Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/300

$\operatorname {Sin} .s={\frac {\operatorname {Sin} .b\operatorname {Cos} .{\frac {1}{2}}C\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Sin} .{\frac {1}{2}}B}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Cos} .{\frac {1}{2}}B}{\operatorname {Sin} .{\frac {1}{2}}C}},\qquad$(xxiv)

$-\operatorname {Cos} .S={\frac {\operatorname {Sin} .B\operatorname {Sin} .{\frac {1}{2}}c\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Cos} .{\frac {1}{2}}b}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Sin} .{\frac {1}{2}}b}{\operatorname {Cos} .{\frac {1}{2}}c}},\qquad$(XXIV)

$\operatorname {Sin} .(s-a)={\frac {\operatorname {Sin} .b\operatorname {Sin} .{\frac {1}{2}}C\operatorname {Cos} .{\frac {1}{2}}A}{\operatorname {Cos} .{\frac {1}{2}}B}}={\frac {\operatorname {Sin} .c\operatorname {Cos} .{\frac {1}{2}}A\operatorname {Sin} .{\frac {1}{2}}B}{\operatorname {Cos} .{\frac {1}{2}}C}},\qquad$(xxv)

$\operatorname {Cos} .(S-A)={\frac {\operatorname {Sin} .B\operatorname {Cos} .{\frac {1}{2}}c\operatorname {Sin} .{\frac {1}{2}}a}{\operatorname {Sin} .{\frac {1}{2}}b}}={\frac {\operatorname {Sin} .C\operatorname {Sin} .{\frac {1}{2}}a\operatorname {Cos} .{\frac {1}{2}}b}{\operatorname {Sin} .{\frac {1}{2}}c}},\qquad$(XXV)

En vertu des formules (29) et (30), les valeurs (vii) et (VII) peuvent être écrites ainsi

$p^{2}={\frac {1}{4}}\left(1-\operatorname {Cos} .^{2}a-\operatorname {Cos} .^{2}b-\operatorname {Cos} .^{2}c+2\operatorname {Cos} .a\operatorname {Cos} .b\operatorname {Cos} .c\right),$ (xxvi)

$P^{2}={\frac {1}{4}}\left(1-\operatorname {Cos} .^{2}A-\operatorname {Cos} .^{2}B-\operatorname {Cos} .^{2}C-2\operatorname {Cos} .A\operatorname {Cos} .B\operatorname {Cos} .C\right).$ (XXVI)

En transformant les cosinus d’angles en cosinus de moitiés, dans la première et en sinus de moitiés dans la seconde, à l’aide des formules (5), on trouvera

$p^{2}=4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Cos} .^{2}{\frac {1}{2}}b\operatorname {Cos} .^{2}{\frac {1}{2}}c$

$-\left(1-\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c\right)^{2},\qquad$(xxvii)

$P^{2}=4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Sin} .^{2}{\frac {1}{2}}B\operatorname {Sin} .^{2}{\frac {1}{2}}C$

$-\left(1-\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C\right)^{2}\,;\qquad$(XXVII)

ou encore

$p^{2}=4\operatorname {Cos} .^{2}{\frac {1}{2}}a\operatorname {Sin} .^{2}{\frac {1}{2}}b\operatorname {Sin} .^{2}{\frac {1}{2}}c$

$-\left(1+\operatorname {Cos} .^{2}{\frac {1}{2}}a-\operatorname {Cos} .^{2}{\frac {1}{2}}b-\operatorname {Cos} .^{2}{\frac {1}{2}}c\right)^{2},\qquad$(xxviii)

$P^{2}=4\operatorname {Sin} .^{2}{\frac {1}{2}}A\operatorname {Cos} .^{2}{\frac {1}{2}}B\operatorname {Cos} .^{2}{\frac {1}{2}}C$

$-\left(1+\operatorname {Sin} .^{2}{\frac {1}{2}}A-\operatorname {Sin} .^{2}{\frac {1}{2}}B-\operatorname {Sin} .^{2}{\frac {1}{2}}C\right)^{2}.\qquad$(XXVIII)

Cela posé, on a, par les formules (xvii) et (XVII),