Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/55

$\left\{r-k\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)\right\}^{2}\operatorname {Cos} .{\frac {\varpi }{m}}-k^{2}\operatorname {Sin} .^{2}{\frac {\varpi }{m}}\operatorname {Sin} .^{2}\left({\frac {2n\varpi }{m}}-\alpha \right)$

ou en développant

$r^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}-2kr\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)$

$+k^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .^{2}\left({\frac {2n\varpi }{m}}-\alpha \right)-k^{2}\operatorname {Sin} .^{2}{\frac {\varpi }{m}}\operatorname {Sin} .^{2}\left({\frac {2n\varpi }{m}}-\alpha \right)$

mais on sait que $\operatorname {Cos} .^{2}t={\frac {1}{2}}(1+\operatorname {Cos} .2t)$ et $\operatorname {Sin} .^{2}t={\frac {1}{2}}(1-\operatorname {Cos} .2t)$ ; en conséquence, ce produit pourra être mis sous cette autre forme

$r^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}-2kr\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)$

$+{\frac {1}{2}}k^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\left\{1+\operatorname {Cos} .\left({\frac {4n\varpi }{m}}-2\alpha \right)\right\}-{\frac {1}{2}}k^{2}\operatorname {Sin} .^{2}{\frac {\varpi }{m}}\left\{1-\operatorname {Cos} .\left({\frac {4n\varpi }{m}}-2\alpha \right)\right\}$

ce qui revient à

$r^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}-2kr\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)$

$+{\frac {1}{2}}k^{2}\left(\operatorname {Cos} .^{2}{\frac {\varpi }{m}}-\operatorname {Sin} .^{2}{\frac {\varpi }{m}}\right)+{\frac {1}{2}}k^{2}\operatorname {Cos} .\left({\frac {4n\varpi }{m}}-2\alpha \right)$

ou encore

$r^{2}\operatorname {Cos} .^{2}{\frac {\varpi }{m}}+{\frac {1}{2}}k^{2}\operatorname {Cos} .{\frac {2\varpi }{m}}-2kr\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)+{\frac {1}{2}}k^{2}\operatorname {Cos} .\left({\frac {4n\varpi }{m}}-2\alpha \right),$

ou enfin

${\frac {1}{2}}r^{2}+{\frac {1}{2}}\left(r^{2}+k^{2}\right)\operatorname {Cos} .{\frac {\varpi }{m}}-2kr\operatorname {Cos} .^{2}{\frac {\varpi }{m}}\operatorname {Cos} .\left({\frac {2n\varpi }{m}}-\alpha \right)+{\frac {1}{2}}k^{2}\operatorname {Cos} .\left({\frac {4n\varpi }{m}}-2\alpha \right).$

Pour avoir l’aire du triangle, il faudra encore multiplier ce développement par ${\frac {1}{2}}\operatorname {Sin} .{\frac {\varpi }{m}},$ ce qui donnera finalement