname {Tang} .{\frac {1}{2}}c,\qquad \qquad$(XXII)

$\operatorname {Sin} .^{2}(s-a)=p\operatorname {Cot} .{\frac {1}{2}}A\operatorname {Tang} .{\frac {1}{2}}B\operatorname {Tang} .{\frac {1}{2}}C,\qquad \qquad$(xxiii)

$\operatorname {Cos} .^{2}(S-A)=P\operatorname {Tang} .{\frac {1}{2}}a\operatorname {Cot} .{\frac {1}{2}}b\operatorname {Cot} .{\frac {1}{2}}c.\qquad \qquad$(XXIII)

Les formules (ii, II) reviennent (7) à

${\begin{aligned}{\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}B}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .b\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}b}},&{\frac {\operatorname {Sin} .a\operatorname {Cos} .{\frac {1}{2}}C}{\operatorname {Sin} .{\frac {1}{2}}A}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}C}},\\\\{\frac {\operatorname {Sin} .A\operatorname {Cos} .{\frac {1}{2}}b}{\operatorname {Sin} .{\frac {1}{2}}a}}={\frac {\operatorname {Sin} .B\operatorname {Cos} .{\frac {1}{2}}a}{\operatorname {Sin} .{\frac {1}{2}}b}},&{\frac {\operatorname {Sin} .A\operatorname {Sin} .{\frac {1}{2}}c}{\operatorname {Cos} .{\frac {1}{2}}a}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Cos} .{\frac {1}{2}}c}}\,;\end{aligned}}$

au moyen desquelles les formules (xiii), (XIII), (xiv), (XIV) prennent cette nouvelle forme