Page:Annales de mathématiques pures et appliquées, 1813-1814, Tome 4.djvu/377

361

DE LA CIRCONFÉRENCE.

$P-p=2.^{m+1}n{\operatorname {Tang} .{\frac {\varpi }{2.^{m}n}}-\operatorname {Sin} .{\frac {\varpi }{2.^{m}n}}}.$

Mais on sait que, $x$ étant un arc quelconque ; on a

${\begin{aligned}\operatorname {Sin} .x=&2\operatorname {Cos} .{\frac {x}{2}}\operatorname {Sin} .{\frac {x}{2}}\\=&4\operatorname {Cos} .{\frac {x}{2}}\operatorname {Cos} .{\frac {x}{4}}\operatorname {Sin} .{\frac {x}{4}}\\=&8\operatorname {Cos} .{\frac {x}{2}}\operatorname {Cos} .{\frac {x}{4}}\operatorname {Cos} .{\frac {x}{8}}\operatorname {Sin} .{\frac {x}{8}}\\=&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\=&2^{m}\operatorname {Cos} .{\frac {x}{2}}\operatorname {Cos} .{\frac {x}{4}}\operatorname {Cos} .{\frac {x}{8}}\operatorname {Cos} .{\frac {x}{16}}\ldots \operatorname {Cos} .{\frac {x}{2^{m}}}\operatorname {Sin} .{\frac {x}{2^{m}}}\,;\\\end{aligned}}$

D’où

$2^{m}\operatorname {Sin} .{\frac {x}{2^{m}}}={\frac {\operatorname {Sin} .x}{\operatorname {Cos} .{\frac {x}{2}}\operatorname {Cos} .{\frac {x}{4}}\operatorname {Cos} .{\frac {x}{8}}\ldots \operatorname {Cos} .{\frac {x}{2^{m}}}}}\,;$

En faisant dans cette formule $x={\frac {\varpi }{n}}$, elle deviendra

$2^{m}\operatorname {Sin} .{\frac {\varpi }{2^{m}n}}={\frac {\operatorname {Sin} .{\frac {\varpi }{n}}}{\operatorname {Cos} .{\frac {\varpi }{2n}}\operatorname {Cos} .{\frac {\varpi }{4n}}\operatorname {Cos} .{\frac {\varpi }{8n}}\ldots \operatorname {Cos} .{\frac {\varpi }{2^{m}n}}}}\,;$

valeur qui, substituée dans celles de $p$ et $P$, les change en celles-ci